Multiple Attractor Bifurcation in Three-Dimensional Piecewise Linear Maps

2018 ◽  
Vol 28 (10) ◽  
pp. 1830032 ◽  
Author(s):  
Mahashweta Patra
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Chaotic Orbit ◽  
Stable And Unstable Manifolds ◽  
Period 2 ◽  
Coexistence Of Attractors ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Unstable Manifolds ◽  
Fundamental Uncertainty

Multiple attractor bifurcations lead to simultaneous creation of multiple stable orbits. This may be damaging for practical systems as there is a fundamental uncertainty regarding which orbit the system will follow after a bifurcation. Such bifurcations are known to occur in piecewise smooth maps, which model many practical and engineering systems. So far the occurrence of such bifurcations have been investigated in the context of 2D piecewise linear maps. In this paper, we investigate multiple attractor bifurcations in a three-dimensional piecewise linear normal form map. We show the occurrence of different types of multiple attractor bifurcations in the system, like the simultaneous creation of a period-2 orbit, a period-3 orbit and an unstable chaotic orbit; a mode-locked torus, an ergodic torus and periodic orbits; a one-loop torus and a two-loop torus; a one-loop mode-locked torus and a two-loop mode-locked torus; a one-piece chaotic orbit and a 3-piece chaotic orbit, etc. As orbits lie on unstable manifolds of fixed points, the structure of unstable manifold plays an important role in understanding the coexistence of attractors. In this work, we show that interplay between 1D and 2D stable and unstable manifolds plays an important role in global bifurcations that can give rise to multiple coexisting attractors.

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

2017 ◽  
Vol 27 (02) ◽  
pp. 1730010 ◽  
Author(s):  
David J. W. Simpson ◽  
Christopher P. Tuffley
Keyword(s):  
Periodic Solutions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Stable And Unstable Manifolds ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Continuous Maps ◽  
Piecewise Linear Maps ◽  
Stable Periodic ◽  
Unstable Manifolds

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.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

Estimates of the topological degree of a class of piecewise linear maps with applications

2021 ◽  
pp. 2150073
Author(s):  
Laura Poggiolini ◽  
Marco Spadini
Keyword(s):  
Topological Degree ◽  
Piecewise Linear ◽  
Nonlinear Problems ◽  
Text Structure ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Classical Integral ◽  
Computation Formula

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.

Statistics of a family of piecewise linear maps

2021 ◽  
Vol 427 ◽  
pp. 133019
Author(s):  
J.J.P. Veerman ◽  
P.J. Oberly ◽  
L.S. Fox
Keyword(s):  
Piecewise Linear ◽  
Linear Maps ◽  
Piecewise Linear Maps

scholarly journals On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

2016 ◽  
Vol 16 (03) ◽  
pp. 1660013 ◽  
Author(s):  
Nigel P. Byott ◽  
Congping Lin ◽  
Yiwei Zhang
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Interval Exchange ◽  
Expanding Maps ◽  
Mixing Properties ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Fold Map ◽  
Constant Slope ◽  
Piecewise Expanding Maps

For an integer [Formula: see text], let [Formula: see text] be the partition of the unit interval [Formula: see text] into [Formula: see text] equal subintervals, and let [Formula: see text] be the class of piecewise linear maps on [Formula: see text] with constant slope [Formula: see text] on each element of [Formula: see text]. We investigate the effect on mixing properties when [Formula: see text] is composed with the interval exchange map given by a permutation [Formula: see text] interchanging the [Formula: see text] subintervals of [Formula: see text]. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map [Formula: see text].

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