Problems and Solutions

Nonlinear Dynamics, Chaos and Fractals

Willi-Hans Steeb

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813109940
• Topic: Matemáticas
• Subtopic: Matemática aplicada

This book presents a collection of problems for nonlinear dynamics, chaos theory and fractals. Besides the solved problems, supplementary problems are also added. Each chapter contains an introduction with suitable definitions and explanations to tackle the problems.

The material is self-contained, and the topics range in difficulty from elementary to advanced. While students can learn important principles and strategies required for problem solving, lecturers will also find this text useful, either as a supplement or text, since concepts and techniques are developed in the problems.

Contents:

• One-Dimensional Maps:
  • Notations and Definitions
  • One-Dimensional Maps
• Higher-Dimensional Maps and Complex Maps:
  • Introduction
  • Two-Dimensional Maps
  • Complex Maps
  • Higher-Dimensional Maps
  • Bitwise Maps
• Fractals
  • Introduction
  • Solved Problems
  • Supplementary Problems

Readership: Graduate students who focus on chaos, fractals and nonlinear dynamics.
One-Dimensional Maps;Complex Maps;Bitwise Maps

Chapter 1

One-Dimensional Maps

1.1Notations and Definitions

We consider exercises for nonlinear one-dimensional maps. In particular we consider one-dimensional maps with chaotic behaviour. We first summarize the relevant definitions such as fixed points, stability, periodic orbit, Liapunov exponent, invariant density, topologically conjugacy, etc.. Ergodic maps are also considered.

We use the notation f : D → C to indicate a function f with domain D and codomain C. The notation f : D → D indicates that the domain and codomain of the function are the same set.

We also use the following two definitions: A mapping g : A ↦ B is called surjective if g(A) = B. A mapping g is called injective (one-to-one) when ∀a, a′ ∈ A, g(a) = g(a′) ⇒ a = a′. If the mapping f is surjective and injective, the mapping f is called bijective.

Definition.If B ⊂ C, then f⁽⁻¹⁾(B) is called the inverse image or preimage of B and consists of all elements of D whose image is contained in B. That is

Note that the notation f⁽⁻¹⁾ does not necessarily imply that f is an invertible function.

Definition.Consider a map f : S → S. A point x* ∈ S is called a fixed point of f if

Definition.Let f : A → A and g : B → B be two maps. The maps f and g are said to be topologically conjugate if there exists a homeomorphism h : A → B such that, h ∘ f = g ∘ h.

Definition.Consider a map f : S → S. A point x ∈ S is an eventually fixed point of the function, if there exists N ∈ ℕ such that

whenever n ≥ N. The point x is eventually periodic with period k, if there exists N such that f^(n+k)(x) = f⁽ⁿ⁾(x) whenever n ≥ N.

Definition.Let f be a function and p be a periodic point of f with prime period k. Then...