Limit Cycles and Chaos in Planar Hybrid Dynamical Systems

Core Concepts

This paper studies the family of planar hybrid dynamical systems formed by two linear centers and a polynomial reset map of any degree. It investigates the existence of limit cycles and provides examples of these hybrid systems exhibiting chaotic dynamics.

Abstract

The paper focuses on the study of planar hybrid dynamical systems, which are systems whose dynamics are governed by both continuous and discrete laws. The authors consider a family of such systems, denoted as Xn, where X1 and X2 are linear vector fields with center singularities, and the reset map ϕn is a polynomial of degree n.

The key highlights and insights are:

1. For the case of X1 (i.e., n = 1), the authors prove that if the hybrid system has a regular limit cycle, then it is unique, hyperbolic, and its stability is determined by the derivative of the reset map ϕ at the origin.

2. For the general case of Xn (n ≥ 1), the authors show that there exists a compact set K ⊂ R^2 such that the relevant dynamics of the hybrid system occurs within K. Outside of K, the orbits either escape to infinity or converge to K.

3. The authors provide an example of a chaotic hybrid system in X2, where the first return map is conjugate to the square of the logistic map, which is known to exhibit chaotic behavior.

4. The authors also provide a detailed analysis of the structure of the first return map P, showing that its domain can be partitioned into at most n pairwise disjoint intervals.

Overall, the paper demonstrates that even simple planar hybrid dynamical systems can exhibit rich dynamics, including the existence of limit cycles and chaotic behavior, which contrasts with the relatively simple dynamics of the corresponding Filippov systems studied in previous work.

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Key Insights Distilled From

Limit cycles and chaos in planar hybrid systems

by Jaume Llibre... at arxiv.org 10-02-2024

https://arxiv.org/pdf/2407.05151.pdf

Deeper Inquiries

What are the potential applications of the chaotic hybrid systems studied in this paper, and how could the insights be extended to higher-dimensional or more complex hybrid systems?

The chaotic hybrid systems explored in this paper have significant potential applications across various fields, including engineering, robotics, economics, and biological systems. In engineering, for instance, understanding chaotic dynamics can enhance the design of control systems that are robust to disturbances and uncertainties. In robotics, chaotic behavior can be harnessed for complex motion planning and navigation, allowing robots to adapt to unpredictable environments.
Moreover, the insights gained from studying planar hybrid systems can be extended to higher-dimensional or more complex hybrid systems by employing similar qualitative analysis techniques. For example, the methods used to analyze limit cycles and chaotic behavior in two-dimensional systems can be adapted to higher-dimensional systems by considering the additional complexity introduced by more vector fields and reset maps. This could involve the use of advanced numerical simulations and bifurcation analysis to explore the rich dynamics that arise in higher dimensions, potentially leading to new phenomena not observed in lower-dimensional systems.

How do the dynamics of the hybrid systems studied here compare to the dynamics of other types of hybrid systems, such as those with different types of vector fields or reset maps?

The dynamics of the hybrid systems studied in this paper, characterized by two linear centers and polynomial reset maps, exhibit unique features compared to other types of hybrid systems. In particular, the presence of linear centers allows for predictable flow behavior in the absence of resets, while the polynomial nature of the reset maps introduces nonlinearity that can lead to complex dynamics, including limit cycles and chaos.
In contrast, hybrid systems with different types of vector fields, such as nonlinear or piecewise-linear fields, may exhibit more intricate dynamics due to the potential for multiple equilibria and more complex bifurcation scenarios. Additionally, hybrid systems with different reset maps, such as piecewise constant or discontinuous maps, can lead to different types of dynamical behavior, including sliding modes and chattering phenomena. The interplay between the nature of the vector fields and the reset maps is crucial in determining the overall dynamics, and thus, the specific characteristics of the hybrid systems studied here may not be directly applicable to all hybrid systems.

Can the techniques used in this paper be adapted to study the bifurcation behavior of limit cycles and the transition to chaos in planar hybrid systems?

Yes, the techniques employed in this paper can be effectively adapted to study the bifurcation behavior of limit cycles and the transition to chaos in planar hybrid systems. The authors utilize qualitative methods from the theory of differential equations and dynamical systems, particularly focusing on the first return map and its properties. This approach can be extended to analyze how changes in parameters, such as the coefficients in the reset maps or the characteristics of the vector fields, influence the existence and stability of limit cycles.
By applying bifurcation theory, researchers can systematically investigate how small perturbations in the system parameters lead to qualitative changes in the dynamics, such as the emergence of new limit cycles or the onset of chaotic behavior. The piecewise nature of the first return map, as highlighted in the paper, provides a framework for identifying bifurcation points and understanding the transitions between different dynamical regimes. Thus, the methodologies presented in this study serve as a valuable foundation for further exploration of bifurcations and chaos in more complex planar hybrid systems.