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

2014 ◽  
Vol 24 (06) ◽  
pp. 1430018 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Normal Form ◽  
Periodic Solutions ◽  
Infinite Sequence ◽  
Piecewise Linear ◽  
Coexisting Attractors ◽  
Stable And Unstable Manifolds ◽  
Continuous Map ◽  
Border Collision Bifurcations ◽  
Saddle Type ◽  
Unstable Manifolds

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.

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 Scaling Laws for Large Numbers of Coexisting Attracting Periodic Solutions in the Border-Collision Normal Form

2014 ◽  
Vol 24 (09) ◽  
pp. 1450118 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Solution ◽  
Normal Form ◽  
Periodic Solutions ◽  
Parameter Space ◽  
Scaling Laws ◽  
Infinite Sequence ◽  
Two Dimensions ◽  
Large Numbers ◽  
Stable Periodic ◽  
Saddle Type

A wide variety of intricate dynamics may be created at border-collision bifurcations of piecewise-smooth maps, where a fixed point collides with a surface at which the map is nonsmooth. For the border-collision normal form in two dimensions, a codimension-three scenario was described in previous work at which the map has a saddle-type periodic solution and an infinite sequence of stable periodic solutions that limit to a homoclinic orbit of the saddle-type solution. This paper introduces an alternate scenario of the same map at which there is an infinite sequence of stable periodic solutions due to the presence of a repeated unit eigenvalue in the linearization of some iterate of the map. It is shown that this scenario is codimension-four and that the sequence of periodic solutions is unbounded, aligning with eigenvectors corresponding to the unit eigenvalue. Arbitrarily many attracting periodic solutions coexist near either scenario. It is shown that if K denotes the number of attracting periodic solutions, and ε denotes the distance in parameter space from one of the two scenarios, then in the codimension-three case ε scales with λ-K, where λ > 1 denotes the unstable stability multiplier associated with the saddle-type periodic solution, and in the codimension-four case ε scales with K-2. Since K-2 decays significantly slower than λ-K, large numbers of attracting periodic solutions coexist in open regions of parameter space extending substantially further from the codimension-four scenarios than the codimension-three scenarios.

scholarly journals Locating oscillatory orbits of the parametrically-excited pendulum

1996 ◽  
Vol 37 (3) ◽  
pp. 309-319 ◽  
Author(s):  
M. J. Clifford ◽  
S. R. Bishop
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Numerical Techniques ◽  
Stable And Unstable Manifolds ◽  
Parametrically Excited ◽  
Countable Infinity ◽  
Unstable Manifolds

AbstractA method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

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 Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

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.

SADDLE-TYPE TORUS BRAIDS IN QUASIPERIODICALLY DRIVEN SYSTEMS: CROSSOVER MAPS OF TRANSVERSALLY INTERSECTING MANIFOLDS

2008 ◽  
Vol 18 (10) ◽  
pp. 3001-3012 ◽  
Author(s):  
ZACHARY H. ZIBRAT ◽  
ANDREW J. SZERI
Keyword(s):  
Cross Sections ◽  
Nonlinear Oscillators ◽  
Stable And Unstable Manifolds ◽  
Poincaré Maps ◽  
Driven Systems ◽  
And Topology ◽  
Saddle Type ◽  
Unstable Manifolds ◽  
Slice Method ◽  
Insight Into

In [Spears et al., 2005] it was demonstrated that insight into the geometry and topology of attractors for nonlinear oscillators driven by n incommensurate frequencies may be obtained from the study of n Poincaré maps defined on global cross-sections. The attractors take the form of stable torus braids. Here, attention is focused on saddle-type torus braids in similar systems. Transverse intersections of stable and unstable manifolds are computed using the phase slice method, along with the application of the crossover map. This neatly maps from one global Poincaré section to another. As such, it can be used to compute lobe intersection geometries in the remaining n - 1 Poincaré sections after doing so by the phase slice method in the first.

BCB Curves and Contact Bifurcations in Piecewise Linear Discontinuous Map Arising in a Financial Market

2019 ◽  
Vol 29 (02) ◽  
pp. 1950022 ◽  
Author(s):  
En-Guo Gu ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Financial Market ◽  
Structural Changes ◽  
Piecewise Linear ◽  
Invariant Set ◽  
Market Model ◽  
Stable And Unstable Manifolds ◽  
Model Dynamics ◽  
Discontinuous Map ◽  
Unstable Manifolds

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the [Formula: see text]-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

Heteroclinic bifurcations in damped rigid block motion

1992 ◽  
Vol 439 (1905) ◽  
pp. 155-162 ◽  
Keyword(s):  
Linear System ◽  
Perturbation Methods ◽  
Piecewise Linear ◽  
Exact Expression ◽  
Rigid Block ◽  
Stable And Unstable Manifolds ◽  
Piecewise Linear System ◽  
Block Motion ◽  
Rocking Motion ◽  
Unstable Manifolds

It is shown that heteroclinic bifurcations are present in a piecewise-linear system of ordinary differential equations that describe the rocking motion of a slender rigid block with damping. An exact expression is given for the bifurcation amplitude. Stable and unstable manifolds are analytically extended to explicitly reveal the intersections. As the damping increases, these bifurcations occur only at increasingly large forcing amplitudes, as manifolds move further apart. No perturbation methods are used in this analysis.

Bifurcation of Quasiperiodic Orbit in a 3D Piecewise Linear Map

2017 ◽  
Vol 27 (10) ◽  
pp. 1730033 ◽  
Author(s):  
Mahashweta Patra ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Bundle Method ◽  
Piecewise Linear Map ◽  
Border Collision Bifurcations ◽  
Unstable Manifolds ◽  
Smooth Map ◽  
Smooth System

Earlier investigations have demonstrated how a quasiperiodic orbit in a three-dimensional smooth map can bifurcate into a quasiperiodic orbit with two disjoint loops or into a quasiperiodic orbit of double the length in the shape of a Möbius strip. Using a three-dimensional piecewise smooth (PWS) normal form map, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of creation of tori with multiple loops may result from border collision bifurcations. We also illustrate the occurrence of multiple attractor bifurcations due to the interplay between the stable and the unstable manifolds. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in PWS systems, but fail in some cases — especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by change in eigenvalues of the unstable fixed points.

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