Nonlinear Differential Equations and Dynamical Systems

By Ferdinand Verhulst. 1989

1 Introduction.- 1.1 Definitions and notation.- 1.2 Existence and uniqueness.- 1.3 Gronwall's inequality.- 2 Autonomous equations.- 2.1 Phase-space, orbits.- 2.2 Critical points and linearisation.- 2.3 Periodic solutions.- 2.4 First integrals and integral manifolds.- 2.5 Evolution of a volume element, Liouville's theorem.- 2.6 Exercises.- 3 Critical points.- 3.1 Two-dimensional linear systems.- 3.2 Remarks on three-dimensional linear systems.- 3.3 Critical points of nonlinear equations.- 3.4 Exercises.- 4 Periodic solutions.- 4.1 Bendixson's criterion.- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem.- 4.3 The Poincare-Bendixson theorem.- 4.4 Applications of the Poincare-Bendixson theorem.- 4.5 Periodic solutions in ?n.- 4.6 Exercises.- 5 Introduction to the theory of stability.- 5.1 Simple examples.- 5.2 Stability of equilibrium solutions.- 5.3 Stability of periodic solutions.- 5.4 Linearisation.- 5.5 Exercises.- 6 Linear Equations.- 6.1 Equations with constant coefficients.- 6.2 Equations with coefficients which have a limit.- 6.3 Equations with periodic coefficients.- 6.4 Exercises.- 7 Stability by linearisation.- 7.1 Asymptotic stability of the trivial solution.- 7.2 Instability of the trivial solution.- 7.3 Stability of periodic solutions of autonomous equations.- 7.4 Exercises.- 8 Stability analysis by the direct method.- 8.1 Introduction.- 8.2 Lyapunov functions.- 8.3 Hamiltonian systems and systems with first integrals.- 8.4 Applications and examples.- 8.5 Exercises.- 9 Introduction to perturbation theory.- 9.1 Background and elementary examples.- 9.2 Basic material.- 9.3 Naive expansion.- 9.4 The Poincare expansion theorem.- 9.5 Exercises.- 10 The Poincare-Lindstedt method.- 10.1 Periodic solutions of autonomous second-order equations.- 10.2 Approximation of periodic solutions on arbitrary long time-scales.- 10.3 Periodic solutions of equations with forcing terms.- 10.4 The existence of periodic solutions.- 10.5 Exercises.- 11 The method of averaging.- 11.1 Introduction.- 11.2 The Lagrange standard form.- 11.3 Averaging in the periodic case.- 11.4 Averaging in the general case.- 11.5 Adiabatic invariants.- 11.6 Averaging over one angle, resonance manifolds.- 11.7 Averaging over more than one angle, an introduction.- 11.8 Periodic solutions.- 11.9 Exercises.- 12 Relaxation Oscillations.- 12.1 Introduction.- 12.2 Mechanical systems with large friction.- 12.3 The van der Pol-equation.- 12.4 The Volterra-Lotka equations.- 12.5 Exercises.- 13 Bifurcation Theory.- 13.1 Introduction.- 13.2 Normalisation.- 13.3 Averaging and normalisation.- 13.4 Centre manifolds.- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation.- 13.6 Exercises.- 14 Chaos.- 14.1 Introduction and historical context.- 14.2 The Lorenz-equations.- 14.3 Maps associated with the Lorenz-equations.- 14.4 One-dimensional dynamics.- 14.5 One-dimensional chaos: the quadratic map.- 14.6 One-dimensional chaos: the tent map.- 14.7 Fractal sets.- 14.8 Dynamical characterisations of fractal sets.- 14.9 Lyapunov exponents.- 14.10 Ideas and references to the literature.- 15 Hamiltonian systems.- 15.1 Introduction.- 15.2 A nonlinear example with two degrees of freedom.- 15.3 Birkhoff-normalisation.- 15.4 The phenomenon of recurrence.- 15.5 Periodic solutions.- 15.6 Invariant tori and chaos.- 15.7 The KAM theorem.- 15.8 Exercises.- Appendix 1: The Morse lemma.- Appendix 2: Linear periodic equations with a small parameter.- Appendix 3: Trigonometric formulas and averages.- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3.- Appendix 5: Bifurcations of self-excited oscillations.- Appendix 6: Normal forms of Hamiltonian systems near equilibria.- Answers and hints to the exercises.- References.

Subfields: Network Science & Complex Systems, Pure Mathematics

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Citations in the corpus

A Closer Look at the Optimization Landscapes of Generative Adversarial Networks (2019, methodology): “Verhulst (1989, Theorem 7.1) deﬁnes a LSSP ω ∗ using the eigenvalues of the Jacobian of the game vector ﬁeld ∇ v ( ω ∗ ) at that point. where denote the real part of the eigenvalue λ belonging to the spectrum of ∇ v ( ω ∗ ) .”
A Family of Gradient Descent Grid Frequency Estimators for the SOGI Filter (2018, methodology): “Then, the integral manifold approach to slow adaptation concept along with the averaging theory [42], [43] can be here applied to show that for a periodic input signal”
Ordinary Differential Equations and Dynamical Systems (2012, unknown): “A brief overview can be found in Verhulst [43].”
Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay (2006, unknown): “By Gronwall’s inequality [44], we obtain”
MOMENT EXPLOSIONS AND STATIONARY DISTRIBUTIONS IN AFFINE DIFFUSION MODELS (2010, background): “See, for example, Chapter 7 of Verhulst (1996).”
Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics (2010, background): “Examples are [1] from neurology, [2] from the theory of solitons, [3] from the biology, [4–6] from theory of dynamical systems, chaos theory and ecology, and [7–11] from hydrodynamics and theory of turbulence.”
On the birth of limit cycles for non-smooth dynamical systems (2012, methodology): “5 of Verhulst [29], where instead of (i) is assumed that (j) F, R, DxF, D 2 xF and DxR are defined, continuous and bounded by a constant M (independent of ε) in [0,∞)×D, −εf < ε < εf ; and instead of (ii) it is required that (jj) for a ∈ D with f(a) = 0 we have that Jf (a) 6= 0, where Jf (a) is the ”
Averaging theory at any order for computing periodic orbits (2013, background): “The averaging theory of first order for studying periodic orbits of the differential equations (1) in arbitrary finite dimension is very classical (see for instance [9, 10]), until third order was developed in [1].”