Optimal transfers between unstable periodic orbits using invariant manifolds

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.

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Breakdown mechanisms of normally hyperbolic invariant manifolds in terms of unstable periodic orbits and homoclinic/heteroclinic orbits in Hamiltonian systems

Nonlinearity ◽

10.1088/0951-7715/28/8/2677 ◽

2015 ◽

Vol 28 (8) ◽

pp. 2677-2698 ◽

Cited By ~ 11

Author(s):

Hiroshi Teramoto ◽

Mikito Toda ◽

Tamiki Komatsuzaki

Keyword(s):

Periodic Orbits ◽

Hamiltonian Systems ◽

Invariant Manifolds ◽

Heteroclinic Orbits ◽

Unstable Periodic Orbits ◽

Normally Hyperbolic Invariant Manifolds

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INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741330005x ◽

2013 ◽

Vol 23 (02) ◽

pp. 1330005 ◽

Cited By ~ 11

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.

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ROTATION NUMBERS OF INVARIANT MANIFOLDS AROUND UNSTABLE PERIODIC ORBITS FOR THE DIAMAGNETIC KEPLER PROBLEM

Fractals ◽

10.1142/s0218348x08003831 ◽

2008 ◽

Vol 16 (01) ◽

pp. 11-23

Author(s):

ZUO-BING WU

Keyword(s):

Periodic Orbits ◽

Invariant Manifolds ◽

Kepler Problem ◽

Poincaré Section ◽

Unstable Periodic Orbits ◽

Poincare Section ◽

Rotation Numbers ◽

Original Definition ◽

Symbolic Sequences ◽

Forward Mapping

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincaré section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincaré section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.

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APPROACHING NONLINEAR DYNAMICS BY STUDYING THE MOTION OF A PENDULUM III: PREDICTABILITY AND CONTROL OF CHAOTIC MOTION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000575 ◽

1994 ◽

Vol 04 (04) ◽

pp. 773-784 ◽

Cited By ~ 9

Author(s):

B. HÜBINGER ◽

R. DOERNER ◽

H. HENG ◽

W. MARTIENSSEN

Keyword(s):

Periodic Orbits ◽

Chaotic Motion ◽

Invariant Manifolds ◽

Control Method ◽

Equations Of Motion ◽

Unstable Periodic Orbits ◽

Physical Measure ◽

Control Scheme ◽

Feedback Control Method ◽

And Control

We apply the concepts of predictability and control of chaotic motion to the driven damped pendulum. A physical measure of predictability is defined and determined from experimental data as well as from the equations of motion. The results are presented in predictability portraits which constitute an intrinsic pattern of zones of varying predictability. The origin of these patterns is related to the unstable periodic orbits and their invariant manifolds within the attractor. In order to control the chaotic motion of the pendulum we implement an extension of the OGY feedback control method, which we call “local control method.” With this control scheme any motion of the pendulum which is a solution of the systems equations of motion can be stabilized. We apply the control formalism in order to stabilize experimentally unstable periodic orbits as well as arbitrarily chosen chaotic trajectories.

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Lobe dynamics and the escape from a potential well

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1991.0169 ◽

1991 ◽

Vol 435 (1895) ◽

pp. 659-672 ◽

Cited By ~ 8

Keyword(s):

Periodic Orbits ◽

Invariant Manifolds ◽

Numerical Technique ◽

Basin Of Attraction ◽

Invariant Sets ◽

Potential Well ◽

Unstable Periodic Orbits ◽

Homoclinic Tangle ◽

Lobe Dynamics ◽

Basin Erosion

Periodically-forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled. The trellises formed by such overlapping manifolds separate the Poincaré plane into regions (called lobes ) whose fates at successive time periods are amenable to analysis. The lobe configurations observed in simple escape systems are discussed, together with their interrelation with the location of periodic orbits and other invariant sets. Three applications of this procedure are illustrated: (i) a numerical technique for localizing the non-wandering set; (ii) a demonstration that the existence of a saddle possessing a bounded homoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction; (iii) the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500765 ◽

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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