Stability of Nonlinear Systems MH

8/3/2019 Stability of Nonlinear Systems MH

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Stability of Nonlinear Systems

Linear systems combine two kinds of stability/instability properties:

1)Large scale stability/instability properties:

Solutions of stable linear systems remain bounded Solutions of unstable linear systems diverge to infinity

2) Small scale (local) stability properties:

Solutions of stable linear systems that start with close initial conditions remainclose

Solutions of unstable linear systems drift apart.

In the case of nonlinear systems, the local stability/instability properties are not linked tothe global stability/instability properties. As an example, take the Van der Pol oscillator

( )

12

221 1 2

dxx

dt

dxx x 1 x

dt

=

= (1)

with = 0.3. If a solution starts exactly at the origin, it remains there forever. If,however, it starts close to the origin, it diverges from it, but without diverging toinfinity.In fact it converges to a periodic solution ().


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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 1. Orbits of three solutions of the Van der Poloscillator (1) with = 0.3 in state space. The first solution(blue) remains at the origin. The two other solutions (red

and green) start close to the origin, but diverge from the

origin and converge to a periodic solution.

0 5 10 15 20 25 30 35 40 45 50

-2

-1

0

1

2

0 5 10 15 20 25 30 35 40 45 50

-4

-2

0

2

4

Figure 2. Same solutions as in Figure 1, but the statevariables are represented as functions of time.

Thus we have in this case small scale instability but large scale stability.


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Large-scale notions of stability/instability

Definition 1:

a) The system has bounded solutions if for each0

Nx there is a constant B suchthat for all 0t

( )t Bx (2)

where x(t) is the solution with x(0) = x0.

b) The system has asymptotically uniformly bounded solutions, if there is a constant

B such that for each 0Nx there is a time T such that for all t T

( )t Bx (3)

A system with bounded, but not asymptotically bounded solutions has typicallytrajectories in state space according to Figure 3.

Figure 3. Typical trajectories in state space of asystem with bounded, but not asymptotically

uniformly bounded solutions.

A system with asymptotically bounded solutions has typically trajectories in state spaceaccording to


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B

Figure 4. Typical trajectories in state space of a system

with asymptotically uniformly bounded solutions.

How is it possible to prove these properties for a specific system? The following theorem

gives a method that is most of the time either easy to apply or it is easy to see that thesystem has solutions that diverge to infinity. The method uses auxiliary functions, calledLyapunov functions.

Theorem 1:

a) Suppose there is a function : NW + such that

W is constant along trajectories (W is a conserved quantity) The level sets ( ){ }W Kx x of W are bounded

Then the system has bounded solutions

b) Suppose there is a function : NW + such that

W is decreasing along trajectories as long as 0W W , i.e. for any solutionx(t)

( )( ) ( )( ) 00 ifd

W t W t W dt

< x x (4)

The level sets ( ){ }W Kx x of W are boundedThen the system has asymptotically uniformly bounded solutions.

Proof: Left as an exercise.

Theorem 1 apparently only shifts the difficulty to the existence of the Lyapunov functionW. However, in most examples either it is easy to find such a function, or it is easy to see


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that some solutions diverge to infinity. In mechanical and electrical systems often W hasthe physical meaning of energy. In this case, systems with bounded, but not

asymptotically uniformly bounded solutions typically have conserved energy and systems

with asymptotically uniformly bounded solutions are dissipative outside of a boundedregion in state space.

Most systems with conserved energy in physics areHamiltonian systems. This means that

they are described by a function 2: MH , the Hamilton function or Hamiltonian,from which the state equation are derived as follows:

k

k

k

k

dq H

dt p

dp H

dt q

=

=

(5)

The arguments q1, , qM of H are called generalized coordinates and the arguments p1,, pM of H are called generalized momenta. The fact that along solutions of (5) the value

of H is constant can be seen easily:

( ) ( ) ( ) ( )( )1 11 1

1 1

, , , , ,

0

M Mi i

M Mi ii i

M Mi i i i

i i

dq dpd H HH q t q t p t p t

dt q dt p dt

dp dq dq dp

dt dt dt dt

= =

= =

= +

= + =

(6)

In the conventional setting of mechanics of M point masses interacting through forcesderived from a potential function V, the qi are the coordinates of the particles and theHamiltonian is of the form

( ) ( )2

1 1 11

, , , , , , ,2

Mi

M M M ii

pH q q p p V q q

m=

= + (7)

The variables pi satisfy, according to (7) and (5)

ii i i

i

dqH p m m

q dt

= =

(8)

i.e. pi is indeed the momentum in direction of qi and the total energy H is the sum of thekinetic energy and the potential energy.

