M.G.Goman and A.V.Khramtsovsky (1997 draft) - Textbook for KRIT Toolbox users

The Textbook�draft�

Dr Mikhail Goman Dr Andrew Khramtsovsky

June ��

Contents

� Investigation of aircraft nonlinear dynamics problems �

� Qualitative methods and bifurcation analysis of aircraft dynamics �

�� Equilibrium solutions � � � � � � � � � � � � � � � � � � � � � � � � � �� Periodic solutions � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Poincar�e mapping � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Bifurcational analysis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Bifurcations of the equilibrium points � � � � � � � � � � � � � � � � �� Bifurcations of the periodic trajectories � � � � � � � � � � � � � � � �

� Numerical algorithms and methods ��

�� Continuation method � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Systematic search for solutions of nonlinear system of equations � � � � � �� Computing manifolds of equilibrium solutions and bifurcation sets � � � � �� Computation of the periodic solutions � � � � � � � � � � � � � � � � � � � � ��

�� Time�advance mapping� � � � � � � � � � � � � � � � � � � � � � � � �� Poincar�e mapping� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Algorithms for stability regions computation � � � � � � � � � � � � � � � � � �� Computation of stability region

boundaries � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Direct computation of stability region cross�sections � � � � � � � � ��

�� Example� Global stability analysis in roll�coupling problem � � � � � � � � ��

�

Chapter �

Investigation of aircraft nonlinear

dynamics problems

The desire to advance aircraft maneuverability results in expansion of the region of theallowed �ight regimes� Inevitably critical situations connected with the loss of stabilityand�or controllability� become more probable� This is especially true for high angle ofattack �ight regimes where stall and spin entry are possible�

Critical �ight regimes signi�cantly di�er from normal �ight regimes in aircraft dy�namics and control� Auto�rotation regimes also belong to the class of critical regimes�these regimes can be entered during active spatial maneuvering with fast rotation in roll�

Thus investigation of aircraft dynamics at high angles of attack and fast rotation isclosely coupled with aircraft combat e�ciency and �ight safety problems� It is neces�sary to use nonlinear equations of motion� taking into account nonlinear dependenciesof aerodynamic characteristics on motion parameters as well as nonlinearities in con�trol system� The behavior of the aircraft can be described by an autonomous nonlineardynamic system depending on a number of parameters� The parameters include con�trol inputs �i�e� de�ections of control surfaces or control levers�� aircraft parameters�parameters of the �ight regime etc�

Since aircraft is su�ciently nonlinear dynamic system� it is impossible to get allthe qualitative pattern of aircraft dynamics by means of mathematical modeling only�Aircraft behavior may be very di�erent depending on initial conditions and controlapplied�

Methods of the qualitative theory of dynamical systems and bifurcation analysis arevery e�cient for the investigation of aircraft dynamics� they and direct mathematicalmodeling complement each other�

Basic results of qualitative theory can be found in many publications �� Unfortunately� mathematical models of aircraft dynamics are toocomplex to be investigated analytically� That�s why major advances in the investigationof aircraft dynamics at high angles of attack are achieved using powerful computersand numerical methods of qualitative analysis� A great e�ort was devoted lately forthe development of numerical methods and software packages for qualitative analysisof nonlinear dynamic systems �� Mostfrequently used for aircraft applications are the software packages BISTAB� AUTO�

�

ASDOBI� KRIT �� Methods for the investigation of nonlinear aircraft dynamics problems can be divided

into two groups� The �rst one comprises approximate approaches and methods fromearlier research� the second one includes new numerical methods based on bifurcationanalysis and theory of nonlinear dynamic systems�

Approximate methods

Since the dawn of aviation� many authors investigated stall and spin problems for ev�ery new generation of the aircraft� With the advent of jet planes a new critical regimeappeared� the regime was caused by a loss of motion stability due to inertia couplingbetween longitudinal and lateral motions at fast rotation in roll� Both spin �� and inertia roll�coupling problem �� were investigated using approximateequations of motion and simpli�ed �usually linear� aerodynamic models� Approximatemethods were used as well� Often for simplicity some state variables �roll rate� angle ofattack� were considered as problem parameters� Such methods allowed to a certain de�gree to avoid solving nonlinear problems� Partial analytical investigation was performed�and �nal solution was received by graphical methods� Revealed peculiarities of aircraftmotion �arising of spin and auto�rotation regimes� departures in roll� were not directlylinked with bifurcations of steady�state solutions of equations of motion� Thus generalresults obtained by qualitative theory of dynamic systems couldn�t be applied�

It�s worth noting that in some works a program of qualitative investigation of object�sdynamics was formulated and even realized to the extent achievable with graphical�analytical methods� see for example ��

Application of continuation methods

Nowadays bifurcation analysis methods and theory of dynamic systems are often appliedtowards nonlinear aircraft dynamic problems� and there are enough many publications�� At the same time these methods are seldom usedat early stages of aircraft development�

Common feature of the above�mentioned papers is an application of continuationmethod for the investigation of nonlinear problems�

Carroll and Mehra were the �rst who used the continuation method in �ight dynam�ics and connected the main types of an aircraft instabilities with bifurcation phenomena�For example� the occurrence of wing rock motion was connected with a Hopf bifurcation�the examples of chaotic motions were presented� etc� Later the existence of new typesof bifurcations in an aircraft dynamics was explored� such as origination of stable torusmanifold �� global bifurcation of a closed orbit related with the appearance of homo�clinical trajectory �� so�called �ip or period doubling and pitchfork bifurcations forclosed orbits �� Continuation methodology and bifurcation analysis can be usedfor determining the recovery technique from critical regimes and for control law designfor improving dynamical behavior ��

While �rst works were devoted mainly to theory and corresponding numerical algo�rithms �� in recent publications the results obtained for real aircraft were presented

�

�� F�� German�French Alpha�Jet� �� F�� F�� For a number of aircraft the results of theoretical analysis were compared with �ight

test results� and good quantitative as well as qualitative agreement was found �see ��

Methodology of investigation

For a dynamic system describing an aircraft behavior� its dependence on a number ofparameters is investigated� The parameters can include �ight regime values� mass andinertia data� de�ections of the control surfaces etc�

Qualitative investigation of dynamics is performed at �xed values of parameters sothat the system can be considered as autonomous one� The behavior of the system canbe predicted knowing qualitative changes in the structure of system�s solutions and typeand character of bifurcations� The Lyapunov�s concept of stability suggesting unlimitedtime variation is used�

This validity of such an approach is limited because of usage of approximate math�ematical models of motion� For energetic spatial maneuvers� mathematical model isvalid only for �nite time period� At t� �� the solutions of equations may di�er fromgenuine motion of the object�

Nevertheless the eigenvalues can describe the local character of motion near steady�state conditions within �nite period of time� In that case the ratio between characteristictimes of the aircraft motion� maneuver duration and �validity time� of the mathematicalmodel� is important for the estimation of the motion character� For example� in case ofvery fast maneuver an instability may have no time to manifest itself� or else an approachto a weak attractor may be very slow because of its weakness�

Therefore the usage of Lyapunov�s concept of stability is a su�cient simpli�cationof the problem� That�s why it is necessary to verify the results obtained by means ofmathematical modeling� the veri�cation is especially important in the case of criticaland near�critical conditions�

Relying on experience achieved� the following three�stage investigation plan is sug�gested for the investigation of nonlinear dynamics of an aircraft� the plan is based onbifurcation analysis and continuation technique �

� At �rst stage the parameters of the problem are considered as �xed� Main goalis to �nd out all possible steady�state solutions �equilibrium� periodic and morecomplex ones�� and to determine their local stability� Continuation method andsystematic search method are of great help here�

Global structure of the state space can be revealed after determining of asymptoticstability regions of all the existing attractors� Convenient graphical representationof phase portrait and steady�state solutions is very important for e�cient dataprocessing�

� At the second stage the changes of the behaviour of the system due to variationsof some parameters are investigated� Prediction of the system response is based onthe knowledge of possible bifurcations and dominating attractors that determinesystem�s dynamics immediately after bifurcations�

�

Rate of parameter change is of importance for the prediction too� The higherthe rate� the greater might be the di�erence between transient and steady�statemotions�

� At the last stage mathematical modeling is used for veri�cation of the qualitativeresults obtained� The situations to be checked include di�erent �nite disturbancesof state variables� and di�erent variations of the parameters of the system�

In the next chapters methods of qualitative investigation and bifurcation analysis aredescribed together with corresponding numerical algorithms�

Chapter �

Qualitative methods and

bifurcation analysis of aircraft

dynamics

In this chapter the basic concepts and ideas of the qualitative theory of dynamicalsystems are described� They will be useful further for the investigation of nonlineardynamics of the aircraft ��

Consider a nonlinear autonomous dynamical system depending on parameters� thesystem is described by a set of di�erential equations

dx

dt� F �x� c� � x � Rn� c �M � Rm� ��

where F is a smooth vector function� The vector �eld F de�nes a map Rn�m � Rm� Thesystem �� satis�es to the conditions of the existence and uniqueness of the solutionx�t�xo� with initial condition x��xo� � xo� The solution �t�xo� � x�t�xo� is called a

trajectory or the �ow of the dynamical system � The set of all the trajectories constitutethe phase portrait of the dynamical system �

The success of qualitative analysis of a dynamical system is closely coupled with asuccess in �nding singular trajectories or critical elements � It is also extremely importantto understand the structure of the state space in the neighbourhood of the criticalelements� The data gathered then allow to understand the global structure of the statespace of the system ��

In Fig�� the examples of stable singular trajectories of the system �� are shown�There are equilibrium point� closed trajectory �periodical solution�� invariant toroidalmanifold �non�regular oscillations� and strange attractor �chaotic motion�� Complexbehaviour of the nonlinear dynamical system is often due to the existence of a numberof isolated attracting sets� each attractor has its own region of attraction� All thatbrings about the strong dependence of motion on initial conditions and on the sequenceof variations of parameters�

The following notation will be used ��

Figure �� Singular trajectories of nonlinear dynamic system�

O � equilibrium point�� closed orbit or phase trajectory �i�e� periodic motion ��T � toroidal manifold of trajectories in the state space�Qualitative analysis pays special attention to the simplest attractors �i�e� equilibrium

points and closed orbits� and their dependence on the parameters�Stable equilibrium points and periodic motions are the simplest attractors determin�

ing steady�state regimes of motion of the system� Saddle�type solutions and their stableand unstable invariant manifolds of trajectories are the decisive factors in forming globalstructure of the state space since they de�ne the boundaries of the regions of attraction��

�� Equilibrium solutions

Equilibrium points �or else equilibrium solutions� x � xe of the system �� are thesolutions of the system of equations

F�xe� c� � �� x � Rn� c �M � Rm� ��

At any given values of the parameters c one can have a set of di�erent isolatedequilibrium points� the number of equilibrium points may change while the values of theparameters c are varied�

On linearizing in the neighbourhood of the equilibrium point xe� the system ��takes the form

dx

dt� Ax ��

where

�

x � x� xe � vector of deviations with respect toequilibrium xe�

A ��F

�x

��x�xe

� Jacoby matrix�

If there are complex pairs among eigenvalues f�igi��n with the indexes � and � ��

�c�� i��

�c�� i��

then general solution of �� can be written as

x � xe � a�e��t�� a� cos��t� a�� sin��t�e��t��

�a� sin ��t� a�� cos��t�e��t�� ane�nt�n

��

The values of a�� an are de�ned by the initial deviation from the equilibriumpoint xe�

a�� a�� an�n � xo � xe ��

or� using a matrix Q in which eigenvectors are the columns

Q � jj�� njj

one may express the vector a � �a�� a�� an� as follows

a � Q��xo � xe� ��

Using matrix notation� the solution of �� is

x � xe �Q �t�Q��xo � xe� ��

where �t� is the matrix of fundamental solutions in the eigenvectors� basis� �t� isa solution of matrix equation

d

dt� ! �� E

where E is a unit diagonal matrix� ! � Q��AQ is a Jordan block�diagonal matrixof the form

! �

��

If real parts of all the eigenvalues are negative Ref�igi��n �� then the trajectory�t�xo� will go towards the equilibrium point xe for any xo belonging to small neigh�bourhood of the equilibrium� The equilibrium point becomes unstable if any of theeigenvalues has a positive real part�

The placement of the eigenvalues on the complex plane de�nes the structure of thestate space in the vicinity of the equilibrium point xe�

Suppose that q eigenvalues lie to the right of the imaginary axis and p ones lie to theleft� the sum p � q � n equals to the dimension of the state space� Then two integralmanifolds of trajectories W s

p and W uq connected with the equilibrium point xe can be

de�ned as follows �W s

p � fx � �t�x�� xe as t� ��g

W uq � fx � �t�x�� xe as t� ��g

The stable p�dimensional manifold W sp comprises all the trajectories in the state

space that go to xe while t� �� Similarly� the unstable q�dimensional manifold W uq

comprises all the trajectories in the state space that go to xe while t� ��For linear approximation �� the manifolds W s

p and Wuq lie in the hyperplanes Lp

and Lq� The hyperplane Lp is determined by p stable eigenvectors f�igi�k��kp� whileLq is determined by q unstable ones f�igi�l��lp�

Lp � span ��k��k�

� � � � ��kp�

Lq � span ��l��l�

� � � � ��lq�

When nonlinear terms are taken into account� then the surfaces W sp and W u

q willdeviate from hyperplanes Lp and Lq �the farther from the equilibrium� the more thedeviations�� But the surfaces remain tangent to hyperplanes Lp and Lq at xe �see�g��

The notation Op�q will be used for such an equilibrium point ��

�� Periodic solutions

As for equilibrium points� one can write the equations for deviations with respect to anarbitrary trajectory �t�xo�

dxdt � F�x� c�� x�� xo

d dt

� A�t� � �� E��

where

A�t� ��F

�xjx�t��t�xo� � Jacoby matrix calculated along the trajectory

�t�xo�� t� � matrix of fundamental solutions of the linearized

system� It de�nes the evolution of the small per�turbations�

�

Figure �� Invariant manifolds of trajectories W sp � W

uq for equilibrium and periodic

solutions�

Consider closed trajectory or periodic solution with the period � The trajectory isclosed when the following condition is satis�ed

�� x�� x�

where x� is a point belonging to the closed trajectory ��

If xo in �� is such that xo � �� then the matrix A�t� is one with periodic coe��cients� The value of �t� at time de�nes stability properties of the periodic trajectory�The matrix � � is called a monodromy matrix � The eigenvalues of � � are calledcharacteristic multipliers�

The periodic solution is stable if all the multipliers lie inside the unit circle on thecomplex plane �note that one multiplier always equals to ��

�� Poincar�e mapping

Poincar�e mapping technique is an e�ective tool for investigation of the state space inthe vicinity of the closed trajectory � or the integral invariant manifolds of the higherdimension�

The equivalent n� ��dimensional system with discrete time may be studied insteadof original dynamical system� Poincar�e mapping can be set using a n � ��dimensionalhyperplane " transversal to the closed trajectory � at the point x�� The trajectoriesin the neighbourhood of � also cross "� Thus every point xk � " �belonging to someneighbourhood of x�� can be mapped into some other point of the hyperplane xk�� "which corresponds to the second intersection of the trajectory �t�xk� with hyperplane"�

The type of the mapping P depends on the intersection condition� The mappingcan be one�sided �as above� or two�sided when all the intersection points are taken into

��

account� Poincar�e mapping generates the sequence

xk�� P�xk� ��

where xk��xk � "�

Fixed point x� of the mapping P

P�x�� x�

corresponds to the periodic trajectory � �see �g��

Figure �� Poincare mapping�

The behaviour of the mapping P near x� is described by discrete linear systemderived from ��

xk � Axk��

where

A ��P

�x

��x�x�

� constant matrix�

x � x� x� � deviation vector with respect to �xed point of the map�ping

The orbit of the mapping P in the small neighbourhood of the �xed point O withcoordinates x� is determined from the relationship

xk � x� � a��k�� an��

kn��n��

where f�igi��n�� and f�igi��n�� are the eigenvalues and eigenvectors of the Jacoby

��

matrix A calculated at the �xed point of the mapping�The parameters faigi��n�� are determined using initial condition xo

a�� a�� an��n�� xo � x�

The eigenvalues �i are the characteristic multipliers for the �xed point� The mappingis compressing in the direction of the eigenvector �i if j�ij � �

If there are complex pairs ��e�i�� among eigenvalues then there will be a rotation ofthe points belonging to the orbit of the mapping in the plane de�ned by the eigenvectors�� and �� The equation �� will look like that�

xk � x� � a��k�� a� cos k�� a�� sin k��k��

� �a� sin k�� a�� cos k��k�� an��kn��n��

��

The multipliers f�igi��n�� do not depend on the orientation of the secant hyperplane" as far as it remains transversal to the trajectory� The multipliers coincide with theeigenvectors of the monodromy matrix�

The behaviour of the trajectories in the state space near the closed orbit � is inone�to�one relation with the behaviour of the mapping �� in the vicinity of the �xedpoint O� The type of the �xed point can be determined provided the placement of thecharacteristic multiplies with respect to the unit circle in the complex plane is known�As for an equilibrium solution �� for �xed point� one can �nd the p � dimensionalstable and q � dimensional unstable invariant manifolds W s

p and Wuq of mapping points�

Here p is a number of multipliers inside the unit circle� and q is the number of multipliersoutside unit circle�

The stable manifold W sp is composed of the points being the results of successive

mappings� each sequence of these points converges to the �xed point O� This manifoldcorresponds to the stable multipliers j�ij ��

The unstable manifold W uq is also composed of the points being the results of succes�

sive mappings� but each sequence of these points converges to the �xed point O usinginverse mappings� The manifold corresponds to the unstable multipliers j�ij � ��

Fixed point O corresponds to the closed trajectory � in the state space� Invariantmanifolds of mapping pointsW s

p andWuq for the �xed point O correspond to the invariant

manifolds of the trajectories W sp�� and W u

q�� the di�erence in the dimension of themanifolds is �� That�s why the notation �p��q�� will be used to describe the type ofthe closed trajectory�

Trajectories from W sp�� manifold go closer to � while t�� and those from W u

q doso while t � �� Other trajectories from the neighbourhood of � �rst go closer to �along W s

p�� and then go away of it along the Wuq�� manifold ��g� ��

�� Bifurcational analysis

Investigation of the possible changes in the structure of the state space due to the changesof parameters c is a vital part in the methodology of the qualitative analysis�

��

There are critical or bifurcational values of parameters cb when qualitative typeof the state space structure changes� The most common is the so�called one�parameterbifurcation when in the parameter space near the point cb one can �nd dynamical systemswith only two di�erent types of state space�

Global changes of the state space structure are the result of successive local bifur�cations in the vicinity of the singular trajectories or critical elements �i�e� equilibriumpoints� closed orbits etc�� and bifurcations with the invariant manifolds W s

p and Wuq �

�� Bifurcations of the equilibrium points

There is a condition that allows to distinguish bifurcational situations for the equilibriumpoint O of �� with coordinates x � xe� The point c �in the space of parameters� is nota bifurcational one for the equilibrium O provided there are no eigenvalues �computedat the equilibrium point� on the imaginary axis in the complex plane� In that case�the dimensions p and q of the invariant manifolds W s

p and W uq remain the same for all

c close enough to the initial value� hence the type of the equilibrium Op�q remains thesame� Otherwise the point in space of parameters cb will be bifurcational one�

The two simplest and the most common bifurcational situations for equilibrium so�lutions are

� there is one eigenvalue equal to zero�

� there is a complex pair on the imaginary axis�

The surfaces No and N� of codimension � in the space of parameters M correspondto these situations� The equation for No is

Det

�� F

�x

��x�xe

�A � ��

The surface N� is de�ned by the equation

Det

�� F

�x

��x�xe

� i�E

�A � ��

Consider the characteristic equation of the linearized system in the vicinity of theequilibrium

�n � a��n�� a��n�� an�� an � � ��

Using coe�cients ai � ai �c� � i � �� n� one can express the equations for the

bifurcational surfaces No �� and N� �� as follows

an � �

and

��

#n��

a� � � � � � �a� a� a� � � � �a� a a� a� � � ��

� �

where #n�� is the last but one Raus determinant for the characteristic equation �� When the point in the space of parameters crosses the bifurcational boundary No�

then the following bifurcations may occur with the initially stable equilibrium point

� stable and saddle�type unstable equilibrium solutions merge and vanish� An abruptloss of stability takes place� Bifurcation scheme is

On�o �On��

� stable point becomes saddle�type unstable� at the same time two more stableequilibrium points appear� The $soft� loss of stability occurs� Bifurcation is

On�o � On�� On�o

� the merge of two saddle�type and one stable equilibrium solutions� The loss ofstability is abrupt�

On�o � �On�� On��

When the bifurcational boundary N� is crossed� the Andronov�Hopf bifurcationoccurs� There are two forms of the Andronov�Hopf bifurcation with initially stableequilibrium point � �g� ��

� the equilibrium becomes oscillatory unstable� At that moment stable periodicsolution is detached from the equilibrium� Soft�type loss of stability takes place�

On�o � On�� n��

� the stable equilibrium solution merges with saddle�type unstable periodic trajec�tory and becomes oscillatory unstable� The abrupt loss of stability occurs�

On�o � �n�� On��

The type of the loss of stability mentioned above takes place when the point in thespace of parameters slowly enough passes through the bifurcational boundary�

Soft�type loss of stability means that the trajectory in the state space �started inthe vicinity of the equilibrium� slowly goes away from the equilibrium point while thevalues of parameters are close to the critical ones� If the parameters return to the initialvalues� then the trajectory return to the neighbourhood of the equilibrium point�

When the loss of stability is abrupt� then the trajectory in the state space quicklyleaves the equilibrium point for some other attractor even if the deviations of the pa�rameters from critical values are very small� If the parameters then return to the initialvalues� the trajectory usually would not return to the initial equilibrium point�

��

The bifurcation points where two or more branches of the equilibrium solutionsintersect ��g�� has the codimension greater or equal to �� So at least two conditionsmust be satis�ed� Hence� in general case� such points are structurally unstable� Possiblechanges in the extended state space �the parameter axis is added� in the vicinity of thebranching point are shown in �g��

Figure �� Bifurcations of equilibrium solutions�

�

Figure �� Changes of extended state space structure near branching point�

�� Bifurcations of the periodic trajectories

The stability type of the �xed point O of the mapping P and of the periodic trajectory� may change when the characteristic multipliers cross the unit circle on the complexplane� That�s why only those values of parameters cb are bifurcational when there existsa multiplier belonging to the unit circle�

Consider the bifurcations of the stable closed orbits� Three simplest cases are

� multiplier crosses the unit circle at the point ��

� multiplier crosses the unit circle at the point ��

� a pair of multipliers crosses the unit circle at the point e�i��

In these cases the simplest bifurcational surfaces N��N��N� of codimension � in thespace of parameters are generated �see �g� ��

Bifurcational surfaces N�� and N� for the �xed point are equivalent to the bifurca�tional surfacesNo and N� for the equilibrium point� the surfaceN�� represents somethingnew�

There are three more situations when periodic trajectory vanishes�

�

Figure �� Bifurcations of periodic solutions�

� the periodic trajectory � shrinks into a point�

� an equilibrium point emerges on the closed orbit ��

� some point belonging to � goes to in�nity� thus the curve is no more closed�

When the boundary N�� is crossed� then two periodic trajectories either merge andvanish or emerge simultaneously �in the latter case the abrupt loss of stability occurs��

� �n�� n��

� � �n�� n��

When the boundary N� is crossed then stable closed orbit �n�� becomes unstable�n�� At the same time the stable ��dimensional toroidal invariant manifold T n��

emerges� In the second possible pattern unstable periodic solution �n�� appears whenunstable ��dimensional toroidal invariant manifold T n�� and stable closed orbit �n��

merge� The bifurcations may be written as follows

� �n�� T n�� n�� soft loss of stability�

� �n�� T n�� n�� abrupt loss of stability

If the boundary N�� is crossed� there may be bifurcations when branching of theperiodic solutions takes place�

� �n�� n�� n�� soft loss of stability

� �n�� n�� n�� abrupt loss of stability

��

When the bifurcational boundary N�� is crossed� then the changes in comparisonwith equilibrium bifurcations are somewhat unusual� Stable periodic trajectory withthe period �n�� becomes unstable one �n�� and at the same time stable periodictrajectory with double period �n�� detaches from the initial closed orbit�

� �n�� n�� n��

Similarly� the merge of the saddle�type unstable double period closed orbit �n��

with stable periodic solution �n�� is possible

� �n�� n�� n��

�

Chapter �

Numerical algorithms and methods

�� Continuation method

Continuation method is intended for computing a dependence of a solution of a nonlinearsystem of equations on a parameter�

Consider system of nonlinear equations

F�x� c� � � ��

where vector function F � Rn� x is a vector in n�dimensional state space� c is a scalarparameter�

The system �� de�nes in extended state space �x� c� one�parameter set of solutions�In general case this set consists of a number of spatial curves �some of these curves mayintersect�� see Fig��

Functions Fi� i � �� n are continuous and have continuous partial derivativeswith respect to all variables xi and the parameter c� In general� F is some smoothmapping �i�e� as many times di�erentiable as needed� F � Rn�� Rn� Poincar�e pointmapping for example�

If the parameter c is changed continuously� then the solution x�c� set implicitly byequation �� will also change continuously� To establish the dependence x�c�� thefollowing equation can be considered�

�F

�xdx�

�F

�cdc � � ��

At limit and branching points Jacoby matrix�F

�xbecomes singular� To overcome

the singularity� the trajectory is parameterized using such �natural� parameter as curvelength s ��

When moving along the trajectory of the solution z � �x� c�� the increments of thestate vector and the parameter are de�ned by the projections of the unit tangent vector

s� The vector s is normal to the rows of the matrix�F

�z�

��F

�x��F

�c

�So� if the matrix

��

has full rank rank�F

�z� n �that�s true at all points except branching ones�� then the

rows of this matrix together with tangent vector s constitute the basis of subspace z andcorresponding matrix is non�singular ��

�

�BBBBBBBBB�

�F�

�x�

�F�

�x��

�F�

�xn

�F�

�c� � � � � � �

�Fn

�x�

�Fn

�x��

�Fn

�xn

�Fn

�cs� s� � � � sn sn��

�CCCCCCCCCA�

�BB�

�F

�z

s�

�CCA ��

Since the vector s is normal to the rows of the matrix�F

�z� hence its components si

are determined by the relationships �

si �An��i

det � ��

where An��i are algebraic supplements of the elements of the bottom row of the matrix�and

det �

vuutn��Xj��

A�n��j

Therefore the trajectory of the solution �x� c� in the extended state space can bede�ned in di�erential form

dz

dt� k�s�

or ��

dxidt

� k�si� i � �� n

dc

dt� k�sn�� k�

det

��F

�x

�

det �

��

with initial conditions z � zo� F�zo� � �� Here s is a scalar parameter along the curve�

Integrating numerically equations �� one can obtain a continuous branch of thesolution curve �� in the extended state space z� As seen from the last equation in

�� determinant of the matrix�F

�xchanges its sign in turning points� These points

correspond to the bifurcation points of dynamical system %x � F�x� c��When integrating the di�erential equations �� numerically� one inevitably makes

some error and deviate from the genuine solution curve�To build a continuation method stabilizing the motion along the solution curve ��

for a system of nonlinear equations �� one should transform the di�erential system

