Related papers: Poincare Map Method for Limit Cycles in a Max-Plus…

In this paper, using our bifurcational geometric approach, we solve the problem on the maximum number and distribution of limit cycles in the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic…

Given a family of systems, identifying stabilizing switching signals in terms of infinite walks constructed by concatenating cycles on the underlying directed graph of a switched system that satisfy certain conditions, is a well-known…

We analyze a three-dimensional discontinuous piecewise linear system \(Z=(X,Y)\) whose switching manifold \(\Sigma\) contains visible-visible two-fold intersection lines. Assuming that the matrices \(DX\) and \(DY\) each have one nonzero…

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve first the problem on the maximum number of limit…

Recent results in control systems and numerical integration literature utilize invariant set theory to lift dynamical systems evolving on nonlinear manifolds to those evolving on vector spaces. We leverage this technique to propose an…

We study the statistical properties of piecewise expanding maps in the general setting of metric measure spaces. We provide sufficient conditions for exponential mixing of such systems with explicit estimates on the constants. We also…

This paper presents new results on the limit cycles of a Li\'enard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [33,34]. The results in [34] are only valid for the smooth system. We…

The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we…

Statistics of Poincar\' e recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the…

In the present paper, an essential generalization of the symbolic dynamics is considered. We apply the notions of abstract self-similar sets and the similarity map for a chaos introduction, which orbits are expanded among infinitely many…

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over an arbitrary finite field. For systems that can be…

The first-return map, or the Poincar\'e map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects…

The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30…

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking…

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and…

We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we…

This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous…

In order to obtain functional limit theorems for heavy tailed stationary processes arising from dynamical systems, one needs to understand the clustering patterns of the tail observations of the process. These patterns are well described by…

In this paper, we consider sufficient conditions for an invariant double circle to occur in a one parameter discrete dynamical systems on a cylinder.

For linear control systems with bounded control range, the state space is compactified using the Poincar\'e sphere. The linearization of the induced control flow allows the construction of invariant manifolds on the sphere and of…