Example 1:

Consider the system described by the Hamiltonian


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( ) ( )2

,2

p H p q V q

m= + (9)

where V is the double-well potential (Figure 5).

( ) ( ) ( )2 2

1 1V q q q= + (10)

1-1

V

Figure 5. Double-well potential

The solutions must move along level curves of the Hamiltonian H (Figure 6). They

are bounded.

q

p

Figure 6. Trajectories of the system with the

Hamiltonian (9). They move along the level

curves of the Hamiltonian.

Example 2:

Consider the 1-dimensional system described by

(11)Case a3 < 0:

Consider the function W(x) = x4. Along any solution of (11) we have


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( )( )

( )

3

3 2 30 1 2 3

4

4

d dxW x t x

dt dt

x a a x a x a x

=

= + + +(12)

For

0 1 2

3 3 3

4 4 4, , ,1

a a ax B

a a a

=

(13)

we have

3 033 3 3 3 0

3

3 123 3 3 3 1

3

3 2 22 23 3 3 2

3

41 1 1 1

4 4 4 4

41 1 1 1

4 4 4 4

41 1 1

4 4 4

aa x a B a B a a

a

aa x a B x a B x a x a x

a

aa x a B x a x a x

a

=

=

=

(14)

Therefore,

( )( ) ( )3 2 60 1 2 36 6

3 33

4

dW x t x a a x a x a x

dt

a x a x

+ + +

+(15)

Since, by hypothesis, a3 < 0,

( )( ) 631

04

dW x t a x

dt < (16)

By theorem 1 b), this implies that the solutions are asymptotically uniformlybounded. Actually, the constant B given by (13) is such an asymptotic uniform

bound for the solutions.

Case a3 > 0:

Consider a solution x(t) with x(0) > B, where B is given by (13). Then, as long as

x(t) > B, it follows from (14) that

( )2 30 1 2 33

31

4

dxa a x a x a x

dt

a x

+ + +

(17)


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Since this is a positive quantity, x grows as a function of time as long as x(t) > B,which implies that x(t) > B for all positive times. Furthermore, by (17), the

derivative has the uniform lower bound a3B3/4 for all positive times, which implies

that this solution diverges to infinity.

Small-scale (local) notions of stability/instability

The local notions of stability/instability cannot be defined in any satisfactory way for a

whole nonlinear system. In fact, for the Van der Pol oscillator (1) with = 0.3, theequilibrium point is unstable, whereas the periodic solution (limit cycle) is stable (cf.

Figure 1, Figure 2). Hence, the local stability has to refer to the single solution, and not to

the system as a whole. There are various ways to define local stability. We shall limit ourattention to the most important definition of stability of a solution, also called Lyapunov

stability.

Definition 2:

a) A solution N N: or :+ + x x of a discrete of analog system is stable,

if for any > 0 there exists a > 0 such that for any solution y with

(0) (0) < y x (18)

the inequality

(t) (t) < y x (19)

holds for all t 0. A solution is unstable if it is not stable.

b) A solution is asymptotically stable, if for any > 0 there exists a > 0 such that forany solution y with

(0) (0) < y x (20)

the inequality

(t) (t) < y x (21)

holds for all t 0and

t(t) (t) 0

+ y x (22)


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c) The basin of attraction at time t = 0 of an asymptotically stable solution x is the

set of all N0 y such that the solution y with 0(0) =y y satisfies (22).

d) A solution x is globally asymptotically stable if it is asymptotically stable and if

its basin of attraction is the whole space N .In this case, actually all solutionsare globally asymptotically stable and for any two solutions (22) holds. Thus, this

is a property of the system rather than of the solution and the system is said tohave unique asymptotic behavior.

Example 3

Consider again the Van der Pol oscillator (1) with = 0.3. From our numericalsimulations we can conjecture (and this will be shown rigorously later) that the

constant solution

c (t) 0x (23)

is unstable, whereas the periodic solution xp

p p(t T) (t) for all t+ =x x (24)

is stable. It may seem that the periodic solution is actually asymptotically stable,but this is not the case, because with xp there is an infinity of T-periodic solutions

p, p(t) (t ) = x x (25)

that differ only by a phase shift from xp. Even if is very small, xp, neverconverges to xp. Thus, there is no neighborhood of the initial conditions ofxp such

that all solutions starting in that neighborhood will eventually converge to xp asrequired for asymptotic stability.

