BORDER-COLLISION CRISES IN A TWO-DIMENSIONAL MAP

1995 ◽  
Vol 05 (01) ◽  
pp. 275-279
Author(s):  
José Alvarez-Ramírez
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Piecewise Linear ◽  
Two Dimensional ◽  
Unstable Periodic Orbits ◽  
Straight Lines

We examine crisis phenomena for a map that is piecewise linear and depend continuously of a parameter λ0. There are two straight lines Γ+ and Γ− along which the map is continuous but has two one-sided derivatives. As the parameter λ0 is varied, a periodic orbit Ƶp may collide with the borders Γ+ and Γ− to disappear. While in most reported crisis structures, a chaotic attractor is destroyed by the presence of (homoclinic or heteroclinic) tangencies between unstable periodic orbits, in this case the chaotic attractor is destroyed by the birth of an attracting periodic orbit Ƶp into that of attraction of the chaotic set. The birth of Ƶp is due to a border-collision phenomenon taking place at Γ+ ∪Γ−.

Global Bifurcation Approximation in Controlling Chaotic Systems

1998 ◽  
Vol 08 (05) ◽  
pp. 1013-1023
Author(s):  
Byoung-Cheon Lee ◽  
Bong-Gyun Kim ◽  
Bo-Hyeun Wang
Keyword(s):  
Feedback Control ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Global Bifurcation ◽  
Good Control ◽  
Hysteresis Phenomenon ◽  
Unstable Periodic Orbits ◽  
Tracking Method ◽  
Adaptive Tracking

In our previous research [Lee et al., 1995], we demonstrated that return map control and adaptive tracking method can be used together to locate, stabilize and track unstable periodic orbits (UPO) automatically. Our adaptive tracking method is based on the control bifurcation (CB) phenomenon which is another route to chaos generated by feedback control. Along the CB route, there are numerous driven periodic orbits (DPOs), and they can be good control targets if small system modification is allowed. In this paper, we introduce a new control concept of global bifurcation approximation (GBA) which is quite different from the traditional local linear approximation (LLA). Based on this approach, we also demonstrate that chaotic attractor can be induced from a periodic orbit. If feedback control is applied along the direction to chaos, small erratic fluctuations of a periodic orbit is magnified and the chaotic attractor is induced. One of the special features of CB is the existence of irreversible orbit (IO) which is generated at the strong extreme of feedback control and has irreversible property. We show that IO induces a hysteresis phenomenon in CB, and we discuss how to keep away from IO.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

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.

scholarly journals Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems

2008 ◽  
Vol 15 (4) ◽  
pp. 675-680 ◽  
Author(s):  
Y. Saiki ◽  
M. Yamada
Keyword(s):  
Fluid Dynamics ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Space Physics ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Geophysical Fluid Dynamics ◽  
Chaotic Orbits ◽  
Low Dimensional ◽  
Complex Phenomena

Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

DYNAMICS OF THE MUTHUSWAMY–CHUA SYSTEM

2013 ◽  
Vol 23 (08) ◽  
pp. 1350136 ◽  
Author(s):  
YUANFAN ZHANG ◽  
XIANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Small Perturbations ◽  
Chaotic Phenomena ◽  
Invariant Surfaces

The Muthuswamy–Chua system [Formula: see text] describes the simplest electronic circuit which can have chaotic phenomena. In this paper, we first prove the existence of three families of consecutive periodic orbits of the system when α = 0, two of which are located on consecutive invariant surfaces and form two invariant topological cylinders. Then we prove that for α > 0 if the system has a periodic orbit or a chaotic attractor, it must intersect both of the planes z = 0 and z = -1 infinitely many times as t tends to infinity. As a byproduct, we get an example of unstable invariant topological cylinders which are not normally hyperbolic and which are also destroyed under small perturbations.

DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

2012 ◽  
Vol 22 (01) ◽  
pp. 1230001
Author(s):  
BENJAMIN COY
Keyword(s):  
Learning Community ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Topological Analysis ◽  
Three Dimensions ◽  
Locally Linear Embedding ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Linear Embedding ◽  
Locally Linear

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

Improved Estimations of Unstable Periodic Orbits Extracted From Chaotic Sets

2001 ◽  
Author(s):  
Z. Al-Zamel ◽  
B. F. Feeny
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Duffing Oscillator ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Affine Map ◽  
Saddle Type ◽  
Recurrence Method

Abstract Unstable periodic orbits of the saddle type are often extracted from chaotic sets. We use the recurrence method of extracting segments of the chaotic data to approximate the true unstable periodic orbit. Then nearby trajectories are then examined to obtain the dynamics local to the extracted orbit, in terms of an affine map. The affine map is then used to estimate the true orbit. Accuracy is evaluated in examples including well known maps and the Duffing oscillator.

scholarly journals Computing and Controlling Basins of Attraction in Multistability Scenarios

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
John Alexander Taborda ◽  
Fabiola Angulo
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Control Technique ◽  
Unstable Periodic Orbits ◽  
Control Effort ◽  
Coexistence Of Attractors ◽  
And Control

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

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