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.

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.

Optimal transfers between unstable periodic orbits using invariant manifolds

2011 ◽  
Vol 109 (3) ◽  
pp. 241-264 ◽  
Author(s):  
Kathryn E. Davis ◽  
Rodney L. Anderson ◽  
Daniel J. Scheeres ◽  
George H. Born
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Unstable Periodic Orbits ◽  
Optimal Transfers

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.

scholarly journals Phase Space Structure and Transport in a Caldera Potential Energy Surface

2018 ◽  
Vol 28 (13) ◽  
pp. 1830042 ◽  
Author(s):  
Matthaios Katsanikas ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Potential Energy ◽  
Potential Energy Surface ◽  
Periodic Orbits ◽  
Central Region ◽  
Invariant Manifolds ◽  
Energy Surface ◽  
Unstable Periodic Orbits ◽  
Phase Space Transport ◽  
The Family

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

scholarly journals INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL

2013 ◽  
Vol 23 (02) ◽  
pp. 1330005 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Rotational Axis ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits ◽  
Asymptotic Curves ◽  
Axis Orbit ◽  
Diffusion Speed ◽  
Transition Points ◽  
Sticky Chaotic Orbits

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

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−.

Existence of Homoclinic Cycles and Periodic Orbits in a Class of Three-Dimensional Piecewise Affine Systems

2018 ◽  
Vol 28 (02) ◽  
pp. 1850024 ◽  
Author(s):  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Homoclinic Bifurcation ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

For a class of three-dimensional piecewise affine systems, this paper focuses on the existence of homoclinic cycles and the phenomena of homoclinic bifurcation leading to periodic orbits. Based on the spatial location relation between the invariant manifolds of subsystems and the switching manifold, a concise necessary and sufficient condition for the existence of homoclinic cycles is obtained. Then the homoclinic bifurcation is studied and the sufficient conditions for the birth of a periodic orbit are obtained. Furthermore, the sufficient conditions are obtained for the periodic orbit to be a sink, a source or a saddle. As illustrations, several concrete examples are presented.

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