By numerical simulation it appears that all solutions other than (23) converge to

some periodic solution xp, . Thus, all solutions, except (23) are stable.


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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 7. Three state space trajectories of the Van der Pol

oscillator (1) with = 0.3: Periodic solution (blue) andtwo other solutions (red and green)

0 2 4 6 8 10 12 14 16 18 20

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14 16 18 20

-3

-2

-1

0

1

2

3

Figure 8. State variables as a function of time for thethree solutions of Figure 7.

Stability of a fixed/equilibrium point


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So far, the definition of local stability has been given, but no criterion how to determine

in a specific case the stability or instability of a solution. There is no easy-to-use criterion

for arbitrary solutions. But for fixed/equilibrium points essentially the stability question isresolved by looking merely at the linearized system.

For the terminology, we do not distinguish between the fixed/equilibrium point and theconstant solution remaining in the fixed/equilibrium point. Thus, e.g. a fixed/equilibrium

point is stable if the constant solution starting at this fixed/equilibrium point is stable.

Consider the autonomous discrete system

( ) ( )( )t 1 t+ =x F x (26)

with N N: F . Let x be a fixed point, i.e.

( ) =F x x (27)

Consider a solution N: + x of(26) that starts close to x . As long as the solutionremains close, the following is a good approximation for the time evolution of the

increment with respect to the fixed point:

( ) ( )

( )( ) ( )

( ) ( )( )

( ) ( )

t 1 t 1

t

t

t

+ = +

=

=

x x x

F x F x

Fx x x

x

Fx x

x

(28)

It is therefore plausible that the stability/instability of the fixed point is given by the

eigenvalues of ( )F

xx

Theorem 2:

Consider a discrete autonomous system given by (26) and let x be a fixed point, i.e.a point where (27) holds.

a) The fixed point x is asymptotically stable if all eigenvalues i of the Jacobian

matrix ( )F

xx

satisfy i 1 < .


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b) The fixed point x is unstable if at least one eigenvalue i of the Jacobian matrix

( )F

xx

satisfies i 1 > .

Remark 1: The theorem gives only sufficient conditions The asymptotic stability of the linearized system at the fixed point implies the

asymptotic stability of the fixed point of the nonlinear system.

The instability of the linearized system at the fixed point implies usually theinstability of the fixed point of the nonlinear system.

If the largest eigenvalue of the linearized system at the fixed point is of absolutevalue 1, nothing can be said about the stability of the fixed point of the nonlinear

system. It all depends on the nonlinear terms.

Example 4

Consider the iterations of the logistic function 2f (x) 1 x= for 0 < 2 in theinterval [-1, +1]. Its fixed points satisfy the equation

2x 1 x= (29)

which yields

( )1

x 1 1 42

= +

(30)

The - sign in (30) yields a fixed point outside of the interval [-1, +1] which we

do not consider. The linearized system at the fixed point amounts to themultiplication with the derivative of f at the fixed point. This also coincides with

its only eigenvalue. Thus, the criterion for asymptotic stability of the fixed point

is

( )df

x 2 x 1dx

= < (31)

which is the case for 0 < < 0.75. Similarly, instability is deduced from theorem

2 for 0.75 < 2. This is confirmed by the following simulations:


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0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 50 4 0 0 4 5 0 5 0 0

-0.8

-0.6

-0.4

-0.2

0

0 .2

0 .4

0 .6

0 .8

lam bda = 0 .74

Figure 9. 500 iterations of the logistic function with = 0.74 startingat x(0) = -0.8.

0 50 1 00 1 50 2 00 250 30 0 35 0 40 0 450 50 0

-0.8

-0.6

-0.4

-0.2

0

0 .2

0 .4

0 .6

0 .8

l am bda = 0 .745

Figure 10. 500 iterations of the logistic function with = 0.745

starting at x(0) = -0.8.


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0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0

-0.8

-0.6

-0.4

-0.2

0

0 .2

0 .4

0 .6

0 .8

lam bda = 0.75 5

Figure 11. 1000 iterations of the logistic function with =0.755 starting at x(0) = -0.8.

