Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps

Invariant Sets and Bifurcation Structures

Viktor Avrutin, Laura Gardini;Iryna Sushko;Fabio Tramontana

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789811204715
• Topic: Mathematics
• Subtopic: Linear & Nonlinear Programming
• Index: Mathematics

The investigation of dynamics of piecewise-smooth maps is both intriguing from the mathematical point of view and important for applications in various fields, ranging from mechanical and electrical engineering up to financial markets. In this book, we review the attracting and repelling invariant sets of continuous and discontinuous one-dimensional piecewise-smooth maps. We describe the bifurcations occurring in these maps (border collision and degenerate bifurcations, as well as homoclinic bifurcations and the related transformations of chaotic attractors) and survey the basic scenarios and structures involving these bifurcations. In particular, the bifurcation structures in the skew tent map and its application as a border collision normal form are discussed. We describe the period adding and incrementing bifurcation structures in the domain of regular dynamics of a discontinuous piecewise-linear map, and the related bandcount adding and incrementing structures in the domain of robust chaos. Also, we explain how these structures originate from particular codimension-two bifurcation points which act as organizing centers. In addition, we present the map replacement technique which provides a powerful tool for the description of bifurcation structures in piecewise-linear and other form of invariant maps to a much further extent than the other approaches.

Contents:

• General Concepts and Tools
• Bifurcations in Piecewise Smooth Maps
• Bifurcations Scenarios (Overview)
• Map Replacement
• Skew Tent Map
• Adding Structures
• Incrementing Structures
• Organizing Centers

Readership: Researchers and graduate students working in the field of piecewise-smooth systems.Discontinuous Maps;Piecewise-Smooth Maps;Piecewise-Linear Maps;One-Dimensional Maps;Skew Tent Map;Border Collision Bifurcations;Homoclinic Bifurcations;Degenerate Bifurcations;Crisis Bifurcations;Robust Chaos;Period Adding;Period Incrementing;Map Replacement Technique00

Chapter 1

General concepts and tools

One of the main objects of our study is the family of one-dimensional piecewise linear maps f : I → I, I ⊆

, defined on two partitions by two linear functions:

where a

, a

, µ

, µ

∈

are parameters. The map (1.1) belongs to a class of one-dimensional piecewise smooth maps g : I → I, I ⊆

, defined in two sub-intervals of I, separated by a border point, by two different smooth functions g

and g

depending smoothly on some parameters. Without loss of generality, the border point can be translated to the origin, so that the map can be written as

If the functions g

and g

are strictly monotone on their partitions, the map (1.2) is called piecewise monotone.

If g

(0) = g

(0), the map (1.2) is continuous. In particular, map (1.1) is continuous for µ

= µ

= µ, in which case it represents the well-known skew tent map

discussed in detail in Chapter 5.

If g

(0) ≠ g

(0), the map (1.2) is discontinuous. The value g(0) may be defined as g(0) = g

(0) or g(0) = g

(0), or in any other way, but we do not specify it intentionally. In fact, the overall bifurcation structure of the parameter space of the map is not influenced by the particular choice of this value (see Sec. 2.1.3).

Our book aims to characterize the dynamics of the map (1.1) depending on its parameters, that is, to describe the bifurcation structure of its parameter space. To this end, in the present chapter we summarize some concepts and tools well-known from the theory of smooth one-dimensional maps, which are applicable to piecewise smooth maps without significant changes, and discuss the notions which either require certain extensions for the considered class of maps (as, e.g., the notion of a critical point, Sec. 1.3), or are not defined for smooth maps (e.g., a Cantor set attracto...