Dynamical Systems and Differential Geometry via MAPLE

Dynamical Systems and Differential Geometry via MAPLE

By

Constantin Udriste and Ionel Tevy

Dynamical Systems and Differential Geometry via MAPLE By Constantin Udriste and Ionel Tevy This book first published 2021 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2021 by Constantin Udriste and Ionel Tevy All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-7223-4 ISBN (13): 978-1-5275-7223-2

Contents

Preface xi

1 DYNAMICAL SYSTEMS 11.1 Exponential matrix . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Connection with systems oflinear differential equations . . . . . . . . . . . 5

1.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Dynamical systems . . . . . . . . . . . . . . . . . . . . 81.4 The almost linear expression of

a vector field . . . . . . . . . . . . . . . . . . . . . . . 111.5 Self-evaluation problems . . . . . . . . . . . . . . . . . 11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 INVARIANT SETS, CONSERVATION ANDDISSIPATION 212.1 Liouville Theorem . . . . . . . . . . . . . . . . . . . . 212.2 Invariant sets . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Non-negative orthant invariance . . . . . . . . 242.3 Conservation laws . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Invariance criterion . . . . . . . . . . . . . . . . 262.4 Dissipated quantities . . . . . . . . . . . . . . . . . . . 272.5 The example of

linear mechanical system . . . . . . . . . . . . . . . . . 282.6 Discrete time dynamical systems . . . . . . . . . . . . 282.7 Poincare section and map . . . . . . . . . . . . . . . . 29

2.7.1 Partial derivatives of the Poincare map . . . . 302.8 Phytoplankton dynamical system . . . . . . . . . . . . 31

2.8.1 Bounds for phytoplankton substrate . . . . . . 312.8.2 Bounds for phytoplankton biomass . . . . . . . 32

v

vi Contents

2.8.3 Bounds for intracellular nutrient per biomass . 332.9 Research problems . . . . . . . . . . . . . . . . . . . . 34

2.9.1 Invariant sets, conservation laws, dissipation . . 342.10 Self-evaluation problems . . . . . . . . . . . . . . . . . 36

2.10.1 Energy conservation and dissipation . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 EQUILIBRIUM POINTS STABILITY ANDHOPF BIFURCATION 413.1 Local stability of equilibrium points . . . . . . . . . . 423.2 Global stability of equilibrium points . . . . . . . . . . 433.3 Hopf bifurcation theory . . . . . . . . . . . . . . . . . 443.4 Lotka - Voltera dynamical system . . . . . . . . . . . . 483.5 Lorenz dynamical system . . . . . . . . . . . . . . . . 493.6 Kepler problem as dynamical system . . . . . . . . . . 523.7 Self-evaluation problems . . . . . . . . . . . . . . . . . 55

3.7.1 Stability of equilibrium points forlinear differential systems . . . . . . . . . . . . 55

3.7.2 Stability of equilibrium points fornonlinear differential systems . . . . . . . . . . 56

3.7.3 Lyapunov functions . . . . . . . . . . . . . . . 603.8 Research problems . . . . . . . . . . . . . . . . . . . . 69Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 GYROSCOPIC FORCES ANDCOLLISION AVOIDANCE 734.1 Gyroscopic forces . . . . . . . . . . . . . . . . . . . . . 744.2 Avoidance of obstacles by

gyroscopic forces . . . . . . . . . . . . . . . . . . . . . 754.3 Two types of collisions . . . . . . . . . . . . . . . . . . 784.4 Asymptotic convergence of

the vehicle to a target . . . . . . . . . . . . . . . . . . 784.5 Optimal control problem . . . . . . . . . . . . . . . . . 794.6 Self-evaluation problems . . . . . . . . . . . . . . . . . 82Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 NAMBU DYNAMICAL SYSTEMS 875.1 Nambu dynamical systems . . . . . . . . . . . . . . . . 875.2 Gradient dynamical system . . . . . . . . . . . . . . . 895.3 Nambu-gradient systems in space . . . . . . . . . . . . 905.4 Stability of equilibrium points . . . . . . . . . . . . . . 92

Dynamical Systems and Differential Geometry via MAPLE vii

5.5 The SU(n) - isotropicharmonic oscillator . . . . . . . . . . . . . . . . . . . . 93

5.6 The SO(4)-Kepler Problem . . . . . . . . . . . . . . . 945.7 Self-evaluation problems . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 GEOMETRIC DYNAMICS AND WIND THEORY 1016.1 Geometric Dynamics generated by

a flow and a Riemannian metric . . . . . . . . . . . . . 1016.2 Wind theory and geometric dynamics . . . . . . . . . 103

6.2.1 Classification of winds . . . . . . . . . . . . . . 1046.2.2 Solitonic wind . . . . . . . . . . . . . . . . . . 1056.2.3 Geometric dynamics as wind . . . . . . . . . . 105

6.3 Biochemical geometric dynamics andwind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 Electric geometric dynamics andwind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Double pendulumgeometric dynamics and wind . . . . . . . . . . . . . . 1136.5.1 Variational integrator . . . . . . . . . . . . . . 114

6.6 ABC geometric dynamics and wind . . . . . . . . . . . 1166.7 Economic geometric dynamics and

wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.8 Self-evaluation problems . . . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 MOTIONS OF CURVES 1237.1 Generating surfaces . . . . . . . . . . . . . . . . . . . . 124