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0

- 0 .8

- 0 .6

- 0 .4

- 0 .2

0

0 .2

0 .4

0 .6

0 .8

l am bda = 0 .7 6

Figure 12. 1000 iterations of the logistic function with = 0.76starting at x(0) = -0.8.

At the bifurcation point = 0.75, i.e. at the parameter value where the behavior ofthe system changes qualitatively, we have ( )

dfx 1

dx= and the linearization at the

fixed point does not give any information on its stability. The Taylor series of f

around the fixed point for = 0.75 is

( )2

f (x) x x 0.75 x= (32)


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where x x x = . From this we cannot immediately draw a conclusion about thestability of the fixed point. However, the Taylor series of the once iterated

function f is

( ) ( ) ( ) ( )2 3 3 4

f (f (x)) x x 2 0.75 x 0.75 x= + (33)

Thus, up to third order approximation, we have

( ) ( ) ( )( )3

x k 2 x k 1.125 x k + (34)

This shows that for a sufficiently small initial deviation from the fixed point, in

two time steps the deviation decreases strictly and it is not difficult to show that itin fact converges to zero. Thus, the fixed point is asymptotically stable. Note that

a positive sign of the third order term would have implied instability of the fixed

point. Thus, the stability is not decided by the linear, but by the first nonlinearterm.

The stability of the fixed point is confirmed by simulation. The convergence,

however, is much slower than for values of off the bifurcation point:

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0 9 0 0 0 1 00 0 0

-0.8

-0 .6

-0 .4

-0 .2

0

0 . 2

0 . 4

0 . 6

0 . 8

lam bda = 0 .75

Figure 13. 10000 iterations of the logistic function with = 0.75starting at x(0) = -0.8.

Now we consider equilibrium points of autonomous analog systems, i.e. systems

described by


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( )( )d

tdt

=x

F x (35)

with N N: F and vectors Nx such that

( )x 0=F (36)

Again, the linearized system allows to determine the stability or instability of the

equilibrium points in most cases.

Theorem 3:

Consider the autonomous system given by (35) and let x be an equilibrium point,i.e. a point where (36) holds.

a) The equilibrium point x is asymptotically stable if all eigenvalues i of the

Jacobian matrix ( )F

xx

satisfy ( )iRe 0 < .

b) The fixed point x is asymptotically stable if at least one eigenvalue i of the

Jacobian matrix ( )F

xx

satisfies ( )iRe 0 > .

A similar remark to Remark 1 can be made here.

Example 5:

Consider again the Van der Pol oscillator (1). It has the equilibrium point =x 0 .Its Jacobian matrix is

( )1

21 2 12

0 1x

1 2 x x x 1x

=

F

x(37)

and thus

0 0 1

0 1 =

F

x(38)

whose eigenvalues are

( )21 42

= (39)


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Thus

0 0> >

< 0. This isconfirmed by numerical simulation:

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

lambda = -0.2

Figure 14. Orbit of the solution of the Van der Poloscillator for = -0.2 and the initial conditions x1 = 0.5,x2 = 0.5.


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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

lambda = -0.05

Figure 15. Orbit of the solution of the Van der Pol

oscillator for = -0.05 and the initial conditions x1 = 0.5, x2= 0.5.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

lambda = +0.05

Figure 16: Orbit of the solution of the Van der Pol oscillator for

= +0.05 and the initial conditions x1 = 0.5, x2 = 0.5.

At the bifurcation point = 0, equation (1) is linear and therefore the solutions aresinusoidal. The amplitude and the phase of the oscillation depend on the initial

condition. The solutions are stable, but not asymptotically stable.


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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

lambda = 0

Figure 17. Orbit of the solution of the Van der Pol

oscillator for = 0 and the initial conditions x1 = 0.5, x2= 0.5.

Stability of a periodic solution of a discrete system

The discussion of the stability of periodic solutions is essentially the same as thediscussion of the stability of fixed points.

Theorem 4:

Consider the system (26). Let x be T-periodic solution of the system, i.e.

( ) ( ) for allt T t t + =x x (41)

where T is a natural number. Then x(0), x(1), , x(T-1) are fixed points of the

mapping N N: G defined by

( ) ( )( )( ) ( ) ( )T= = G x F F F F x F x (42)

Vice versa, to any fixed point ofG corresponds a T-periodic solution of the systemgiven by the iterations ofF. The stability properties of the T-periodic solution (41)

are the same as the stability properties of the corresponding fixed points of (42).


