This book provides a comprehensive survey of the Sharkovsky ordering, its different aspects and its role in dynamical systems theory and applications. It addresses the coexistence of cycles for continuous interval maps and one-dimensional spaces, combinatorial dynamics on the interval and multidimensional dynamical systems. Also featured is a short chapter of personal remarks by O.M. Sharkovsky on the history of the Sharkovsky ordering, the discovery of which almost 60 years ago led to the inception of combinatorial dynamics. Now one of cornerstones of dynamics, bifurcation theory and chaos theory, the Sharkovsky ordering is an important tool for the investigation of dynamical processes in nature. Assuming only a basic mathematical background, the book will appeal to students, researchers and anyone who is interested in the subject.
Author(s): Alexander M. Blokh, Oleksandr M. Sharkovsky
Series: SpringerBriefs in Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 113
City: Cham
Preface
Contents
1 Coexistence of Cycles for Continuous Interval Maps
1.1 Introduction
1.2 Proof of Forcing Sh-Theorem
1.2.1 Loops of Intervals Force Periodic Orbits
1.2.2 The Beginning of the Sh-order
1.2.3 Three Implies Everything
1.2.4 Minimal Cycles Imply Sh-weaker Periods
1.2.5 Orbits with Sh-strongest Periods Form Simplest Cycles
1.3 Proof of Realization Sh-Theorem
1.4 Stability of the Sh-ordering
1.5 Visualization of the Sh-ordering
References
2 Combinatorial Dynamics on the Interval
2.1 Introduction
2.2 Permutations: Refinement of Cycles' Coexistence
2.3 Rotation Theory
2.4 Coexistence of Homoclinic Trajectories and Stratification of the Space of Maps
2.4.1 Homoclinic Trajectories, Horseshoes, and L-Schemes
2.4.2 Coexistence (of Homoclinic Trajectories) and Its Stability: Powers of Maps with L-Scheme and Homoclinic Trajectories
References
3 Coexistence of Cycles for One-Dimensional Spaces
3.1 Circle Maps
3.2 Maps of the nn-od
3.3 Other Graph Maps
3.3.1 Graph-Realizable Sets of Periods
3.3.2 Trees
3.3.3 Graphs With Exactly One Loop
3.3.4 Figure Eight Graph
References
4 Multidimensional Dynamical Systems
4.1 Triangular Maps
4.2 Cyclically Permuting Maps
4.3 Multidimensional Perturbations of One-Dimensional Maps
4.4 Infinitely-Dimensional Dynamical Systems, Generated by One-Dimensional Maps
4.5 Final Remarks
4.5.1 Multivalued Maps
4.5.2 Nonlinear Difference Equations
References
5 Historical Remarks
Appendix Appendix
A.1 The Copy of the First Page of the Paper From 1964
A.2 The Copy of the Last Page of the Paper From 1964
A.3 Translation of the Original Paper From 1964
