[Paper Review] Period adding and incrementing gluing bifurcations in one-dimensional piecewise-smooth maps: theory and applications

Albert Granados, Lluı́s Alsedà+1 more|arXiv (Cornell University)|Jul 7, 2014

Mathematical Dynamics and Fractals93 references3 citations

TL;DR

This paper establishes rigorous sufficient conditions for period-adding and period-incrementing bifurcation scenarios in one-dimensional piecewise-smooth maps, demonstrating that periodic orbits emerge with symbolic sequences following a Farey tree structure (period adding) or via block attachment (period incrementing), with periods increasing through consecutive addition or constant addition, respectively, under orientation-preserving conditions in one or both domains.

ABSTRACT

In this review article we provide sufficient conditions for the occurrence of the so-called period adding and period incrementing bifurcations scenarios in piecewise-smooth one-dimensional maps, and we show how these results can be easily applied to real-world applications. The first scenario occurs when the piecewise-smooth map preserves orientation. This scenario refers to the appearance of periodic orbits whose symbolic sequences and rotation numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. The second scenario occurs when the piecewise-smooth map preserves orientation only in one domain. In this case, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. These scenarios have been widely observed in the literature by the non-smooth community when numerically investigating piecewise-smooth discontinuous maps. In this review we assemble and extend classical results for circle maps to discontinuous circle maps and apply them to provide rigorous and complete proofs.

Motivation & Objective

• To identify and formalize the conditions under which period-adding and period-incrementing bifurcation scenarios occur in one-dimensional piecewise-smooth maps.
• To extend classical circle map theory to discontinuous maps, providing a rigorous foundation for observed bifurcation patterns.
• To unify and generalize existing numerical observations of periodic orbit sequences in non-smooth dynamical systems.
• To offer complete and rigorous proofs for the emergence of periodic orbits with specific symbolic sequences and rotation numbers in piecewise-smooth settings.
• To demonstrate the applicability of the theoretical framework to real-world systems through illustrative examples.

Proposed method

• Derivation of sufficient conditions for period-adding bifurcations when the map preserves orientation globally.
• Analysis of period-incrementing bifurcations when orientation is preserved only in one domain, leading to symbolic sequences formed by block attachment.
• Application of Farey tree structure to describe the symbolic sequences and rotation numbers of periodic orbits in the period-adding scenario.
• Use of symbolic dynamics and rotation number theory to characterize the sequence of periodic orbits in discontinuous maps.
• Extension of classical results from continuous circle maps to discontinuous, piecewise-smooth maps via rigorous topological and combinatorial arguments.
• Construction of proofs based on the structure of the symbolic sequences and the behavior of the map across discontinuity points.

Experimental results

Research questions

• RQ1 Under what conditions does a period-adding bifurcation scenario occur in a piecewise-smooth one-dimensional map?
• RQ2 How do symbolic sequences and rotation numbers evolve in the period-adding scenario, and what role does the Farey tree structure play?
• RQ3 What are the sufficient conditions for a period-incrementing bifurcation when orientation is preserved in only one domain?
• RQ4 How does the block attachment mechanism generate periodic orbits with incremented periods in the second scenario?
• RQ5 To what extent can classical circle map theory be extended to discontinuous piecewise-smooth maps?

Key findings

• Period-adding bifurcations occur when the map preserves orientation globally, leading to periodic orbits whose symbolic sequences follow the Farey tree structure.
• In the period-adding scenario, the periods of periodic orbits are generated by consecutive addition, corresponding to the sum of previous periods in the Farey sequence.
• Period-incrementing bifurcations arise when orientation is preserved only in one domain, resulting in symbolic sequences formed by repeated attachment of a fixed symbolic block.
• In the period-incrementing scenario, the periods of periodic orbits increase by a constant term, reflecting a linear growth pattern in the sequence of orbits.
• The theoretical framework successfully generalizes classical results from continuous circle maps to discontinuous, piecewise-smooth maps with rigorous proofs.
• The derived conditions provide a complete and systematic method for predicting and analyzing periodic orbit sequences in real-world applications of non-smooth dynamical systems.