7.1.1 Ruled surfaces . . . . . . . . . . . . . . . . . . 1247.1.2 Surfaces of revolution . . . . . . . . . . . . . . 124

7.2 Motion of a curve in space . . . . . . . . . . . . . . . . 1257.2.1 The integration of

the natural equations of a curve . . . . . . . . 1267.2.2 A motion of an inextensible curve of

constant torsion . . . . . . . . . . . . . . . . . 1287.2.3 A motion of an inextensible curve of

constant curvature . . . . . . . . . . . . . . . . 1297.3 Motion of curve along

the tangent vector . . . . . . . . . . . . . . . . . . . . 1307.4 Motion of curve along normal vector . . . . . . . . . . 130

viii Contents

7.4.1 Existence of the PDE solution thatdescribes the movement along normal . . . . . 132

7.5 Korteweg-de Vries (KdV) equation . . . . . . . . . . . 1347.6 Motion of curve along

the binormal vector . . . . . . . . . . . . . . . . . . . . 1357.6.1 Nonlinear Schrodinger equation (NSE) . . . . . 1367.6.2 Solitonic Hasimoto surface . . . . . . . . . . . . 137

7.7 Evolution of implicit plane curves . . . . . . . . . . . . 1387.7.1 Deformations by infinitesimal transformation . 140

7.8 Self-evaluation problems . . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8 MOTIONS OF SURFACES 1458.1 Evolution of a surface in space . . . . . . . . . . . . . 145

8.1.1 Geometry of a surface . . . . . . . . . . . . . . 1468.1.2 Variation of a surface . . . . . . . . . . . . . . 149

8.2 Motion of a surface alongnormal vector . . . . . . . . . . . . . . . . . . . . . . . 151

8.3 Minimal surface PDE . . . . . . . . . . . . . . . . . . 1558.4 Evolution of implicit surfaces . . . . . . . . . . . . . . 156

8.4.1 Normal velocity . . . . . . . . . . . . . . . . . . 1588.4.2 Tangential velocity . . . . . . . . . . . . . . . . 1598.4.3 Deformation of surfaces by

an infinitesimal transformation . . . . . . . . . 1608.5 Self-evaluation problems . . . . . . . . . . . . . . . . . 161Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 165

9 EVOLUTIVE RIEMANNIAN METRICS 1679.1 Geometric vector fields . . . . . . . . . . . . . . . . . . 1689.2 Flows of Riemannian metrics . . . . . . . . . . . . . . 1699.3 Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . 1729.4 Riemannian metric dynamics . . . . . . . . . . . . . . 1759.5 Ricci wave . . . . . . . . . . . . . . . . . . . . . . . . . 1779.6 Flows of metrics determined by

geometric vector fields . . . . . . . . . . . . . . . . . . 1789.7 Dynamics of metrics determined by

geometric vector fields . . . . . . . . . . . . . . . . . . 1799.8 Metric flow determined by a Hessian . . . . . . . . . . 1809.9 Metric dynamics determined by

a Hessian . . . . . . . . . . . . . . . . . . . . . . . . . 181

Dynamical Systems and Differential Geometry via MAPLE ix


10 EVOLUTION OF TZITZEICA HYPERSURFACES 18910.1 Introduction and motivation . . . . . . . . . . . . . . . 19010.2 Convexity of Tzitzeica hypersurfaces . . . . . . . . . . 19010.3 Normal evolution of

Tzitzeica hypersurfaces . . . . . . . . . . . . . . . . . . 19110.3.1 Steepest increase lines . . . . . . . . . . . . . . 19210.3.2 Evolution along the normal vector field . . . . 192

10.4 Infinitesimal normal transformationof Tzitzeica hypersurface . . . . . . . . . . . . . . . . . 194

10.5 Evolution alonga centro-affine vector field . . . . . . . . . . . . . . . . 194

10.6 Evolution along an affine vector field . . . . . . . . . . 19510.7 Tzitzeica hypersurfaces as invariants

with respect to excess demand flow . . . . . . . . . . . 19710.8 General evolution of

Tzitzeica surfaces . . . . . . . . . . . . . . . . . . . . . 19810.9 Self-evaluation problems . . . . . . . . . . . . . . . . . 199Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 200

11 EVOLUTIVE OPTIMIZATION PROBLEMS 20311.1 Evolution of

an optimization problem . . . . . . . . . . . . . . . . . 20411.1.1 Evolution of objective function . . . . . . . . . 20411.1.2 Evolution of a constraint manifold . . . . . . . 20511.1.3 Evolution of minimum value . . . . . . . . . . . 207

11.2 Evolution ofthe catastrophe manifoldassociated to a Lagrange potential . . . . . . . . . . . 20811.2.1 Evolution preserving

the catastrophe manifold . . . . . . . . . . . . 20811.2.2 All ingredients are evolutive . . . . . . . . . . . 209

11.3 Evolution ofan optimal control problem . . . . . . . . . . . . . . . 21111.3.1 Evolution of a curve described by

a second order ODE . . . . . . . . . . . . . . . 21211.3.2 Evolution of a surface described by

a second order PDE . . . . . . . . . . . . . . . 212

x Contents

11.3.3 Evolution ofa curve satisfying a first order PDE . . . . . . 215


12 TASKS WITH MAPLE 22112.1 Report on ”dynamical systems” . . . . . . . . . . . . . 22112.2 Report on ”Invariant sets,

conservation and dissipation” . . . . . . . . . . . . . . 22212.3 Report on ”Local stability of

equilibrium points andHopf bifurcation” . . . . . . . . . . . . . . . . . . . . . 224