��

Figure �� One�parameter family of solutions of the system F�x� c� � � in extendedstate space z � �x� c��

�� such that trajectory curve becomes an attractor in modi�ed system �� One canappend the equations �� with linear equation

dF

dt� k�F � ��

The solution of this auxiliary equation is in the form F � Foe�k�t� where positive

constant k� sets the speed of decreasing the error

kFk �

vuutn��Xj�i

F �i

while increasing t�Making substitutions in the left side of �� and using z variables� we obtain

�F

�x

dx

dt��F

�c

dc

dt�

�F

�z

dz

dt� �k�F

Scalar product of this equation with s vector may be appended to eqs�� resultingin joint system of equations

dz

dt�

��k�Fk�

��

At the turning points on the curve when matrix�F

�xbecomes singular� matrix

remains regular� and system of equations �� also remains regular� Thus one can

��

rewrite �� in the following form�

dz

dt� ��

��k�Fk�

��

When integrating equations �� with k� � �� not only $following the curve�� butalso an asymptotic convergence to the solution curve starting from rather wide regionin z space is assured� If k� � � then only convergence to the solution curve z�t� alongsubspace normal to a $family� of curves F�x� c� � Fo is done� Fo being a constant vector�

Equations �� enable the computation of the desired solution with automatic elim�ination of the error kFk � � due to numerical integration procedure with �xed step �t�and the errors in the initial conditions�

Assuming coe�cient k� ��

�t� where �t is a parameter increment �i�e� step of

numerical integration along the curve�� the correction vector �z can be obtained fromthe system of linear equations

�F

�z�z � �F

s��z � �

��

Correction vector�z is calculated using a condition that it is normal to unit tangentvector s� the condition ensures the uniqueness of correction vector�

To calculate the solution curve x�c� one can use a numerical method which includesthe following steps�

�� Start� Calculation of the initial point xo� co � F�xo� co� � � either using knownmethods� with �xed c � co or using eqs �� Iterations continue until error jjFjjbecomes less than some prede�ned value�

In many cases convergence is much better if the parameter c is allowed to vary�

�� A unit tangent vector s is calculated using ��

�� Finite step along trajectory �t is obtained from a condition� that increments ofall variables �zi do not exceed some prede�ned values �z�i � �t � min

k��z�i sk��

�� The next point on a curve is calculated using equations �� After that a correc�tion is made �see �� until required accuracy is reached�

� Go to ��

Problem of passing through branching point is due to non�uniqueness of thecontinuation direction s at such point� The matrix becomes singular �det � ��and rankFz � �n� �� In the case of two intersecting branches� rankFz � �n� �� andthe intersecting curves touch �in linear approximation� lay in� some ��dimensional planepassing through the branching point� The plane is de�ned by two eigenvectors of the�n�� n� ��matrix FT

zFz corresponding to two zero eigenvalues� It is not too di�cult

to �nd the second branch using ��dimensional search method�

��

�� Systematic search for solutions of nonlinear sys�

tem of equations

Method for systematic search of the set of solutions of nonlinear system of equations�SSNE method� systematic search method� is a regular algorithm that looks for unknownsolutions of nonlinear system of equations of the order N � �� At least one solutionshould be known in advance� One cannot guarantee full success� however� in manycases the method allows to �nd all the solutions� The SSNE algorithm is based oncontinuation technique �� There are only a few papers devoted to systematic searchalgorithms ��

Consider a system of nonlinear equations �

f �x� � �� where f � RN � x � RN � ��

and suppose that xo is a known solution of the system ��

f �xo� � �

First one can choose an arbitrary unit vector �n� in the most general case �n mightdepend on vector x� Then a set of mutually�normal unit vectors �k� k � �� N � ��normal to �n is computed� Together �n and �k form basis B of N �dimensional linear space �

��n � � �k� � � � k � �� N � �

��n � � �n� � ��

��i � �j� �

�� i � j

�� i � jB � jj�� N�� njj � B� �B � EN

��

Consider auxiliary system of equations Gn� �

Gn� �x� � � � where Gk� � RN�� x � RN ��

and scalar function g n� �x� obeying the following conditions �

�� Gn��x�

g n��x�

�� B� � f�x� � or

g n��x� � ��n � � f�x��

Gn�k �x� � ��

�k � f�x�� k � �� N � �

��

From �� it follows that if xo is a solution of basic system �� then it will be asolution of auxiliary system too �

Gn� �xo� � �

g n� �xo� � ��

��

If basic system �� is non�singular and has only isolated solutions� then the systemGn��x� � � �consisting of �N � �� equations and depending on N unknowns� determinessome spatial curves of co�dimension �� There may be only one curve or a number ofthem� di�erent curves can intersect at branching points�

Now choose xo as initial point and apply continuation method towards auxiliarysystem �� As a result ��dimensional spatial curve L

n�� will be computed �or a

number of them� since advanced continuation method allows to compute all the branchesintersecting at branching points�� The curve is a set of solutions of auxiliary system� Ateach point belonging to the curve the value of the function g n� �x� is determined�

If there exists such a point x� � Ln�� belonging to L

n�� that the function g n� �x��

changes sign in its vicinity� then x� is the solution of equation g n� �x�� Hence x� is�might be� unknown� solution of basic system � f �x��

Computing numerically points belonging to the curve Ln�� and selecting the points

where the function g n� changes sign� one can determine a number of new solutions ofthe basic system of equations ��

When is the SSNE algorithm able to �nd all the solutions& The following Theoremis valid �� if there exists such an auxiliary system Gn� that it de�nes connected set of

curves Ln�� then �� SSNE algorithm will �nd all the solutions of the basic system� and

�� it will be enough to consider only this one auxiliary system Gn��

If the set of the curves Ln�� is a connected one� continuation method easily computes

all these curves� Since all the solutions of the basic system also belong to this set� all ofthem can be found�

There is one important special case when known solution xo of the basic system ��is close to bifurcation with some other solutions� In the small neighbourhood of xo basicequations are equivalent to a system consisting of one nonlinear and �N � �� linearequations provided proper system of co�ordinates is chosen� Selecting vector normalto the rows of �N � �� N �matrix determining linear equations� as �n vector� one can

get linear auxiliary system� Hence� the set Ln�� consists of only one spatial straight

line� That means that SSNE method will �nd all the solutions belonging to close�to�bifurcation family�

In practice it is reasonable to try a number of di�erent �n vectors� One simple ap�proach showed good results �� when the vectors are selected from the following family �

�n � fe�� eNg � where eki �

�� i � k

�� i � k

As an example� consider the system of equations ��

fi ��k � ai��x� � � � �� ai�NxN

fk � � �x�

where A � faijg is an �N�� N�matrix of the rank N�� and � is an arbitrary nonlinearfunction�

Linear auxiliary system Gn� � Ax de�ne straight line in state space� and again SSNEmethod will easily �nd all the solutions determined by the equation ��x� � ��

��

u

u

u

u

C

D

B

A

L�� L

��

L��

L��

y

x

f� �x� y� � �x� �� x� ��

f� �x� y� � �y � �� y � ��

� solutions�

A � �x� y� � �� B � �x� y� � ��C � �x� y� � �� C � �x� y� � ��

Figure �� System with four connected solutions�

Experience with SSNE applications shows that method works much more e�cientlyif a series of auxiliary systems are used consecutively� Ability of continuation methodto detect and analyze branching points is very important� Another way to increase thee�ciency of the method is to re�start it using each newly�found solution as the initialone�

Three simple examples� presented in Fig�� show the peculiarities of the SSNEalgorithm� The top plot shows rather simple situation� The system of equations hasfour solutions� each solution is connected with another two solutions via L

n�� n � ek

curves� Working with only one auxiliary system� one cannot �nd more that one newsolution� To get all the solutions� it is necessary to consider two auxiliary systems andto re�start the algorithm from newly�found solutions�

In the middle plot of Fig�� only computation of all the branches intersecting atbranching point allows to �nd the second solution� In the case of bottom plot thealgorithm will fail because the solutions are not connected by the curves L

n�� it is

necessary to select better values of unit vector �n�So� SSNE algorithm is not a universal one� it cannot guarantee a success� At the

same time it is very useful and e�cient in di�erent applications� Example in Fig�� shows computation of the set of solutions for the aircraft roll�coupling problem � th

order system�� dashed lines are the projections of the curves Ln�� onto the plane of the

plot�One more example� taken from � �� is shown in Fig�� The steady�state conditions

of ��reactor chemical plant were investigated� This time SSNE algorithm managed to�nd �� new solutions�

�

u

u

B

A

L�� L�

�

L��

L��

y

x

f� �x� y� � �x� �� y � ��

f� �x� y� � �x� �� y � ��

� solutions�

A � �x� y� � ��B � �x� y� � ��

Figure �� System with two solutions connected via branching point�

u

u

B

A

L�� L�

�

L��

L��

y

x

f� �x� y� � x� � �

f� �x� y� � xy � �

� solutions�

A � �x� y� � ��B � �x� y� � ��

Figure �� System with two non�connected solutions�

�

Figure �� Example of SSNE method application

Figure �� Steady�state conditions of ��reactor chemical plant

��

�� Computing manifolds of equilibrium solutions

and bifurcation sets

A number of coexisting equilibrium solutions under the same conditions �i�e� values ofparameters� may be di�erent� One can consider regions in the space of the parameterswhere under the conditions de�ned by each point of the region there exists the samenumber of equilibrium solutions�

