Piecewise-linear Continuous Maps Research Articles

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.

Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here, we contribute to this topic by investigating the existence of n-cycles (nge 3) and border-collision bifurcations in a class of m-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is shown that these n-cycles collide with the switching boundary in a border-collision bifurcation, and parametric regions for the existence of both stable and unstable n-cycles and border-collision bifurcations will be derived theoretically. We then discuss how our results can be extended and applied to PLRNNs. Finally, numerical simulations demonstrate the implementation of our results and are found to be in good agreement with the theoretical derivations. Our findings thus provide a basis for understanding periodic behavior in PLRNNs, how it emerges in bifurcations, and how it may lead into chaos.

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

In two-parameter bifurcation diagrams of piecewise-linear continuous maps on , mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and the N-dimensional map is fixed within each sector. The skew sawtooth maps are continuous, degree-one, and piecewise-linear, with two different slopes. They approximate the stable dynamics of the N-dimensional map with an error that goes to zero with the distance from the shrinking point. The results explain the complicated radial pattern of periodic, quasi-periodic, and chaotic dynamics that occurs near shrinking points.

Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for [Formula: see text]-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

The structure of mode-locking regions of piecewise-linear continuous maps: I. Nearby mode-locking regions and shrinking points

The mode-locking regions of a dynamical system are subsets of parameter space within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a distinctive chain structure with points of zero width called shrinking points. In this paper a local analysis about an arbitrary shrinking point is performed. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional centre manifolds in order to build a comprehensive theoretical framework for the local dynamics. The main results are universal quantitative descriptions for the shape of nearby mode-locking regions, the location of nearby shrinking points, and the key properties of these shrinking points. The results are applied to the three-dimensional border-collision normal form, a model of an oscillator subject to dry friction, and a model of a DC/DC power converter.

Border-Collision Bifurcations in $\mathbb{R}^N$

For piecewise-smooth maps, new dynamics can be created by varying parameters such that a fixed point collides with a surface on which the map is nonsmooth. If the map is continuous and has bounded derivatives near but not on the surface, this is referred to as a border-collision bifurcation. Dynamics near border-collision bifurcations are well approximated by piecewise-linear continuous maps. A lack of differentiability allows for substantial complexity and a wide variety of invariant sets can be created in border-collision bifurcations, such as invariant circles and chaotic sets, and several attractors may be created simultaneously. Yet many calculations that would be intractable for smooth nonlinear maps can be performed exactly for piecewise-linear maps and in recent years several new results have been obtained for border-collision bifurcations. This article reviews border-collision bifurcations with a general emphasis on results that apply to maps of any number of dimensions. The paper covers applicat...

Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form

The border-collision normal form is a piecewise-linear continuous map on ℝN that describes the dynamics near border-collision bifurcations of nonsmooth maps. This paper studies a codimension-three scenario at which the border-collision normal form with N = 2 exhibits infinitely many attracting periodic solutions. In this scenario there is a saddle-type periodic solution with branches of stable and unstable manifolds that are coincident, and an infinite sequence of attracting periodic solutions that converges to an orbit homoclinic to the saddle-type solution. Several important features of the scenario are shown to be universal, and three examples are given. For one of these examples, infinite coexistence is proved directly by explicitly computing periodic solutions in the infinite sequence.

Analytical expressions for power spectral density issued from one-dimensional continuous piecewise linear maps with three slopes

During the two last decades, chaotic signals have been increasingly considered in telecommunications, signal processing or secure transmissions. Despite the importance of frequency in the telecommunications and transmission security there are few works that explore the spectrum of chaotic signals by considering the bandwidths. Previous works have concerned skew tent maps. In this paper, we derive analytical expressions for the autocorrelation sequence (ACS) and power spectral density (PSD) of chaotic signals generated by one-dimensional continuous piecewise linear maps with three slopes. We obtain similar results to those obtained using the skew-tent map. Our results permit to conjecture that the bandwidth of chaotic signals obtained from piecewise linear continuous map can have the same behaviour whatever be the number of slopes in the map.

A bound on the number of periodic orbits of certain piecewise linear maps

Let f: [0, 1] → [0, 1] be a piecewise-linear continuous map. Let the slopes m i , i = 1, …, n, satisfy the condition that m i ϵ {± m j } j = 0 ∞, MϵN, ¦m¦ >1. Let θ N = { a bp N : (a, p) = 1, p is prime a bp N ϵ [0, 1]} , N ⩾ 1 , where we assume ( p, m) = 1. The main result states that f¦ θ N has a bounded number of periodic orbits independent of N.