12.4 Report on ”Gyroscopic forces andcollision avoidance” . . . . . . . . . . . . . . . . . . . . 225

12.5 Report on”Nambu dynamical systems” . . . . . . . . . . . . . . 226

12.6 Report on ”Geometric dynamics andwind theory” . . . . . . . . . . . . . . . . . . . . . . . 226

12.7 Report on ”Motions of curves” . . . . . . . . . . . . . 22812.8 Report on ”Motions of surfaces” . . . . . . . . . . . . 22912.9 Report on

”Riemannian metrics of evolution” . . . . . . . . . . . 230Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 231INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Preface

Our book ”Dynamical Systems and Differential Geometry via MAPLE” is designed starting from the course of the same name taught (by the first author) to engineering students, year III, from the Faculty of Ap-plied Sciences, the Mathematical-Computer Section, in the University Politehnica of Bucharest, since 2005. It condenses the offer and the importance of the mutual impact and interconditioning of the three mathematical disciplines. Some subjects are taught at other facul-ties or universities in the world, especially in the master or doctorate courses, being un-missed from the papers that can be published in journals now categorized as good or very good.

The subject of the course includes twelve Chapters: Dynamical systems; Invariant sets, conservation and dissipation; The stability of equilibrium positions and Hopf’s bifurcation; Gyroscopic forces and collision avoidance; Nambu dynamical systems; Geometric dynamics and wind theory; Motions of a curve; Motions of a surface; Evolutive Riemannian metrics; Evolution of Tzitzeica hypersurfaces; Dynamic of evolutive optimization problems; Tasks with Maple. The problems (self-evaluation or research) in each Chapter, while not at all trivial, tremendously enhance one’s understanding of the material.

All Chapters were structured by importance, accessibility and im-pact of theoretical notions capable of shaping a future mathematician-computer scientist possessing knowledge of evolutionary dynamical systems. The intermediate variants of the manuscript have extended over the years, leading to the selection of the most important and manageable notions and reaching maturity through this version of the book that we have decided to publish.

The scientific authority we have has allowed us to impose the point of view on the thematic and on the types of rations that deserve to beoffered to readers of texts of dynamical systems, differential geometry

xi

xii Preface

and Maple. Everything was structured for the benefit of optimizingevolutionary geometric aspects that describe significant physical orengineering phenomena. It is known that to avoid blocking the stu-dents’ minds, a simplified language is preferred, specific to the appliedmathematics, as we have done as a rule, sending the pure mathematicsreaders to bibliography. Our long experience as a mathematics pro-fessors at a technical university has helped us to fluently expose ideasstripped of excessive formalization, with great impact on students’understanding of natural phenomena encountered in engineering andeconomics as well as their dressing in mathematical clothes for the in-tellectual holiday. In this sense, we preferred evolutionary models thatintelligently present the real world and we have avoided the hashishof abstract notions specific to theorists.

The novelty problems were finalized following repeated discussionswith the mathematics and physics professors from The Faculty of Ap-plied Sciences, especially with those interested in dynamical systemsand differential geometry, with Maple simulations, to whom we thankand thoughts of appreciation. We are open to any kind of remarksor criticisms that bring didactic benefits to this course of dynamicevolutionary systems.

The area of Dynamical Systems and Differential Geometry viaMAPLE is a mixed field which has become exceedingly technical inrecent years. Everything was structured for the benefit of optimizingevolutionary geometric aspects that describe significant physical orengineering phenomena. Also, Maple has a large number of subrou-tine libraries which have been expressly written to provide solutionsto standard homework problems for students.

With this book, the authors provide a self-contained and acces-sible introduction for graduate or advanced undergraduate studentsin mathematics, engineering, physics, and economic sciences. Bothclassical and modern methods used in the field are described in de-tail, concentrating on the model cases that simplify the presentationwithout compromising the deep technical aspects of the theory, thusallowing students to learn the material in a short period of time. Thisbook is suitable for self-study for students and professors with a back-ground in differential geometry both, and for teaching a semester-longintroductory graduate course in Dynamical Systems and DifferentialGeometry via MAPLE. Copious exercises are included in each Chap-ter, and applications of the theory are also presented to connect dy-namical systems and differential geometry with the more general areas

Dynamical Systems and Differential Geometry via MAPLE xiii

of dynamical systems, differential geometry and mathematical physics,in large.

The mottos used at Chapters are stanzas from George Cosbuc’spoems (in Romanian), selected by the authors and translated in En-glish by Associate Prof. Dr. Brandusa Prepelita-Raileanu, UniversityPolitehnica of Bucharest. George Cosbuc (20 September 1866 - 9 May1918) was a Romanian poet, translator, teacher, and journalist, bestremembered for his verses describing, praising and eulogizing rurallife, its many travails but also its occasions for joy.

The authors thank Miss Associate Prof. Dr. Oana-Maria Pastae,”Constantin Brancusi” University of Tg-Jiu, for the English improve-ment of the manuscript.

