Improved Estimations of Unstable Periodic Orbits Extracted From Chaotic Sets

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.

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

Nonlinear Processes in Geophysics ◽

10.5194/npg-15-675-2008 ◽

2008 ◽

Vol 15 (4) ◽

pp. 675-680 ◽

Cited By ~ 2

Author(s):

Y. Saiki ◽

M. Yamada

Keyword(s):

Fluid Dynamics ◽

Periodic Orbit ◽

Periodic Orbits ◽

Space Physics ◽

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.

Dynamic Stabilization of Unstable Periodic Orbits in a CO2 Laser by Slow Modulation of Cavity Detuning

10.1142/s0218127498001492 ◽

1998 ◽

Vol 08 (09) ◽

pp. 1783-1789 ◽

Cited By ~ 6

Author(s):

A. N. Pisarchik ◽

R. Corbalán ◽

V. N. Chizhevsky ◽

R. Vilaseca ◽

B. F. Kuntsevich

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Transient Response ◽

Co2 Laser ◽

Dynamic Stabilization ◽

Dynamical Regime ◽

Unstable Periodic Orbit ◽

Periodic Attractor ◽

Slow Modulation

We demonstrate numerically and experimentally that a slow modulation of cavity detuning in a loss-modulated CO 2 laser can stabilize unstable periodic orbits even when the system remains in a particular dynamical regime for adiabatic changes of the detuning. When the parameter changes faster than the transient response of deformation of the original periodic attractor, the system can evolve toward an unstable periodic orbit.

Stabilizing Periodic Orbits of the Fractional Order Chaotic Van Der Pol System

Volume 8: Dynamic Systems and Control, Parts A and B ◽

10.1115/imece2010-40165 ◽

2010 ◽

Author(s):

Mohammad A. Rahimi ◽

Hasan Salarieh ◽

Aria Alasty

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Fractional Order ◽

Linear Feedback ◽

Order System ◽

Fractional Order Systems ◽

Unstable Periodic Orbit ◽

Van Der Pol ◽

Linear Feedback Controller

In this paper, stabilizing the unstable periodic orbits (UPO) in a chaotic fractional order system called Van der Pol is studied. Firstly, a technique for finding unstable periodic orbit in chaotic fractional order systems is stated. Then by applying this technique to the van der Pol system, unstable periodic orbit of system is found. After that, a method is presented for stabilization of the discovered UPO based on theories stability of the linear integer order and fractional order systems. Finally, a linear feedback controller was applied to the system and simulation is done for demonstration of controller performance.

CHAINS OF ROTATIONAL TORI AND FILAMENTARY STRUCTURES CLOSE TO HIGH MULTIPLICITY PERIODIC ORBITS IN A 3D GALACTIC POTENTIAL

10.1142/s0218127411029823 ◽

2011 ◽

Vol 21 (08) ◽

pp. 2331-2342 ◽

Cited By ~ 9

Author(s):

M. KATSANIKAS ◽

P. A. PATSIS ◽

A. D. PINOTSIS

Keyword(s):

Phase Space ◽

Hamiltonian System ◽

Periodic Orbit ◽

Periodic Orbits ◽

Color Variation ◽

Space Structures ◽

Unstable Periodic Orbit ◽

Surface Of Section ◽

Autonomous Hamiltonian System

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis & Zachilas, 1994] in order to visualize them. As examples, we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition, we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

Global Bifurcation Approximation in Controlling Chaotic Systems

10.1142/s0218127498000826 ◽

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 ◽

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.

Computing and Controlling Basins of Attraction in Multistability Scenarios

Mathematical Problems in Engineering ◽

10.1155/2015/313154 ◽

2015 ◽

Vol 2015 ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

John Alexander Taborda ◽

Fabiola Angulo

Keyword(s):

Fixed Point ◽

Periodic Orbit ◽

Periodic Orbits ◽

Basin Of Attraction ◽

Basins Of Attraction ◽

Control Technique ◽

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

10.1142/s0218127493000593 ◽

1993 ◽

Vol 03 (03) ◽

pp. 685-691 ◽

Cited By ~ 4

Author(s):

J.W.L. McCALLUM ◽

R. GILMORE

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Chaotic Attractor ◽

Geometric Model ◽

Duffing Oscillator ◽

Chaotic Attractors ◽

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.

THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

10.1142/s0218127411029811 ◽

2011 ◽

Vol 21 (08) ◽

pp. 2321-2330 ◽

Cited By ~ 11

Author(s):

M. KATSANIKAS ◽

P. A. PATSIS ◽

G. CONTOPOULOS

Keyword(s):

Hopf Bifurcation ◽

Periodic Orbits ◽

Transition Point ◽

Initial Conditions ◽

Unstable Periodic Orbit ◽

Surface Of Section ◽

Autonomous Hamiltonian System ◽

Hamiltonian Hopf Bifurcation ◽

Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

BORDER-COLLISION CRISES IN A TWO-DIMENSIONAL MAP

10.1142/s0218127495000235 ◽

1995 ◽

Vol 05 (01) ◽

pp. 275-279

Author(s):

José Alvarez-Ramírez

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Chaotic Attractor ◽

Piecewise Linear ◽

Two Dimensional ◽

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−.

Spatiotemporal Distributions of Unstable Periodic Orbits in Noisy Coupled Chaotic Systems

10.1142/s021812740300817x ◽

2003 ◽

Vol 13 (09) ◽

pp. 2673-2680

Author(s):

Kevin Dolan ◽

Annette Witt ◽

Jürgen Kurths ◽

Frank Moss

Keyword(s):

Periodic Orbits ◽

Coupling Structure ◽

Spatially Extended ◽

Experimental Time Series ◽

Linear Correlation Analysis ◽

Spatially Extended Systems ◽

Recurrence Method ◽

Dynamical Information ◽

Insight Into

Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rössler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a linear correlation analysis of the spatiotemporal occurrence of encounters with these low period UPOs. This analysis can provide an insight into the coupling structure of such a spatially extended system.