Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics 0521475724, 9780521475723

Table of contents :
Contents
Preface
0 - Hyperbolicity, stability and sensitive chaotic
dynamical systems
§1 Hyperbolicity and stability
§2 Sensitive chaotic dynamics
1 - Examples of homoclinic orbits in dynamical systems
§1 Homoclinic orbits in a deformed linear map
§2 The pendulum
§3 The horseshoe
§4 A homoclinic bifurcation
§5 Concluding remark
2 - Dynamical consequences of a transverse homoclinic intersection
§1 Description of the situation - linearizing coordinates
and a special domain R
§2 The maximal invariant subset of R - topological analysis
§3 The maximal invariant subset of R - hyperbolicity and
invariant foliations
§4 The maximal invariant subset of R - structure
§5 Conclusions for the dynamics near a transverse homoclinic orbit
§6 Homoclinic points of periodic orbits
§7 Transverse homoclinic intersections in arbitrary dimensions
§8 Historical note
3 - Homoclinic tangencies: cascades of bifurcations,
scaling and quadratic maps
§1 Cascades of homoclinic tangencies
§2 Saddle-node and period doubling bifurcations
§3 Cascades of period doubling bifurcations and sinks
§4 Homoclinic tangencies, scaling and quadratic maps
4 - Cantor sets in dynamics and fractal dimensions
§1 Dynamically defined Cantor sets
§2 Numerical invariants of Cantor sets
§3 Local invariants and continuity
5 - Homoclinic bifurcations: fractal dimensions
and measure of bifurcation sets
§ 1 Construction of bifurcating families of diffeomorphisms
§2 Homoclinic tangencies with bifurcation set of small relative
measure - statement of the results
§3 Homoclinic tangencies with bifurcation set of small relative
measure - idea of proof
§4 Heteroclinic cycles and further results on measure of
bifurcation sets
6 - Infinitely many sinks and homoclinic tangencies
§ 1 Persistent tangencies
§2 The tent map and the logistic map
§3 Hénon-like diffeomorphisms
§4 Separatrices of saddle points for diffeomorphisms near
a homoclinic tangency
§5 Proof of the main result
§6 Sensitive chaotic orbits near a homoclinic tangency
7 - Overview, conjectures and problems - a theory of
homoclinic bifurcations - strange attractors
§1 Homoclinic bifurcations and nonhyperbolic dynamics
§2 Strange attractors
§3 Summary, further results and problems
Appendix 1 - Hyperbolicity: stable manifolds and foliations
Appendix 2 - Markov partitions
Appendix 3 - On the shape of some strange attractors
Appendix 4 - Infinitely many sinks in one-parameter families
of diffeomorphisms
Appendix 5 - Hyperbolicity and the creation of homoclinic orbits,
reprinted from Annals of Mathematics 125 (1987)
References
Index

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