Periodic orbit scar in wavepacket propagation

2019 ◽  
Vol 30 (04) ◽  
pp. 1950026
Author(s):  
Mitsuyoshi Tomiya ◽  
Shoichi Sakamoto ◽  
Eric J. Heller
Keyword(s):  
Periodic Orbit ◽  
Energy Range ◽  
Periodic Orbits ◽  
Semiclassical Approximation ◽  
Stationary States ◽  
Unstable Periodic Orbits ◽  
Time Average ◽  
Wavepacket Propagation

This study analyzed the scar-like localization in the time-average of a time-evolving wavepacket on a desymmetrized stadium billiard. When a wavepacket is launched along the orbits, it emerges on classical unstable periodic orbits as a scar in stationary states. This localization along the periodic orbit is clarified through the semiclassical approximation. It essentially originates from the same mechanism of a scar in stationary states: piling up of the contribution from the classical actions of multiply repeated passes on a primitive periodic orbit. To achieve this, several states are required in the energy range determined by the initial wavepacket.

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.

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.

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.

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−.

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

1998 ◽  
Vol 08 (09) ◽  
pp. 1783-1789 ◽  
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 Orbits ◽  
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

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 Orbits ◽  
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.

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

2011 ◽  
Vol 21 (08) ◽  
pp. 2331-2342 ◽  
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 Orbits ◽  
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.

Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

2015 ◽  
Vol 25 (13) ◽  
pp. 1550185 ◽  
Author(s):  
Tetsushi Ueta ◽  
Daisuke Ito ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Transient Behavior ◽  
Control Input ◽  
Stable Limit ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Saddle Node

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

scholarly journals Using Lagrangian Descriptors to Uncover Invariant Structures in Chesnavich’s Isokinetic Model with Application to Roaming

2020 ◽  
Vol 30 (05) ◽  
pp. 2050076
Author(s):  
Vladimír Krajňák ◽  
Gregory S. Ezra ◽  
Stephen Wiggins
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Unstable Periodic Orbits ◽  
Invariant Structures ◽  
Text Model ◽  
Model Subject ◽  
Lagrangian Descriptors

Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case, we use the parametrization of a periodic orbit to construct a Lagrangian descriptor that will be locally minimized on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH[Formula: see text] model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational requirements, the method enables us to study the invariant structures responsible for roaming in the isokinetic Chesnavich CH[Formula: see text] model.

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