February 2021

Prof. Emeritus Dr. Constantin Udriste1, Prof. Dr. Ionel TevyDepartment of Mathematics and Informatics, Faculty of Applied

Sciences, University Politehnica of Bucharest, Splaiul Independentei313, RO-060042, Bucharest, 6, Romania; [email protected] ;[email protected]

1 Academy of Romanian Scientists, Ilfov 3, RO-050044 Bucharest5, Romania.

Chapter 1

DYNAMICALSYSTEMS

Motto:Some people hold dear

What’s vain in the other men’s eye

But He who can scan both the earth and the sky

And set up a bridge between life and death.

George Cosbuc - The Poet

In this Chapter we present some concepts regarding the exponential matrix,

flows and related dynamical systems that have direct applications in science and

engineering. Explicit links: (1) the connection between the exponential matrix and

the solutions of linear systems of differential equations with constant coefficients;

(2) the relationship between flows and vector fields; (3) the correlation between

dynamical systems and vector fields. Suggestions for further reading: [1]-[9].

1.1 Exponential matrix

LetMn×n(R) be the set of square matrices of order n, endowed witha norm ‖ · ‖. Let I be the unit matrix of order n. For each matrixA ∈Mn×n(R), the power series

∞∑k=0

Ak

k!= I +

A

1!+A2

2!+ ...

1

2 Dynamical systems

is absolutely convergent and hence convergent. For a matrix of func-tions, A, the series is uniform convergent on each subset A | ||A|| ≤a, a ∈ R+. The sum of the previous series is called the exponentialmatrix and is denoted by eA or expA.

If O means zero matrix, then eO = I. Also, the absolute con-vergence implies the fact that if A,B ∈ Mn×n(R) are commutablematrices, i.e., AB = BA, then eAeB = eA+B . It follows that thematrix eA is invertible and its inverse is e−A.

Here we are interested by the matrix function

ϕ : R→Mn×n(R), ϕ(t) = etA =

∞∑k=0

tkAk

k!,

also called the exponential matrix. The Cayley-Hamilton Theoremensures that the exponential matrix etA is a polynomial of degreen − 1, where n is the order of the matrix A. The coefficients of thepolynomial in A are series of functions. Explicitly

etA =

∞∑k=0

tkAk

k!=

n−1∑k=0

ck(t)Ak.

Theorem 1.1.1. The matrix function ϕ(t) = etA is the solution ofthe Cauchy problem

dϕ

dt(t) = Aϕ(t), ϕ(0) = I.

Proof. The series

ϕ(t) = etA =

∞∑k=0

tkAk

k!, and

∞∑k=0

d

dt

tkAk

k!= A

∞∑k=0

tkAk

k!

converge absolutely and uniformly in any domain of R ×Mn×n(R)characterized by |t| ≤ b and ||A|| ≤ a. That is why there exists dϕ

dt (t)(derivative of the sum of initial series) and it is equal to the sum ofthe series formed deriving term by term.

The matrices A and etA are commutable. This remark allows usto give a nice proof for

Theorem 1.1.2. The matrix etA is invertible and its inverse is e−tA.

Dynamical Systems and Differential Geometry via MAPLE 3

Proof. Let us consider the matrix E(t) = etAe−tA, t ∈ R. By deriva-tion, we find

dE

dt(t) = AetAe−tA + etA(−A)e−tA = O, ∀t ∈ R.

It follows E(t) = E(0) = I.

The direct calculation of the matrix etA is cumbersome or evenimpossible. That is why we explain some helpful techniques.

Case of a diagonal matrix For a diagonal matrix

D =

λ1 0 ... 00 λ2 ... 0... ... ... ...0 0 ... λn

,we have

Dk =

λk1 0 ... 00 λk2 ... 0... ... ... ...0 0 ... λkn

and hence

etD =

eλ1t 0 ... 0

0 eλ2t ... 0... ... ... ...0 0 ... eλnt

.Case of a Jordan cell Let Jp be a Jordan cell of order np, i.e.,

Jp =

λp 1 0 ... 00 λp 1 ... 0... ... ... ... ...0 0 0 ... 10 0 0 ... λp

= λpIp + Ep,

where Ip is the unit matrix of order np, and the matrix

Ep =

0 1 0 ... 00 0 1 ... 0... ... ... ... ...0 0 0 ... 10 0 0 ... 0

4 Dynamical systems

is nilpotent of order np. Taking into account that the matrices Ip andEp are commutable and computing the powers of Ep, it follows

etJp = eλptIpetEp = eλpt

[Ip +

tEp1!

+t2E2

p

2!+ ...+

tnp−1Enp−1p

(np − 1)!

]

= eλpt

1 t

1!t2

2! ... tnp−1

(np−1)!

0 1 t1! ... tnp−2

(np−2)!

... ... ... ... ...0 0 0 ... 1

.Generally, a Jordan matrix has the form

J =

J1 0 ... 00 J2 ... 0... ... ... ...0 0 ... Jk

,where Jp, p = 1, ..., k are Jordan cells of orders np and n1+...+nk = n.It follows

etJ =

etJ1 0 ... 0

0 etJ2 ... 0... ... ... ...0 0 ... etJk

.Case of diagonalizable and Jordanizable matrices First, we

formulate a general result regarding the similar matrices. This givesrise to a consequence that is helpful to us in the calculations.

