Chaotic Dynamics in Two-Dimensional Noninvertible Maps

The following sections are included:

• CONTENTS

• PREFACE

The following sections are included:

• Continuous Dynamic Systems and Discrete Systems

• Some Considerations on Maps and Some Definitions

• Diffeomorphism and Noninvertible Maps. The Hadamard's Theorem
  • The Hadamard's Theorem
  • The Two Classes of Noninvertible Maps. Critical Manifolds

• Imbedding of a Noninvertible Map into a Diffeomorphism with a Higher Dimension

• The Valiron's Results

• Consequences of the Valiron's Results

• Some Definitions and Properties

• Maps and the Schroder's Equation
  • Generalities
  • Basin Boundary, Singular Points and the Schroeder's Equation
  • One-dimensional noninvertible maps and chaotic behaviours. Generalization of Chebyshev's polynomials
  • One-dimensional noninvertible maps. Properties of generalized Chebyshev's polynomials
  • One-dimensional noninvertible maps. Examples of functions H_k giving rise to chaotic behaviours
  • Generalization to m-dimensional noninvertible maps m > 1

The following sections are included:

• Introduction

• Critical points. Properties
  • Definitions
  • Different determinations of the inverse map
  • Folding of the real line. Definition of preimages. Generalities

• Some Properties of One-Dimensional Maps

• Other Definitions

• Maps Defined by a Function with only one Extremum. Rotation Sequences and Boxes in Files Structures
  • Rotation sequences
  • Diffeomorphism of the circle. Fractal “boxes in files” bifurcation structure
  • Quadratic algebraic rotation numbers

• Maps Defined by a Function With Only One Extremum. Fractal “Box-Within-a-Box” Bifurcation Structure
  • Generalities
  • Properties and fundamental bifurcations of unimodal noninvertible maps
  • Succinct description of the “box-within-a-box” bifurcations structure
  • Accumulation points of boxes
  • Generalizations

• Maps Defined by a Function with Two Extrema

• Basins and their Bifurcations
  • Generalities
  • Simply connected basin
  • Nonconnected basin with finite boundary. Bifurcation
  • First type of basin with an infinite (fractal) boundary. chaotic transient. Bifurcation
  • Second type of basin with an infinite (fractal) boundary. Fuzzy boundary. Bifurcation
  • Remarks

• Properties of Continuous Piecewise Linear Maps

• Properties of Piecewise Continuous Maps
  • General Considerations
  • Remark
  • Piecewise linear maps with only one discontinuity point
  • Maps with several points of discontinuity

The following sections are included:

• Introduction

• Critical curves. Types of Noninvertible Maps
  • Critical curves and inverses of a noninvertible map
  • Types of noninvertible maps with critical curves. Their symbolic representation
  • Characterization of the different determinations of the inverse map
  • Critical set of a power of the map T.

• Foliation of the Plane, and Definition of Preimages. Generalities

• Some Properties Related to Images and Preimages
  • Mapping of the intersection between a curve segment and a curve of merging preimages
  • Images and preimages of a segment of critical curve
  • Some properties of critical curves (general case)

• Foliation of Different Types of Noninvertible Maps
  • The problem of a three-dimensional representation of the foliation
  • (Z₀ – Z₂) maps
  • (Z₁ – Z₃ – Z₁) maps
  • (Z₁ < Z₃) maps
  • (Z₁ < Z₃ >) map
  • (Z₀ – Z₂ ≪ Z₄) map
  • (Z₂ ≪ Z₄ ≥) maps

• Bifurcations of the Foliation Associated with the Inverse Map
  • Preliminary considerations and the simplest bifurcation
  • The basic bifurcation
  • Bifurcation by “dovetail-lip transition”
  • Bifurcation by “dovetail-non adjacent cusps” transition
  • Bifurcation by disappearing of the two cusps of a dovetail

• Properties of Piecewise Continuous Maps
  • General considerations
  • Properties of critical curves
  • Remarks
  • Properties of preimages and critical curves
  • Example

The following sections are included:

• Introduction

• Definitions and General Properties
  • Absorbing areas. Chaotic areas
  • Connection of a set

• Simple Examples
  • Simple examples of (Z₀ – Z₂) maps, or equivalent to such maps
  • Simple examples with other maps
  • Two basic propositions

• Properties of (Z₀ – Z₂) Maps
  • First set of properties
  • Different kinds of absorbing areas (nonmixed ones)

• Properties of Absorbing Areas and of Invariant Areas. General Case
  • Basic properties of areas with finite boundary
  • Some properties of invariant areas with infinite boundary
  • Points of nonsmoothness in boundaries of invariant areas. Bifurcation of nonsmoothness
  • Some other properties of simply connected invariant nonmixed absorbing areas
  • New set of examples of simply connected invariant nonmixed absorbing areas

• Invariant Areas with Infinite Boundary
  • Generalities on contact bifurcations
  • First example of contact bifurcation. Homoclinic and heteroclinic bifurcations
  • Second example of contact bifurcation

• Multiply Connected (Nonmixed) Invariant Absorbing Areas
  • First results on (Z₀ – Z₂) maps. Bifurcation of an annular absorbing area
  • On the bifurcation destroying holes in (Z₀ – Z₂) maps
  • Cyclic holes in (Z₀ – Z₂) maps