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Example 6:

Consider again the iterations of the logistic map2

f (x) 1 x= . A periodic solution

of period 2 corresponds to a fixed point of

( ) ( )( )

( )2

2 2 2 3 4

g x f f x

1 1 x 1 2 x x

=

= = + (43)

or, equivalently, to a zero of the polynomial

( ) 3 4 2 2P x x 2 x x 1= + + (44)

Note that the fixed points of f are also fixed points of g. Since the fixed points of f

satisfy

2Q(x) x 1 x 0= + = (45)

the ploynomial P(x) must contain the factor Q(x). Indeed

( ) ( )( )2 2P x Q x x x 1= + (46)

Thus, the cycle of order 2 is given by the solutions of

2 2x x 1 0 + = (47)

i.e.

( )1

x 1 4 32

= (48)

This shows that the system has a cycle of period 2 for > 0.75. Furthermore, thiscycle is asymptotically stable if

( ) ( ) ( ) 2 1 2dg df df

x x x 4 x x 4 1 1dx dx dx

+ = = = < (49)

which is the case for 0.75 < < 1.25.


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Variational equations

Before discussion the stability of periodic solutions, we introduce the concept ofvariational equations. The variational equations describe the time evolution of the

dependence of a solution on the initial conditions. For this purpose we use the notation

(t,x0) for the family of all solutions, whereas we continue to use x(t) for a singlesolution, i.e. a particular member of the family.

Let us first consider a solution x(t) of a discrete time system with initial condition x(0).Then the solutions starting close to x(0) at time 0 can be approximated by

( ) ( ) ( ) ( )( )

( ) ( )( ) ( )( )

0 0

00

, , , 0

, 0 0

t t t t

t t

= +

+

x x x x

x x x x

x

(50)

Thus, if a solution ( )tx starts close to x(0) and if we define its increment with respect tothe solution x(t) by

( ) ( ) ( )t t t = x x x (51)

then the first order approximation to this increment is

( ) ( )( ) ( )0

, 0 0t t

x x x x

(52)

Note that ( )00

,t

xx

is the Jacobian matrix

( ) ( )

1 1

01 0N

0 0

0N N

01 0N

x x

t, t,

x x

=

x x

x(53)

The question is then how to calculate this derivative. Let us consider again the particular

solution x(t) of the system and denote

( ) ( )( )0

, 0 M t t

=

x

x(54)

If we differentiate the equation


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( ) ( )( )0 01, ,t t+ = x F x (55)with respect to x0 at x(0), we obtain

( ) ( )( ) ( )1t t t

+ =

F M x M

x(56)

This is the variational equation of the discrete system along the solution x(t). Its solution

is, since M(0) is the identity matrix.

( ) ( )( ) ( )( ) ( )( ) ( )( )1 2 1 0t t t

=

F F F F M x x x x

x x x x (57)

The variational equations for continuous time systems are obtained in an analog way.

Let x(t) be a solution of an with initial condition x(0). If we differentiate the equation

( ) ( )( )0 0, ,t tt =x F x (58)

with respect to x0 at x(0), we obtain the variational equation for the analog system along

the solution x(t):

( )( ) ( )d

t tdt

=

M F

x Mx

(59)

This is a linear time-dependent differential equation for the matrix function M(t). Thetime dependence is caused by the argument x(t) of the Jacobian matrix of F. If we

combine it with the original system equation, we obtain the time-independent nonlinearsystem of N + N2 differential equations

( )( )

( )( ) ( )

dt

dt

dt t

dt

=

=

xF x

M Fx M

x

(60)

with the initial conditions x(0) and M(0) = I. For numerical calculations, it isadvantageous to solve this combined system, rather than the original and the variational

equations sequentially.

Finally, for some theoretical considerations it is useful to note that the solutions of thevariational equations are not only first order approximations for solution increments, butan exact expression can actually be given. The price to pay for this is that a whole family

of variational equations have to be solved. The following holds for both discrete and

analog systems.