Theorem 1.1.3. If S is a nonsingular matrix and if S−1AS = B,then etA = SetBS−1.

Proof. The relation A = SBS−1 implies Ak = SBkS−1. That is why

etA =

∞∑k=0

tkAk

k!=∞∑k=0

tkSBkS−1

k!= S

( ∞∑k=0

tkBk

k!

)S−1 = SetBS−1.

Corollary 1.1.4. (1) If A is a diagonalizable matrix, D is the diag-onal matrix associated to A, and S is the diagonalizing matrix (thematrix of eigenvectors placed on columns), then etA = SetDS−1.


(2) If the matrix A has multiple eigenvalues and is reduced to theJordan canonical form J , and S is the jordanizing matrix (the matrixof eigenvectors and principal vectors placed on columns), then etA =SetJS−1.

1.1.1 Connection with systems oflinear differential equations

First, we refer to systems of linear differential equations with constantcoefficients. Let A ∈Mn×n(R) be a real square matrix of order n.

Theorem 1.1.5. The solution of the Cauchy problem x(t) = Ax(t),x(0) = x0 is x(t) = etAx0, t ∈ R.

Proof. The existence and uniqueness theorem together with the ex-tension theorem show that the solution exists, is unique and definedon the whole set R. The rest is just checks by calculations.

Second, we refer to systems of linear differential equations withnonconstant coefficients.

Theorem 1.1.6. The matriceal differential equation X(t) = A(t)X(t),X(0) = X0 has the solution

X(t) = exp

(∫ t

0

A(τ)dτ

)X0

iff the matrices A(t) and

∫ t

0

A(τ)dτ commute.

Theorem 1.1.7. The solution of Cauchy problem X(t) = A(t)X(t),X(0) = X0 can be written in the form

X(t) = exp(Ω(t))X0 ,

where Ω(t) and A(t) are similar and Ω(0) = 0. If the matrices A(t)and Ω(t), or Ω(t) and Ω(t), commute, then Ω(t) = A(t).

Proof. Looking for a solution as X(t) = exp(Ω(t))X0 , we compute,

X(t) =

(d

dtexp(Ω(t))

)X0 =

∫ 1

0

exp(αΩ(t)) Ω(t) exp((1− α)Ω(t)) dαX0.

6 Dynamical systems

But X0 = exp(−Ω(t))X(t). Then,

X(t) =

∫ 1

0

exp(αΩ(t)) Ω(t) exp((−α)Ω(t)) dαX(t).

Comparing with the given system and applying the mean-valueTheorem, we find a function α : [0, 1]→ R such that

A(t) = exp(α(t)Ω(t)) Ω(t) exp(−α(t)Ω(t)) .

This means that the matrices Ω(t) and A(t) are similar.If the matrices A(t) and Ω(t) commute, then A(t) and exp(Ω(t))

commute and there results Ω(t) = A(t), as in the previous theorem.Similarly, if Ω(t) and Ω(t) commute.

Example 1.1.8. Find a fundamental matrix of the system of differ-ential equations x(t) = x(t) + ty(t), y(t) = tx(t) + y(t) making surethat the coefficient matrix

A(t) =

(1 tt 1

), t > 0.

commutes with its integral.The integral of the matrix A(t) is found by elementwise integration.

For simplicity, we take the lower boundary of integration to be zero.Then, ∫ t

0

A(τ)dτ =

(t t2

2t2

2 t

)By computation, we verify A(t) ·

∫ t0A(τ)dτ =

∫ t0A(τ)dτ · A(t). So,

the commutative property of the matrix product is true. Therefore,the fundamental matrix is given by

Φ(t) = exp

(∫ t

0

A(τ)dτ

)= exp

(t t2

2t2

2 t

).

We compute the matrix exponential by converting the matrix to thediagonal form. In this case, the eigenvalues and the eigenvectors de-pend on the variable t. Since the matrix of reduction to the diagonalform is given by

S =

(1 −11 1

),


we find

Φ(t) = exp

(t t2

2t2

2 t

)= et

(cosh t2

2 sinh t2

2

sinh t2

2 cosh t2

2

).

Example 1.1.9. Consider the homogeneous linear differential systemx(t) = A(t)x(t), where

A(t) =

(4t − 4

t2

2 − 1t

), t > 0.

Here the vectors x1(t) = T [2t2, t3], x2(t) = T [1, t] are particular solu-tions. Since ∣∣∣∣ 2t2 1

t3 t

∣∣∣∣ = t3 > 0,

the solutions x1(t) and x2(t) are linearly independent. The generalsolution of the system is x(t) = c1x

1(t) + c2x2(t).

Let us write the solution using the exp matrix. Here, the idea ofsolving is based on knowing the solution

x(t) =

(2t2 1t3 t

)(c1c2

).

Identifying, we obtain

exp(Ω(t)) =

(2t2 1t3 t

), t > 0.

1.2 Flows

Let D be an open subset in Rn. A family of functions T t : D →Rn, t ∈ (−ε, ε) is called local flow on D or local group with one param-eter of diffeomorphisms on D if the following conditions are satisfied:

(1) The function T : (−ε, ε) ×D → Rn, (t, x) → T t(x) is of classC∞.

(2) For each t ∈ (−ε, ε), the function T t : D → T t(D) is a diffeo-morphism.