Boundary surfaces of these regions at the parameter space correspond to bifurcationsof dynamical system� In most cases bifurcation results in $birth� or $death� of a pairof equilibrium solutions� one of these solutions always being unstable�

It is possible to reconstruct qualitatively the dynamics of the system at the conditionsof varied parameters if manifolds of equilibrium solutions and corresponding bifurcationsets can be determined�

If a dynamical system is in the form

dx

dt� F�x� c�� x � Rn� c � Rm ��

then a manifold of equilibrium solutions �equilibrium surface� Ee of the system �� is

de�ned as follows

Ee � fx � F�x� c� � �� x � Rn� F � Rn� c �M � Rmg

When two parameters are varied� then it is convenient to express the dependence ofthe solution on parameters as a number of two�dimensional surfaces in ��dimensionalspace �two parameters and one of state variables��

A huge amount of information about system�s behaviour can be gathered throughanalysis of the smooth maps of these surfaces onto the plane of parameters�

The most common types of bifurcations are the so�called $fold�� $cusp�� $butter��y� and $swallow�s tail� bifurcation patterns well�known from catastrophe theory forgradient systems � ��

The equilibrium surfaces can be calculated using either continuation method or byscanning across the plane of two selected parameters c� and c��

A set of bifurcation points Be of the equilibrium surface Ee is de�ned by a condition

Be �nx � G�x� c� � �� x � Rn� G � Rn�� c �M � Rm

o

where

G �

�BBB�

F�x� c�

det

��F�x��

�CCCA

If a dependence of one state variable on two parameters c� and c� is studied� thenBe is a manifold of dimension �� The projection of Be onto plane of parameters givesboundaries N�� which separate regions with di�erent number of equilibrium solutions�

�

Boundaries N� are also called bifurcational portrait or diagrams� These diagramsmay have cusp points� The curve Be and bifurcational diagrams N� can be computedusing continuation method�

Continuation method employs the extended Jacoby matrix�G

�z� where z � �x� c��

and thus requires evaluation of gradient of the determinant # � detkFxk of matrix Fx�

Partial derivatives of the determinant of Jacoby matrix�#

�zk� k � �� n� � may

be expressed using partial derivatives of F�x� c� vector function

�#

�zk�

nXi��

nXj��

��Fi

�zk�xjAij� k � �� n� � ��

where Aij are algebraic complements to the elements of the Jacoby matrix

��Fi

�xj

i��n

j��n

�

If there are no analytic expressions for partial derivatives�#

�zk� they can be computed

using known multipoint numerical schemes�

�� Computation of the periodic solutions and their

stability

Periodic solutions of the system of nonlinear equations

dx

dt� F�x� c�� x � Rn� c �M � Rm ��

corresponding to closed orbits in the state space� can be found using di�erent iterativenumerical algorithms�

The problem is reduced to a search for �xed point of appropriate mapping� Time�advance mapping �t�x� and mapping of hypersurface of codimension � onto itself maybe used�

�� Time�advance mapping�

The requirement for a trajectory in state space to be closed results in a search for zerosof nonlinear vector function H

H�x� � � �� x�� x ��

where �� x� is� for example� time�advance mapping of initial point x with time interval

�Thus we need to �nd a period of the solution and some point x belonging to the

closed orbit ��

��

The system �� comprise n equations depending on n�� unknowns� So it de�nesa set of solutions x � � depending on one parameter�

If x � � and H � �� one can obtain equations in increments �x and � in thevicinity of the closed orbit ��g� ��

��

�x� E

��x� F�� x�� H�x� � ��

Matrix��

�x

��x�

� calculated in some point belonging to the curve � while being

the period� is called the monodromy matrix� Monodromy matrix always has a uniteigenvalue� and the corresponding eigenvector �� coincides with tangent vector to aclosed trajectory at x� �� F�x��

Figure �� Time�advance mapping

Consider a small shift along the closed trajectory� #x � �F�x�� The result of amapping applied to a new point �� x�#x� so we have

x�#x � �� x�#x� ��

hence

��

F�x� ��

�x� F�x� ��

Thus the matrix

��

�x� E

�in the left hand side of the equation �� is singular

in the vicinity of the closed orbit �� i�e� it has zero eigenvalue �with a correspondingeigenvector F�x��

For �nite solutions #x of the equations �� to exist� it is necessary that a vectorin the right hand side of the equations

��

�x�E

�#x � �H�x� �� F�� x��# ��

is normal to vector F�x�� which is eigenvector of��

�xmatrix �� i�e�

F� �x� �H�x� � � F��x��# � � � ��

where F� denotes a transposed vector F�

This condition may be used for calculation of the increment of time period #

# � �F��x�H�x� �

F��x�F�� x��

Geometrically the relationship �� means that a �nal point on the trajectory ��x�must belong to a hyperplane "F � hyperplane "F being a set of vectors normal to F�x�vector�

If �� x� � "F � then period doesn�t change during convergence to a closed orbit�Thus hyperplane "F de�nes mapping with a constant period in a vicinity of a closedorbit�

It follows also that all other eigenvectors �i� i � �� n of the monodromy matrix��

�xlie in the hyperplane "F and hence are normal to vector F�x��

General solution #x of the system of equations �� will include a certain shift inthe "F hyperplane and some arbitrary displacement along vector F� In order to minimizeabsolute value of #x� it is reasonable to append equations �� with a condition thatrequires #x to be normal to the tangent vector F�#x � � �

��

�x� E F�� x�� #x �H�x� �

�F� � # �

��

The resulting system of equations �� is regular� Thus one can remain in thehyperplane normal to a closed orbit during convergence to the orbit�

��

Eigenvalues �� n�� and eigenvectors F�x�� n�� of the matrix

��

�x

��x�

de�ned in a point belonging to a closed orbit provide an information about stability ofthe periodic solution and about the structure of the state space in the vicinity of theclosed orbit�

Monodromy matrix��

�xis computed by means of numerical integration of equations

�� with initial conditions being varied successively with �xi� i � �� n

��

�x�

��x� �xi�� x� �xi�

��xi

�i��n

��

One can check the accuracy of computation knowing that always there is an eigen�value equal to � with corresponding eigenvector F�x��

�� Poincar�e mapping�

Poincar�e point mapping P� "� " of secant �n��dimensional hyperplane " �transver�sal to phase trajectories in the vicinity of closed orbit� can also be used for �ndingperiodic solutions� Let�s consider the properties of that mapping�

Transversal intersection of the trajectory in the state space with secant hyperplane "results in a sequence of points� One can distinguish these points according to directionof intersection with "� It is possible to consider the subsequence which includes pointswith the same direction of intersection fx��x��x�� g when trajectory leaves half�space"� for half�space "�� Full sequence form an orbit of bidirectional Poincar�e mappingfx��x��x��x�x�� g �see �g� ��

Also K�fold iterated Poincar�e mapping can be considered� In that case point of theeach K�th intersection in the chosen direction is included into the sequence� K�foldmapping Pk �x� is used to study periodical solutions with K�fold period�

" hyperplane is set using two vectors�

" � fx � h��x� xo� � �g ��

where vector xo is some point on hyperplane� and vector h is normal to hyperplane "�

When integrating the trajectory one can determine in what half�space current point xis located�

"� � fx � h��x� xo� � �g ��

or

"� � fx � h��x� xo� �g ��

��

The point of intersection x with " can be re�ned using linear interpolation betweentwo successive point on the trajectory x� � "� and x� � "�

x � x�H�

H� �H�� x�

H�

H� �H�

H� � h��x� � xo� � �

H� � �h��x� � xo� � �

��

To satisfy the required accuracy for intersection point

h��x � xo� � � ��

iterative procedure can be used� It comprises a backward step along trajectory �in

reverse time� of the size #t � t� � t� where

t � t�H�

H� �H�� t�

H�

H� �H��

and once more linear interpolation ��

It is possible to look for �xed points of the point mapping xk�� P �xk� or for zerosof vector function H�x� � P �x� � x� x � " using variations �x that keep the vectorx in the hyperplane "� A basis �i of linear independent vectors �i� i � �� n � �for " is to be constructed �rst� the orthogonalization procedure starting with normal tohyperplane vector h may be used�

Matrix�P

�xeigenvalues do not depend on the choice of hyperplane " until it remains

transversal to the closed orbit�

Let �i� i � �� n � � be eigenvectors of the matrix�P

�x� If �xk � ��

where �� are constants� and �� are eigenvectors forming complex eigenvector�c � �� i�� corresponding to a complex pair of eigenvalues �e

�i�� then the deviationwill be de�ned as follows

�xk�� cos�� sin�� sin�� cos��

Suppose that the other secant hyperplane "h is de�ned� vector h being normal toit� Let the trajectory to intersect with "h in points �x�k and �x�k�� If this trajectoryintersects with "F in �xk and �xk�� then

�x�k � �xk ��x�kh

F�h� F

�x�k�� xk�� x�k��h

F�h� F

��

��

The map �� is linear� hence

�x�k � �� cos�� sin�� sin�� cos��

��

where

��i � �i �

��ih

F�h� F

It means that Jacoby matrix for the point mapping de�ned on "h hyperplane hasthe eigenvectors ��

�� and eigenvalues as the matrix for mapping de�ned on "F

hyperplane� These eigenvalues are the characteristic multipliers for the periodic solution�Jacoby matrix of the point mapping can also be computed using arbitrary variations

�x of initial point� so that the variations leaving the hyperplane "h are allowed� Theresult of the mapping P �x� �x� belongs to " hyperplane�

In this case the matrix of linearized map�P

�xcan be linked with monodromy matrix

��

�x

��

�x��P

�x�Fh�

F�h��

Let F�� n�� be the eigenvectors and �� n�� be corresponding eigen�

values of��

�xmatrix� One can check that F vector is an eigenvector of

��

�x� but

corresponds in this case to zero eigenvalue�Indeed�

��

�xF � F

�P

�xF � F�

Fh�

F�hF � �

��

All other eigenvalues of�P

�x� as well as those of

��

�xmatrix� correspond to charac�

teristic multipliers of the periodic solution� One may get the eigenvectors using lineartransformation of this shift along trajectory

�P

�x��i � �i�

�i

��i � �i �

�

�i

h��i

F�h

��

��

Iterative search for a periodic solution is performed in accordance with �#x � xi��xi� �

�P

�x�E

�#x � �H�x� � �P �x� � x ��

Unlike �� the matrix in the left hand side �� is regular� The value of #x canbe obtained directly solving linear system of equations ��

Continuation algorithm may be e�ciently used to investigate the dependence of theperiodic solutions and their stability on parameters� zeros of the vector function

H�x� c� � P �x� c�� x � � ��

can be studied�

�� Algorithms for stability regions computation

Every stable attractor �equilibrium point� periodic trajectory� etc�� has its own domain

of attraction� also referred to as stability region� Domain of attraction is a set of initialpoints in state space� trajectories starting from these initial conditions approach theattractor as t��

In many cases boundaries of stability regions are formed by stable invariant manifoldsgenerated by saddle equilibrium and saddle periodic trajectories ��

For planar dynamical systems the boundaries of stability regions are either $sta�ble� trajectories of saddle equilibrium points or unstable limit cycles� They de�ne theboundaries of all the domains of attraction�

If the dimension of the state space n � � then stable manifolds of trajectories W s ofthe codimension � passing through saddle equilibrium points or saddle periodic trajec�tories are the most widespread boundary �components��

Qualitative analysis of dynamical system is done before the investigation of stabilityregions� In particular� the analysis of stable and unstable invariant manifolds of criticalelements like saddle equilibria and saddle closed orbits must be performed� First thosesaddle critical elements must be found whose unstable manifolds W u have trajectoriesapproaching to the stable equilibrium point or closed trajectory�

�� Computation of stability region

boundaries

Consider the algorithm of searching for the boundary of stability region formed by stableinvariant manifold corresponding to saddle equilibrium with one positive eigenvalue ��

The boundary of stability region W sn�� in the state space can be represented by a

number of its cross�sections by two�dimensional planes P� �see Fig�� The intersectionbetween P� and W s

n�� results in the curves �Sp �

�SP � P� �W sn��

�

Figure �� Algorithm computing stability region cross�sections

These curves are the boundaries of the stability cross section�Trajectories passing through the points of the curve �SP belong to W s

n�� and theyapproach the saddle point as t�� If trajectory initial point is moved aside from thecurve �SP �but still remains it in the plane P�� then the trajectory will miss the saddlepoint�

One can introduce a frame of reference on the cross plane P� � xk� xi� Now it isnecessary to de�ne a measure of miss� Consider the hyperplane Ln�� passing throughthe saddle point and tangent to theW s

n�� Ln�� is a linear n��dimensional hyperplanewith a basis formed by stable eigenvectors of Jacoby matrix �computed at saddle point��

The hypercylinder CR is de�ned as a set of points being on the same distance R fromthe straight line passing through saddle point and parallel to unstable eigenvector ��

There exists a neighbourhood in the vicinity of a point on the curve �SP � P�� allthe trajectories started from the points belonging to the neighbourhood cross the surfaceof the cylinder CR� The maximum size of that neighbourhood is depends on the radiusR of the cylinder� A distance from the intersection point �where trajectory crosses thesurface of CR� to the hyperplane Ln�� is used as a measure of miss� see Fig��

The measure of miss H is de�ned as

H�x� � �x� xe��

where xe is a saddle point vector� and �� is a unit vector normal to Ln�� hyperplane

�it can be calculated as unstable eigenvector of transposed Jacoby matrix��

�

Scalar function r�x� evaluates the distance between current point on the trajectoryx and the surface of the cylinder CR�

r�x� � kx� xe �H�x��k �R � ��

where kxk is euclidean norm of the x vector�If r�x� changes its sign� it means that cylinder CR is crossed by the trajectory�The mapping of a point of a plane P� onto the surface of the cylinder CR is determined

by the condition r�x� � �� For every point belonging to some neighbourhood of the curve�SP � P�� the mapping de�nes a scalar value H�xR� which is denoted as HR�xk� xi��The mapping HR�xk� xi� is performed by numerical integration of dynamical system� thesign of r�x� is checked during integration�

The problem of computing the curve �SP �i�e� the section of stability region boundaryW s

n�� by the plane P�� is reduced to �nding and continuation of a solution of the scalarequation

HR�xk� xi� � � ��

The equation �� depends on two parameters xk and xi � the coordinates of the initialpoint of the mapping with respect to the frame of reference de�ned on the P� plane�

To use algorithm e�ciently� the radius of the cylinder R is to be chosen properly�The radius must be small enough for the tangent hyperplane Ln�� to be a reasonableapproximation of the invariant manifolds W s

n�� inside the cylinder CR� At the sametime� the radius must be large enough to minimize calculation time and to establishreasonable domain of algorithm convergence�

Condition �� de�nes an approximate boundary of stability region� This boundarytends to the exact boundary of the section of invariant manifolds W s

n�� when R� ��The solution of �� is calculated using the continuation method� First the method

converges to a solution of �� and then a continuation of the curve �SP is performed �

dxkdt

��kHRH

xkR �Hxi

R

�HxkR �

� � �HxiR �

�

dxidt

�Hxi

R � kHRHxkR

�HxkR �

� � �HxiR �

� �

��

Required accuracy of computations is reached by means of proper selection of the co�e�cient k and integration step #t in �� The partial derivatives Hxk

R and HxiR are

obtained numerically�

�� Direct computation of stability region crosssections

If computer is powerful enough� then the direct computation of two�dimensional crosssections of stability region very often is more e�cient than other methods� The algorithmincludes the following stages �

�� Known attractors are written into memory� Periodical solutions and more complexattractors are represented by an arbitrary point belonging to the attractor�

��

Figure �� Direct computation of stability region boundaries

For each attractor some reasonably small region S�i is ether determined or selected�The region S�i must be a subset of attractor�s stability region� The regions S

�i � i �

�� for example� can be estimated by means of Lyapunov function method�� see �g��

To determine all the attractors in advance is desirable but not obligatory for thisalgorithm�

�� Maximum integration time and boundaries of an allowed state space region arespeci�ed� Maximum integration time should be big enough �not less that ��characteristic periods of time for the system in question��

�� Two�dimensional secant plane is selected� and some frame of reference on thatplane is de�ned�

�� A grid on a secant plane is selected� the grid de�nes initial points for numericalintegration of the dynamical system�

� System of equations is integrated starting from every grid point� Integration stopsif a trajectory enters inside anyone of the regions of guaranteed stability S�i � i �� or if maximum integration time is exceeded� or if the trajectory leavesallowed state�space region�

If maximum integration time was exceeded� the process of automatic adding of newattractors may be trigged� Last trajectory point is registered as a new attractorand some small region of guaranteed stability is assigned to it�

�

� Grid points are classi�ed according to the results of integration� Either they belongto i�th attractor stability region or situation is uncertain� As s result� a section ofstability region may be determined with desired accuracy�

The ability of the algorithm to look for new attractors may be handy for the analysisof the state space structure�

The computation time for this algorithm depends on the number of grid points andon the size of the regions S�i � i � �� Though usually time�consuming� this methodis simple� reliable and e�cient�

�� Example Global stability analysis in

roll�coupling problem

The roll coupling problem at fast roll rotation is usually described by the following thorder system of equations �supposing that gravitational terms and velocity variation arenegligible��

%� � q � �p cos�� r sin�� tan � � z

%� � p sin � � r cos�� y

%p � �i�qr � l

%q � i�rp �m

%r � �i�pq � n �

��

where p� q� r are angular velocity components in body�axis� �� are angle of attack andsideslip� i� � �Iz�Iy� Ix� i� � �Iz�Ix� Iy� i� � �Iy�Ix� Iz are non�dimensional inertiacoe�cients� l � L Ix� m � M Iy� n � N Iz� y � Y MV � z � Z MV � The reducednormal and side forces z� y� pitch� roll and yaw moments m� l and n in equations �� can be represented as a functions of state variables and control parameters both in linearor nonlinear forms�

z � z�� zq��q � z�e ��e � � � �

y � y� � y�� yr��r � y�r ��r � � � �

m � m�� mq��q �m�e��e ��m��

l � l�� lp��p� lr��r

� l�r ��r � l�a ��a ��l��

n � n�� np��p� nr��r

� n�r��r � n�a��a ��n��

��

The equations �� complemented by �� form the closed nonlinear autonomoussystem of �fth order with the state vector x � �� p� q� r�T � The nonlinearities in theseequations can be divided into three groups� kinematic� inertia moments and aerodynam�ics terms� All these terms can lead to the cross coupling between the longitudinal andlateral modes of motion�

��

The main feature of the considered problem is that even in the case of linear represen�tation of aerodynamic coe�cients the existence of multiple stable steady�state solutions�equilibrium and periodic� is possible� The bifurcational analysis of all the possiblesteady state solutions and their local and global stability analysis can show the genesisof stability loss and explain in many cases the very strange aircraft behaviour�

The validity of the system �� is con�ned in time� Therefore� in the casesof weakness or lack of stability of the steady states the conclusions resulting from theconsideration of the asymptotic stability in Lyapunov sense� when t � �� can bewrong� The similar problem can arise for short�term control inputs� The prediction ofthe bifurcation analysis will be more consistent when the considered steady states havethe su�cient margin of asymptotic stability� In any case the numerical simulation ofaircraft motion using the complete set of equations is necessary for �nal veri�cation ofthe bifurcational analysis results�

To demonstrate the possibilities of bifurcation and global stability analysis� roll�coupling problem is studied for the hypothetical swept�wing �ghter� Flight conditionsare high altitudes and supersonic velocities�

The simpli�ed model considered in �� is taken as an example for demonstrationof stability region boundaries computation� Equations �� can be reduced to �rdorder system provided the characteristic frequency of the longitudinal motion su�cientlyexceeds the characteristic frequency of the isolated motion in yaw� This may be validat high supersonic �ight� Angle of attack in such cases remains practically constant atthe disturbed motion and q � p�� Approximate equations governing the variations ofangular rates p� r and sideslip angle � in this case have the following form�

%p � a�� a� p� a� r � lc

%r � �b� � b� p�� b� r � b p

%� � �r cos�� p sin�� c� �

��

In the calculations the following values of the parameters were assumed�

a� � �� s�� a� � �� s�� a� � �

b� � � s�� b� � �� b� � �� s��

b� � � �� c� � �

During bifurcation analysis and stability regions computation the roll control lc is con�sidered as a parameter�

At any value of the control parameter lc there are three equilibrium points� Fig��left plot� shows the dependence of roll rate p on the control parameter lc for all threeequilibrium points� Two branches with non�zero coordinates at lc � � are unstable atany value of the control parameter� These equilibria have one real positive eigenvalueand a complex pair with negative real part� The branch of solutions passing throughthe origin comprises the stable equilibria at lc �� and oscillatory unstable equilibriaat lc � �� At Hopf bifurcation point �lc � �� the stable closed trajectories orlimit cycles appear� Fig�� right plot� shows the projections of the closed trajectoriesonto �p� r� plane for several values of lc� For each cycle the corresponding values of the

��

control parameter lc� time period of oscillations T and the value of the main characteristicmultiplier of the Jacoby matrix of the Poincar�e point mapping at the �xed point of themapping are shown in �g�� Time period of oscillations increases with the growth ofthe amplitude of the cycle� The increase is especially dramatic when the cycle approachessaddle�type unstable equilibrium point� In Fig�� one can see how the cycle changes itsform and stretches towards saddle point� The change in the locations of the equilibriumpoints is insigni�cant while the control parameter remains in the region where cyclesexist� At lc � �� closed trajectory merges with saddle equilibrium point and disappear�At lc � �� there are no cycles and all three equilibrium points are unstable�

At step�wise control input with lc �� the motion parameters approach theirequilibrium values aperiodically� At �� lc �� steady oscillations arise after sometime of transient motion� If critical value lc � �� is exceeded� the parameters of themotion increase sharply and irreversibly after several oscillations�

Every stable equilibrium or limit cycle has a certain region of attraction� which isbounded by two separating surfaces passing through saddle unstable equilibrium points�Fig�� shows two sections of the stability region at lc � � de�ned by thier boundariescomputation� Cross plane P� in both cases was chosen normal to the p axis� In the �rstcase �left plot� the plane passes through the stable solution located at the origin� and inthe second case �right plot� the cross plane P� passes through saddle equilibrium point�

Change in the shape of the separating surfaces and corresponding deformation ofattraction region of stable equilibrium point with the increase of the control parameterlc are demonstrated in Fig�� the cross plane is de�ned by a condition r � � andpasses through zero point��

At lc � �� the section of the region of attraction becomes double�linked due tospatial deformation and bending of the separating surface passing through saddle pointto which stable equilibrium is approaching� Further growth of the control parameter�lc � �� results in the appearance of another internal instability region at � ��p �� At lc � �� the internal instability regions merge leaving only narrow �capture�area�

The periodic solutions of equations �� shown in Fig�� appear due to Hopfbifurcation at lc � �� and disappear at lc � �� because of the homoclinical bifurcationof saddle point unstable trajectory� when limit cycle merges with saddle point�

The performed analysis reveals the sequence of bifurcations of steady�state solutionsof equations �� bringing about possible irreversible loss of stability at critical controlinput in roll� Excess of the critical value corresponding to the disappearance of stableoscillations� may result in unproportional rotation�

The �fth order system �� which takes into account longitudinal dynamics� isconsidered for computation of stability region cross sections by direct method� Thefollowing linear aerodynamic coe�cients were used in the numerical analysis��

�Angles of attack and sideslip are in radians rotation rates are in radians per second control surfacede�ections are in degrees

��

Figure �� Bifurcations of equilibrium and periodic solutions�

��

Figure �� Bifurcations of equilibrium and periodic solutions� Closed orbit time periodand multiplier�

Figure �� Stability region cross sections in the plane �� r� obtained by their bound�aries computation�

��

Figure �� Stability region cross sections in the plane �� p� for di�erent control pa�rameters lc� obtained by thier boundaries computation�

zo � � yo � � i� � ��

z� � �� y� � �� i� � ��

z�e � �� y�r � �� i� � ��

l� � �� mo � �� n� � ��

lp � �� m� � �� np � ��

lr � �� m�e � �� nr � ��

l�a � �� mq � �� n�a � ��

l�r � �� m �� n�r � ��

In this case two critical roll rates exist and are very close to Phillips� approximatevalues� As a result the equilibrium solutions are divided by this critical lines intodi�erent unconnected families� which are shown for di�erent normal factor values inFig�� at the bottom right plot� The dependencies of equilibrium roll rateon aileron de�ection are presented for di�erent elevator de�ections� corresponding tonormal factor parameters az � �� Di�erent line types indicate local stability ofthe computed equilibriums� solid lines correspond to stable solutions� dashed lines � todivergent solutions� dash�dotted lines � to oscillatory unstable solutions�

There are di�erent families of equilibrium curves� some of them are located betweenthe critical rates p� p� and in the outer regions� Some of them generate the stableautorotational rolling regimes at zero aileron de�ection� The autorotational rolling so�lutions exist at all aileron de�ections �see Fig�� but at �pro�roll� aileron de�ections�p � �� a � �� the autorotational equilibrium solutions become oscillatory unstable

��

�dash�doted lines�� and after Hopf bifurcation point on each curve the family of stableclosed orbits appear�

A similar oscillatory instability of equilibrium solutions appears at subcritical equi�librium curves at large aileron de�ections �see Fig�� The Hopf bifurcation pointsin this case also give birth to the families of stable closed orbits �they are shown inFig��

�

Figure �� Stability region cross sections for level �ight �az � �� without rotation�

Figure �� Stability region cross sections for �ight with zero normal factor �az � ��without rotation�

�

Figure �� Stability region cross sections for �ight with negative normal factor �az �� without rotation�

The global stability analysis or investigation of asymptotic stability regions is veryimportant in the case� when there are multiple locally stable equilibrium points andclosed orbits� The reconstruction of these stability regions provides global informationabout state space of system in question and determines the critical disturbances of thestate variables leading to the loss of stability� As already was noted� more convenientway of reconstruction of multidimensional stability region is the direct computation of itstwo�dimensional cross sections� The disturbances are only in two selected state variables�with other state variables �xed�� Cross�section is often selected so as to pass throughthe locally stable equilibrium point under consideration�

At zero aileron and rudder de�ections �a � �r � � and �e � � �� az � �� forsupersonic �ight regime �see Fig�� there are �ve equilibrium points� three of them� are locally stable �black circles� and two � are locally unstable �white cycles�� Threedi�erent cross sections of the asymptotic stability regions of stable equilibrium points arepresented in Fig�� All the cross sections pass through the central stable equilibriumpoint without rotation� The �rst cross section is placed in the plane of angle of attack �and sideslip �� the second one is placed in the plane of roll rate p and sideslip �� and thethird one is placed in the plane of yaw rate r and sideslip �� The region of attractionfor central equilibrium is marked with light points� the regions of attraction for criticalequilibria are marked with more darker points� One can see� that more critical level ofperturbation for central equilibrium is in yaw rate � rcr � �� s� Far from centralequilibrium the boundary of stability region has thin structure with multiple folds�

The size of asymptotic stability region of central equilibrium becomes signi�cantly