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Consider two solutions ( )tx , ( )tx and their difference ( ) ( ) ( )t t t = x x x . In

addition, let ( ),t x be the solution with initial condition

( ) ( ) ( ) ( )0, 0 1 0 = + x x x . Then, trivially,

( ) ( )( ) ( )( )( )( ) ( )( )

( )( )

( )( ) ( )

( )( ) ( )

1

0

1

00

1

00

, 0 , 0

, 0,1 , 0,0

, 0,

, 0, 0,

, 0, 0

t t t

t t

dt d

d

dt d

d

t d

= =

=

=

=

x x x

x x

x

x xx

x x

x

(61)

Thus, ifM(t,) is the solution of the variational equation along the solution of the systemequation x(t,) then

( ) ( ) ( )1

0

, 0t t d = x M x (62)

Stability of a periodic solution of an analog system

Consider the system

( )( )d

tdt

=x

F x (63)

with N N: F and suppose it has a T-periodic solution

( ) ( )t T t for all t+ = (64)Now consider a hypersurface S in

N that intersects the periodic solution transversally at

the point (0). If the hypersurface is described by h(x) = 0 with Nh : , then thecondition of transversal intersection means


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( )( )

( )( ) ( )T

h 0 0

h d0 0 0

dt

=

x

(65)

In a neighborhood U of (0) we define the first return map R, or Poincar map asfollows. Let 0 Ux and let x(t) be the solution with x(0) = x0. This solution will intersectS again, after approximately time T, in a point R(x0).

x0 (0) R(x0)

S

Figure 18. Return map (Poincar map).

Clearly, R((0)) = (0), i.e. (0) is a fixed point of the return map. The following theoremis given without proof.

Theorem 5:

Consider an analog system with a periodic solution (t). Consider the return map Rofa Poincar section S through (0). Then the periodic solution (t) is stable if and onlyif the fixed point (0) of the return map is stable.

Corollary 1:

Consider an analog system with a periodic solution (t). Consider the return map Rofa Poincar section S through (0). Suppose that all eigenvalues i of the Jacobianmatrix of R at (0) satisfy |i| < 1. Then the periodic solution (t) is stable.

Proof: This follows from theorems 3 and 5.

Remark 2:

a) Why is it not possible to conclude in Corollary 1 that the periodic solution (t) isasymptotically stable? The fact is that the fixed point (0) of the return map isindeed asymptotically stable, but for variations of the initial condition in thedirection transversal to the Poincar section, the corresponding solutions do not


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converge to (t) as t . In particular, if we choose an initial condition( ) ( )0 =x , then the corresponding solution is ( ) ( )t t = +x and even for

arbitrarily small the solution x(t) never converges to (t).b) The problem with the criterion for the stability of a periodic solution given in

Corollary 1 is that the return map is defined only implicitly and therefore it is noteasy to calculate the eigenvalues of its Jacobian matrix directly. The followingtheorem indicates an indirect method.

Theorem 6

Consider an analog system with a periodic solution (t) of period T. Consider thereturn map R of a Poincar section S through (0) and the solution M(t) of thevariational equations (60) along (t). If 1, ... , N-1 are the eigenvalues of theJacobian matrix of Rat (0) then 1, ... , N-1, 1 are the eigenvalues of M(T), andvice versa.

Remark 3:

a) The eigenvalues ofM(T) are called Floquet multipliers.

b) The eigenvector of the additional eigenvalue 1 ofM(T) is ( )0d

dt

. This follows

from the reasoning given in Remark 2.

Corollary 2:Consider an analog system with a periodic solution (t) of period T. Consider thereturn map R of a Poincar section S through (0) and the solution M(t) of thevariational equations (60) along (t). If the all Floquet multipliers i (eigenvalues ofM(T)) except one (whose value is 1) satisfy |i| < 1, the periodic solution (t) isstable. In addition, any solution starting sufficiently close to (0) converges to a time-shifted version of the periodic solution (t).

Proof: The stability follows from theorems 6 and Corollary 1. The last property can be

proved from Remark 3 b).

Example 7:

Numerical solutions of the variational equations (60) of the Vanderpol oscillator with

= 0.3 along the periodic solution yields the Floquet multipliers 1 and 0.15. Thus,solutions starting sufficiently close to the periodic orbit converge to a time-shifted

version of the periodic orbit. Actually all solutions except for the unstable equilibriumpoint have this property. This is illustrated in .


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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 19. State space trajectories of three solutions of

the Vanderpol oscillator for = 0.1. The closedtrajectory (blue) and two trajectories that converge to theperiodic orbit.

0 2 4 6 8 10 12 14 16 18 20

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14 16 18 20

-3

-2

-1

0

1

2

3


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Figure 20. The same solutions as in Figure 19. The two state variables arerepresented as functions of time. Clearly, the red and the green solution

converge to a time-shifted version of the periodic (blue) solution.

Stability of Nonlinear Systems MH

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