(3) For each pair t, s ∈ (−ε, ε), with t+ s ∈ (−ε, ε) and T s(x) ∈ D,we have T t+s(x) = T t(T s(x)), for each x ∈ D. It follows T 0(x) = x.

8 Dynamical systems

The adjective ”local” refers to t ∈ (−ε, ε). If (−ε, ε) = R, i.e.,ε =∞, then this adjective is replaced by ”global”.

A flow T t defines a vector field X(x) = ddtT

t(x)|t=0 on D. Con-versely, any vector field of class C∞ defines a flow by its field lines.This thing will be proved in the following Section.

Let us consider the affine group of transformations

xi′

= aij(t)xj + ai(t), det(aij(t)) 6= 0, t ∈ (−ε, ε), aij(0) = δij , a

i(0) = 0

on Rn. The associated affine vector field is defined by

dxi′

dt(0) =

daijdt

(0)xj +dai

dt(0).

Denoting A =(daijdt (0)

), a =

(dai

dt (0))

, we find X(x) = Ax+ a. The

flow generated by the vector field X(x) = Ax + a consists in certainaffine transformations.

1.3 Dynamical systems

The evolution over time of some physical systems can be modelled asdynamical systems.

Definition 1.3.1. A semigrup G which acts on the space Rn, in thesense that there exists a function Φ : G×Rn → Rn such that ΦsΦt =Φst is called dynamical system. The function Φ is called flow.

If G is a group, then we say that the dynamical system is invertible.The discrete dynamical systems correspond to G = N \ 0 = N0

or G = Z and they have the prototype a recursive sequence describedby Φ : G→ G, Φn = Φ Φn−1, n ∈ N0 or x(t+ 1) = X(x(t)), t ∈ N0.

The continuous dynamical systems correspond to G = R+ orG = R and they have the prototype described by a first order dif-ferential system of the form x(t) = X(x(t)), where X is a vector fieldon Rn of class C∞. The maximal curve that satisfies the Cauchyproblem x(t) = X(x(t)), x(0) = x0 is defined on the interval Ix0

=(T−(x0), T+(x0)) and is called field line. Abandoning the index 0 andentering the set W = ∪x∈RnIx × x ⊂ R × Rn, we define the flowΦ : W → Rn, (t, x) → Φ(t, x), where t → Φ(t, x) is an integral curvestarting from the point x. In other words, a solution α = α(t), α(0) =


x, t ∈ (−ε, ε) of the differential system x(t) = X(x(t)) generates alocal diffeomorphism

T t : U ⊂ Rn → Rn, T t(x) = Φ(t, x), t ∈ I,

solution of the operatorial differential equation

dT t

dt= X T t, T 0 = id,

called local flow on Rn generated be the vector field X (local groupwith one parameter of diffeomorphisms). The linear approximationx′ = x+tX(x) of T t(x) is called infinitesimal transformation generatedby the vector field X.

The image set in Rn of the curve (field line, integral curve) αx(t) =Φ(t, x) is called orbit passing through the point x ∈ Rn. A pointx ∈ Rn which verifies Φ(t, x) = x, for any t ∈ I, is called equilibriumpoint for the system. In this case the orbit passing through the pointx is reduced to this point, x = αx.

If any solution α = α(t), α(0) = x is defined for each x ∈ Rn andeach t ∈ R, then the vector field X is called complete, and the flow iscalled global (group with a parameter of diffeomorphisms).

A graphical representation of the equilibrium points and of somegeneric orbits of the dynamical system is called the phase portrait.

To point things out, let’s take the example X(x) = x3, x(t) = x3(t)defined on R. Here we find the flow

Φ(t, x) =x√

1− 2x2t, W = (t, x) | 2tx2 < 1,

T−(x) = −∞, T+(x) =1

2x2.

Example 1.3.1. We consider the linear vector field

X(x, y, z) = (x− y + z, 2y − z, z).

The field lines are solutions of the linear differential system (withconstant coefficients)

dx

dt= x− y + z,

dy

dt= 2y − z, dz

dt= z.

The third equation has the general solution z = cet, t ∈ R. Replacingin the second equation, we find a linear differential equation dy

dt = 2y−

10 Dynamical systems

cet, with the general solution y = be2t + cet, t ∈ R. Introducing y andz in the first equation, we get dx

dt = x− be2t, with the general solutionx = aet−be2t, t ∈ R. Obviously (0, 0, 0) is the only equilibrium point.

A modified flow generated by the previous vector field is

T t : R3 → R3, T t(x, y, z) = (xet − ye2t, ye2t + zet, zet), t ∈ R.

In order to study the effect of the flow on the volume, we use theevolution velocity, i.e., the divergence of the vector field X,

div(X) =∂X1

∂x+∂X2

∂y+∂X3

∂z= 1 + 2 + 1 = 4.

Since div(X) > 0, the flow associated to this vector field increases thevolume.

Example 1.3.2. Study the completeness of the non-linear vector field

X(x, y, z) = (xz, yz,−(x2 + y2)).

The orbits of the vector field X, solutions of the symmetric differ-ential system

dx

xz=dy

yz=

dz

−(x2 + y2),

are x = c1y, x2 +y2 +z2 = c2. These orbits can thus be parameterized

x =√c2 cosu cos v, y =

√c2 cosu sin v, z =

√c2 sinu, u ∈ R,

cos v = c1 sin v. We parametrize again u = u(t) such that dzdt = −(x2 +

y2). Since dzdt = dz

dududt , it follows

√c2 cosududt = −c2 cos2 u and hence

ducosu = −√c2 dt. So,

ln∣∣∣tg(π

4− u

2

)∣∣∣ = −√c2 t

or

tg(π

4− u

2

)= e−

√c2 t, t ∈ R.