��

less when aircraft is trimmed at zero or negative angle of attack �Figs�� Toillustrate this� similar cross sections of the stability regions for �e � �� az � �� arepresented in Fig��

�

Bibliography

�� Poincar�e� Henri $Les methodes nouvelles de la mechaniqes celestre�� vol� I�III� Paris��

�� Arnold V�I� $Geometrical Methods in the Theory of Di�erential Equaitions�� Berlin�Springer�Verlag� ��

�� Arnold V�I� $Additional chapters of the theory of Ordinary Di�erential Equations��Nauka publ�� Moscow� �� in Russian��orArnold V� Chapitres Suppl�ementaires de la Th�eorie des �Equations Di��erentielles

Ordinaires� �Editions Mir� ��

�� Iooss G�� and Joseph D��Elementary Stability and Bifurcation Theory�� Berlin�Springer�Verlag� ��

� � Iooss G�� and Aldermeyer� �Topics in Bifurcation Theory and Applications�� in theseries �Advanced Series in Nonlinear Dynamics�� Vol�� Word Scienti�c� ��

�� Marsden J�E� Mc Cracken M� �The Hopf Bifurcation and its Applicatins�� SpringerVerlag� ��

�� Gukenheimer J� and Holmes P�� Nonlinear Oscillations� Dynamical systems andBifurcations of Vector Fields�� Springer Verlag� ��

�� Carr J�� Applications of Center Manifold Theory�� Springer Verlag� ��

�� Langford W�F�� A review of interactions of Hopf and steady state bifurcations�in �Nonlinear Dynamics and Turbulence�� Pitman Advanced Publishing Program��

�� Seydel R�� BIFPACK� a program package for continuation� bifurcation and stabilityanalysis�� University of Wurzburg� ��

�� Rosendorf P�� Orsag J�� Schreiber I� and Marec M� �Interactive system for studies innon�linear dynamics� in Continuation and bifurcations� numerical techniques andapplications� NATO ASI Series� Vol �� Kluwer Academic Publishers� ��

�� Meija R�� CONKUB� A coversational path�following for systems non�linear equa�tions�� J�of Comp� Physics� Vol� ��

��

�� Morgan A�� Solving polynomial systems using continuation�� Prentice Hall� En�glewood� ��

�� Watson L�T�� Billups S�C� and Morgan A� �HOMEPACK� A suite of codes forglobally convergent homotopy algorithms�� ACM transactions on Math� Software�Vol ��

�� Khibnik A�I� �LINLBF� A program for continuation and bifurcation analysis ofequilibria up to codimension three�� in Continuation and bifurcations� numericaltechniques and applications� NATO ASI Series� Vol� �� Kluwer Academic Pub�lishers� ��

�� Kass�Petersen C� �PATH� Users�s guide�� University at Leeds� Center for non�linearstudies� ��

�� Bank R��E�� PLTMG User�s guide � Edition �� University of California� La Jolla��

�� Phillips� W�H�� E�ect of Steady Rolling on Longitudinal and Directional Stability��NASA TN �� June ��

�� Gates� O�B�� and Minka� K�� Note on a Criterion for Severity of Roll�InducedInstability�� Journal of the Aero�Space Sciences� May �� pp��

�� Hacker� T�� and Oprisiu� C� �A Discussion of the Roll Coupling Problem�� Progressin Aerospace Sciences� Vol� � � Pergamon Press� Oxford� ��

�� Biushgens� G�S�� and Studnev� R�V�� $Aircraft Dynamics� Spatial Motion�� Moscow�Mashinostroenie Publishers� �� in Russian��

�� Young� J�W�� Schy� A�A�� and Johnson� K�G�� Pseudosteady�State Analysis ofNonlinear Aircraft Maneuvers�� NASA TP �� Dec� ��

�� Dolzenko� N�N�� Investigation of equilibrium aircraft spin�� TsAGI ProceedingsNo� �� in Russian��

�� Adams� W�M��Analytic Prediction of Airplane Equilibrium Spin Characteristics��NASA TN D��

�� Tischler� M�B�� and Barlow� J�B�� Determination of Spin and Recovery Charac�teristics of a General Aviation Design�� Journal of Aircraft� Vol�� No�� pp��

�� Mehra�R�K�� Bifurcation analysis of aircraft high angle of attack �ight dynamics��AIAA N �� August ��

�� Carroll�J�V�� and Mehra�R��K�� Bifurcation Analysis of Nonlinear Aircraft Dynam�ics�� Journal Guidance� Vol� � No� � �� pp� ��

�

�� Guicheteau� P�� Bifurcation Theory Applied to the Study of Control Losses onCombat Aircraft�� La Recherche Aerospatiale� Vol� �� pp��

�� Guicheteau� P�� Bifurcation theory in �ight dynamics� An application to a realcombat aircraft�� ICAS Paper � �� Sept��

�� Guicheteau� P�� Stability Analysis Through Bifurcation Theory �� and �� and�Nonlinear Flight Dynamics�� in Nonlinear Dynamics and Chaos� AGARD LectureSeries �� June� ��

�� Zagaynov� G�I�� and Goman� M�G�� Bifurcation analysis of critical aircraft �ightregimes�� ICAS�� September� ��

�� Planeaux� J��B�� and Barth� T�J�� High Angle of Attack Dynamic Behavior of aModel High Performance Fighter Aircraft�� AIAA Paper �� Aug� ��

�� Planeaux� J��B�� Beck� J�A�� and Baumann� D��D�� Bifurcation analysis of a model�ghter aircraft with control augmentation�� AIAA Paper �� Aug� ��

�� Jahnke� C�C�� Culick� F�E�C�� Application of Dynamical Systems Theory to Non�linear Aircraft Dynamics�� AIAA Paper �� Aug� ��

�� Jahnke� C�C�� Culick� F�E�C�� Application of Bifurcation Theory to High�Angle�of�Attack Dynamics of the F�� Journal of Aircraft� Vol� �� No� �� Jan��Feb��

�� Goman� M�G�� Fedulova� E�V�� and Khramtsovsky� A�V�� Maximum Stability Re�gion Design for Unstable Aircraft with Control Constraints�� AIAA Guidance Nav�igation and Control Conference� Paper N �� July� ��San Diego� CA�

�� Kubichek M� and Marek M��Computational Methods in Bifurcation Theory andDissipative structures�� Springer Verlag� New York� ��

�� Genesio P�� Tartaglia M�� Vicino A� $On the estimation of asymptotic stabilityregions� State of the art and new proposals�� IEEE Trans� on Aut� Contr�� vol�AC�� no� �

�� Hsiao�Dong Chiang� Morris W� Hirsch� Felix F� Wu $Stability Regions of NonlinearAutonomous Dynamical Systems�� IEEE trans� on Automatic Control� vol�� no��Jan� ��

�� Chao� K�S�� Liu� D�K�� and Pan� C�T�� A systematic search method for obtain�ing multiple solutions of simultaneous nonlinear equations�� IEEE Transaction�vol�CAS�� No��

�� Goman� M�G�� $Di�erential method for continuation of solutions to systems of �nitenonlinear equations depending on a parameter�� Uchenye zapiski TsAGI� vol� XVII�no� � �� in Russian��

�

�� Goman� M�G�� and Khramtsovsky� A�V�� Analysis of the Asymptotic StabilityRegion Boundary for a Dynamic System�� TsAGI Journal� Vol�� No��

�� Wood E�F�� Kempf J�A� and Mehra R�K� �BISTAB� A portable bifurcation andstability analysis package�� Appl�Math�Comp�� vol � � ��

�� Doedel E� and Kernevez J�P�� AUTO� Software for continuation and bifurcationproblems in ordinary di�erential equations�� California Institute of Technology�Pasadena� ��

�� Guicheteau P�� Notice d�utilisaton du code ASDOBI�� Pre�print ONERA� ��

�� Lowenberg� M�H�� Bifurcation analysis for aircraft dynamics�� University of Bris�tol� Contract Report No�� for DRA� Bedford� UK� June� ��

�� Goman� M�G�� and Khramtsovsky� A�V�� KRIT � Scienti�c Package for continua�tion and bifurcation analysis with aircraft dynamics applications�� TsAGI� ��

�� Thomas S� Parker� Leon O� Chua $INSITE � A Software Toolkit for the Analysisof Nonlinear Dynamical Systems�� Proc� of the IEEE� vol�� n��

�� Yu�I�Neymark $Method of point mapping in the theory of nonlinear oscillations��Nauka publ�� Moscow� �� in Russian��

� �� Yu�I�Neymark� P�S�Landa $Stochastic and chaotic oscillations�� Nauka publ��Moscow� �� in Russian��

� �� Poston T�� Stewart I�N� $Catastrophe Theory and Its Applications�� London� Pit�man� ��

� �� Martin Holodniok� Alois Kl�'(c� Milan Kub�'(cek� Milo(s Marek� �Methods of Analysisof Nonlinear Dynamic Models�� Academia� Praha� �� in Czech�� Mir� Moscow�� in Russian�� Pp� �� russian ed� �Section �� Task ��

�

M.G.Goman and A.V.Khramtsovsky (1997 draft) - Textbook for KRIT Toolbox users

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Transcript of M.G.Goman and A.V.Khramtsovsky (1997 draft) - Textbook for KRIT Toolbox users