• Generalization to Maps Different from (Z₀ – Z₂) Type

• Example of Mixed Absorbing Areas and Mixed Chaotic Areas
  • Critical curves
  • Singularities at finite distance
  • Hypotheses H
  • Properties of saddle sets
  • Attracting set as a mixed chaotic area

• Piecewise Continuous Maps

The following sections are included:

• Introduction

• Case of (Z₀ – Z₂) Maps.
  • Definitions and some properties
  • Conditions related to the existence of a non-connected basin, and a multiply connected basin
  • Basin bifurcations

• Study of a First Type of (Z₀ – Z₂) Quadratic Map. General Properties
  • Cycles of period k = 1, 2
  • Critical curves

• Basin Behaviour
  • General properties
  • Nature of the fixed point P belonging to the basin boundary
  • Preimages of the fixed point P

• Bifurcations of the “Basin” Boundary of Unbounded Iterated Sequences
  • Two bifurcations giving rise to a non-connected basin
  • Bifurcation giving rise to a multiply connected basin. Weak fractal basin boundary
  • Bifurcation giving rise to a sharp fractal boundary
  • Bifurcation giving rise to a strange repulsor (or repeller)
  • Bifurcation creating lakes inside the holes of a chaotic area
  • Bifurcations occuring when the external boundary is smooth (the Parameter a is positive)
  • Remarks on homoclinic and heteroclinic situations

• Bifurcation Leading to a Discontinuous Increase of a Basin (Second Form of the Quadratic Z₀ – Z₂ Map)
  • General properties
  • Non classical bifurcation
  • First example of islands aggregation

• Third Form of (Z₀ – Z₂) Quadratic map
  • General properties
  • Formation of a singular set, support of the fractal structure of basin islands
  • Bifurcations giving rise to infinitely many islands from a simply connected basin. Symbolism of islands

• Cascade of Bifurcations Leading to Islands Aggregation in the Third Form of (Z₀ – Z₂) Quadratic Map
  • First bifurcation by contact island-immediate basin
  • Cascade of bifurcations by contact island-immediate basin leading to islands aggregation

• Basin Bifurcations Generated by a (Z₁ < Z₃ >) Map (Lip Type)
  • Bifurcation leading to a discontinuous change of basins
  • Bifurcations leading to lakes aggregation

• Basin Bifurcations Generated by a (Z₂ ≪ Z₄ ≥) Map
  • General properties
  • Chaotic areas. Bifurcation by meeting of two chaotic areas, leading to only one chaotic area
  • Bifurcation giving rise to a chaotic area presenting different dynamics
  • Fractal basin boundary and half fractal basic boundary

• Classification of Bifurcations of Two-Dimensional Noninvertible Maps
  • Recall of the one-dimensional case
  • First classification of bifurcations of chaotic areas. Two-dimensional case
  • Second classification of two-dimensional bifurcations

• Bifurcations Related to a Mixed Chaotic Area
  • Bifurcations “period two mixed chaotic area↔connected mixed chaotic area”
  • Bifurcation “mixed chaotic area↔mixed chaotic area” of another type
  • Definition of the mixed chaotic area near its limit of existence
  • Basin of the mixed chaotic area and bifurcation destroying it

• Basins of Piecewise Continuous Maps
  • Generalities
  • Examples

The following sections are included:

• Introduction

• Interaction of Line Segments with Critical Curves. Basic Situations (General Case)

• Bifurcations of the Unstable Set of a Saddle in the Case of (Z₀ – Z₂) Maps
  • General considerations
  • First bifurcation creating self-intersections of the unstable set
  • Change of properties of the unstable set
  • Bifurcation creating a new type of self intersections of the unstable set

• Bifurcations of an Invariant Closed Curve. Case of (Z₀ – Z₂) Maps
  • General considerations
  • Qualitative changes of the invariant closed curve
  • Qualitiative change creating a “weakly chaotic ring” from an invariant closed curve
  • Case of piecewise smooth maps

• Intersections of an Invariant Closed Curve with the Unstable Set of a Saddle, in the Case of (Z₀ – Z₂) Maps

• Coexistence of an “External k-Pair” with an Invariant Closed Curve in the Case of (Z₀ – Z₂) Maps. Cycles Properties

• First Quadratic Noninvertible (Z₀ – Z₂) Map
  • General properties
  • Basin bifurcation by creation of an heteroclinic orbit
  • Bifurcations leading to a weakly chaotic ring
  • Bifurcations leading to a complex homoclinic situation

• Second Quadratic Map
  • General properties
  • Parametric interval of existence of a stable period 7 cycle
  • The fundamental global contact bifurcation
  • Other parametric intervals of existence of stable cycles

• Consequences for Three-Dimensional Invertible Maps

• Example of Oscillations of an Invariant Closed Curve in Absence of an External Pair

• Piecewise Linear Maps
  • First example
  • Second piecewise linear map

• Example of (Z₁ – Z₃ – Z₁) Maps

• Conclusion

The following sections are included:

• REFERENCES

• INDEX