Consequently, the vector field X is complete.Shortcut The orbits of X belong to the spheres x2 + y2 + z2 =

c2, and these surfaces are compact. Therefore the vector field X iscomplete.


1.4 The almost linear expression ofa vector field

We consider a real function f of class C1 defined on a convex neigh-bourhood V of a point x0 ∈ Rn. Then it appears the almost linearexpression

f(x) = f(x0) +

∫ 1

0

d

dtf(x0 + t(x− x0))dt

= f(x0) +

∫ 1

0

fxi(x0 + t(x− x0))(xi− xi0)dt = f(x0) + gi(x)(xi− xi0),

where gi(x0) = fxi(x0). This result is known as Hadamard’s Lemma.Extending this remark to a vector field X of class C1, with the

property X(x0) = 0, there is a quadratic matrix A(x) such thatX(x) = A(x)x, x ∈ V . We combine with the remark that a Cauchyproblem x(t) = A(x(t))x(t), x(t1) = x1 is equivalent to an integral

equation x(t) = x(t1) +

∫ t

t1

A(x(s))x(s)ds.

1.5 Self-evaluation problems

Problem 1. Show that T (t, x) = et(1 +x)− 1 is a flow and write thedynamical system that corresponds to this flow.

Solution. We show that T (0, x) = x and T (t + s, x) = T (t, T (s, x))(or, if we denote T (t, x) with T t(x), then T 0(x) = id(x) = x andT t+s(x) = (T t T s)(x)). The relationships are obviously true, so theapplication T is a flow.

Since

d

dtT t(x)|t=0 =

d

dtαx(t)|t=0 = αx(0) = X(αx(0)) = X(x),

the vector field corresponding to this flow is

X(x) =d

dtT t(x)|t=0 =

d

dt(et(1 + x)− 1)|t=0 = 1 + x,

and the associated differential equation is x(t) = x(t) + 1.

Problem 2. Describe the flow generated by the vector field X(x) = x3

on R.


Solution. X being a vector field on R, the field lines are describedby those functions α : I → R, α(t) = x(t) which verify the ODEx(t) = X(x(t)), i.e., x(t) = x3(t). By integration, we find the solution

x2 =1

c− 2t, c ∈ R. The field lines are the functions

α(t) = x(t) =1

±√c− 2t

, c ∈ R.

Let x ∈ R+. We are looking for that field line that ”leaves” from x,

i.e., α(0) = x, hence c =1

x2. Therefore, the flow associated with the

vector field in the statement is the application T t(x) =x√

1− 2tx2.

Problem 3. Find out the flow generated by the linear vector field

X(x, y, z) = (x+ y − z, x− y + z,−x+ y + z)

and investigate its effect on volume.

Solution. The field lines are the solutions of the differential system

α(t) = X((α(t)),

x = x+ y − zy = x− y + zz = −x+ y + z.

This is a linear system, that can be solved by eigenvalues and eigen-vectors method.

The eigenvalues and eigenvectors of the associated matrix are:λ1 = 1, whose eigenvector is v1 = (1, 1, 1); λ2 = 2, with v2 = (1, 0,−1)and λ3 = −2, with v3 = (1,−2, 1). Then the general solution of theODE system is x

yz

= c1et

111

+ c2e2t

10−1

+ c3e−2t

1−21

.

Equivalently, the field lines are

α(t) = (c1et + c2e

2t + c3e−2t, c1e

t − 2c3e−2t, c1e

t − c2e2t + c3e−2t),

c1, c2, c3 ∈ R. The only equilibrium point is (0, 0, 0).


By definition, the flow is the map T t which associates to each arbi-trary point (x, y, z) of R3 the field line α(x,y,z)(t), satisfying α(x,y,z)(0)= (x, y, z). For t = 0, we obtain the linear algebraic system

c1 + c2 + c3 = x, c1 − 2c3 = y, c1 − c2 + c3 = z,

with unique solution c1 =x+ y + z

3, c2 =

x− z2

, c3 =x− 2y + z

6.

Hence the flow is

T t(x, y, z) =

(x+ y + z

3et +

x− z2

e2t +x− 2y + z

6e−2t ,

x+ y + z

3et − 2

x− 2y + z

6e−2t,

x+ y + z

3et − x− z

2e2t

+x− 2y + z

6e−2t

).

To determine the effect of the flow on the volume, we calculate thedivergence of the vector field X, i.e.,

div(X) =∂X1

∂x+∂X2

∂y+∂X3

∂z= 1− 1 + 1 = 1.

Since div(X) > 0, the flow increases the volume.

Problem 4. Study the effect of flow generated by the non-linear vectorfield

X(x, y, z) = (x2 + y2, 2xy,−z2)

on the volume. Write the straightening diffeomorphism and the cor-responding infinitesimal transformation.

Solution. The field lines are the solutions of differential system

x = x2 + y2, y = 2xy, z = −z2.

