# Introduction to Applied Nonlinear Dynamical Systems and Chaos

@inproceedings{Wiggins1989IntroductionTA, title={Introduction to Applied Nonlinear Dynamical Systems and Chaos}, author={Stephen Wiggins}, year={1989} }

Equilibrium Solutions, Stability, and Linearized Stability * Liapunov Functions * Invariant Manifolds: Linear and Nonlinear Systems * Periodic Orbits * Vector Fields Possessing an Integral * Index Theory * Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows * Asymptotic Behavior * The Poincare-Bendixson Theorem * Poincare Maps * Conjugacies of Maps, and Varying the Cross-Section * Structural Stability, Genericity, and Transversality * Lagrange's…

## 5,032 Citations

Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation

- Mathematics
- 1992

Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

- MathematicsNonlinearity
- 2021

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter ɛ > 0 and give several necessary conditions…

On the control of nonlinear dynamical systems

- Mathematics
- 2001

Nonlinear dynamical systems undergo complex or even chaotic behavior which is often not desired in applications. Therefore, one tries to control the system performance by using small feedback…

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

- MathematicsRegular and Chaotic Dynamics
- 2018

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of…

Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows

- Mathematics
- 1998

Third-order explicit autonomous differential equations in one scalar variable or, mechanically interpreted, jerky dynamics constitute an interesting subclass of dynamical systems that can exhibit…

On the Asymptotic Determination of Invariant Manifolds for Autonomous Ordinary Differential Equations

- Mathematics
- 2002

A methodology to calculate the approximate invariant manifolds of dynamical systems
defined through an m-dimensional autonomous vector field is presented. The technique is based on the calculation of…

Inariant manifolds and chaotic vibrations in singularly perturbed nonlinear oscillators

- Mathematics
- 1998

Chaotic motions near homoclinic manifolds and resonant tori in quasiperiodic perturbations of planar Hamiltonian systems

- Physics, Mathematics
- 1993

Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid

- MathematicsInt. J. Bifurc. Chaos
- 2016

This paper studies the integrability and dynamics of quadratic polynomial differential systems defined in ℝ3 having an elliptic paraboloid as an invariant algebraic surface and proves the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers.

Homoclinic orbits and chaos in three– and four–dimensional flows

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 2001

We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations.…