The dynamical system has only one equilibrium point, (0, 0, 0).We write the associated symmetric differential system

dx

x2 + y2=

dy

2xy=

dz

−z2,

to find the first integrals. The first two quotients defines an homoge-neous differential equation, with unknown y = y(x). With the change


of unknown function u =y

x, we obtain a separable variables equation

1 + u2

u− u3du =

dx

x, with the unknown u = u(x). The fraction in the

left member is broken down into simple fractions. As a result of in-

tegration, we find the equalityu

1− u2= xc, c ∈ R, whence it follows

y

x2 − y2= c1, c1 ∈ R. The expression

y

x2 − y2defines a first integral.

To find another first integral, we use the method of integrablecombinations. By adding the numerators and denominators of the

first two reports, we reachd(x+ y)

(x+ y)2=

dz

−z2. Denote x + y = v.

Integrating, we obtain1

v+

1

z= c, c ∈ R, i.e.,

1

x+ y+

1

z= c2, c2 ∈ R.

In this way, the expression1

x+ y+

1

zdefines also a first integral.

We study the functional independence of the two first integrals.The Jacobi matrix is −

2xy

(x2 − y2)2

x2 + y2

(x2 − y2)20

− 1

(x+ y)2− 1

(x+ y)2− 1

z2

.

The rank of the matrix is two on R3 \Oz. Hence, on this set, the twofirst integrals are functional independent.

The implicit Cartesian equations of field lines arey

x2 − y2= c1

1

x+ y+

1

z= c2, c1, c2 ∈ R.

The straightening diffeomorphism (change of variables) is

φ : R3 → R3, φ(x, y, z) = (x, y, z),

˙x = 0˙y = 0˙z = 1.

We define x =y

x2 − y2and y =

1

x+ y+

1

z(the right hand members

are constant along the field lines, so their derivatives with respect tothe parameter t is equal to zero).


We are looking for z. Fromdz

dt= 1, it follows

dz

dz· dzdt

= 1, i.e.,

dz

dz·(−z2) = 1, equation with separable variables which by integration

leads to z =1

z.

Let A = (x, y, z) |x− y = 0 or x+ y = 0 or z = 0. It follows thestraightening diffeomorphism

φ : R3 \A→ R3, φ(x, y, z) =

(y

x2 − y2,

1

x+ y+

1

z,

1

z

).

Remark. By a diffeomorphism xi′

= xi′(xi), i = 1, 2, 3, the com-

ponents Xi of a vector field are changed after the rule (tensorial law)

Xi′ =∂xi

′

∂xiXi.

The infinitesimal transformation determined by the vector field Xis

(x′, y′, z′) = (x, y, z) + tX(x, y, z),

i.e.,

x′ = x+ t(x2 + y2), y′ = y + 2txy, z′ = z − tz2, t ∈ (−ε, ε).

Problem 5. Find the straightening diffeomorphism and the infinites-imal transformation for the vector field X(x, y, z) = (z,−xy, 2xz).

Solution. The field lines are the solutions of the differential system

x = z, y = −xy, z = 2xz.

The equilibrium points are the solutions of the algebraic system

z = 0, xy = 0, 2xz = 0,

i.e., all the points of the form (α, 0, 0) and (0, α, 0), with α ∈ R. Inother words, the axes Ox and Oy consist of equilibrium points.

To find the first integrals, we use the associated symmetric system

dx

z=

dy

−xy=

dz

2xz.


The first ratio and the last one determine a differential equationwith separable variables, which, by integration, leads to x2 − z =c1, c1 ∈ R.

The second and third ratios also determine an equation with sep-arable variables, with the implicit solution zy2 = c2, c2 ∈ R.

The functional independence of the two first integrals, x2 − z andzy2, is established using the rank of the Jacobi matrix(

2xy 0 −10 2yz y2

).

This matrix has rank 1 on the axis Oy and for the points in the planexOz, and in the other points its rank is 2, so the two first integralsfound are functionally independent on R3\Oy ∪ xOz. Cartesian equa-tions of orbits are x2 − z = c1, zy

2 = c2.The straightening diffeomorphism is the map φ : R3 → R3,

φ(x, y, z) = (x, y, z), where ˙x = 0, ˙y = 0, ˙z = 1.Define x = x2 − z and y = zy2, using the first integrals, which are

constant along the field lines, so their derivative with respect to theparameter t equal to zero.

Let us find z. From the conditiondz

dt= 1, it follows

dz

dz· dzdt

= 1,

i.e.,dz

dz· 2xz = 1. We express the variable x in function of z, from the

Cartesian equations of field lines (using the first integrals). It followsx =√z + c1, with the condition z + c1 > 0. The equation for finding

z becomes dz =1

2z√z + c1

dz, which has separable variables. For the

calculation of the integral of the right member, a change of variable

is made,1√z + c1

= u, which ultimately leads to the result

z =1

2√x2 − z

ln|x| −

√x2 − z

|x|+√x2 − z

.

The straightening diffeomorphism is the map

φ : D ⊂ R3 → R3, φ(x, y, z) =

(x2 − z, zy2,

1

2√x2 − z

ln|x| −

√x2 − z

|x|+√x2 − z

).

The infinitesimal transformation determined by X is (x′, y′, z′) =(x, y, z)+ tX(x, y, z), i.e., x′ = x+ tz, y′ = y− txy, z′ = z+2txz, t ∈(−ε, ε).

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