Mel'nikov analysis of phase space transport in a N-degree-of-freedom Hamiltonian system

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

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Tilting and Squeezing: Phase Space Geometry of Hamiltonian Saddle-Node Bifurcation and its Influence on Chemical Reaction Dynamics

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420300086 ◽

2020 ◽

Vol 30 (04) ◽

pp. 2030008 ◽

Cited By ~ 6

Author(s):

Víctor J. García-Garrido ◽

Shibabrat Naik ◽

Stephen Wiggins

Keyword(s):

Phase Space ◽

Hamiltonian System ◽

Potential Energy ◽

Invariant Manifolds ◽

Reaction Dynamics ◽

Potential Energy Function ◽

Space Structures ◽

Stable And Unstable Manifolds ◽

Saddle Node Bifurcation ◽

Saddle Node

In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

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WEAK DISSIPATION IN A HAMILTONIAN MAP

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000654 ◽

1994 ◽

Vol 04 (04) ◽

pp. 921-932 ◽

Cited By ~ 1

Author(s):

RAÚL J. MONDRAGÓN C. ◽

PETER H. RICHTER

Keyword(s):

Phase Space ◽

Hamiltonian System ◽

Periodic Orbits ◽

Adiabatic Invariance ◽

Bifurcation Tree ◽

Bouncing Ball ◽

Different Types ◽

Weak Dissipation ◽

Hamiltonian Map

The dynamics of a bouncing ball reflected off a harmonic spring is investigated, with weak dissipation of three different types. The phase space is found to be organized into a system of tubes that wind around the branches of the bifurcation tree of periodic orbits of the Hamiltonian system. Instead of attraction towards special periodic orbits we observe a kind of piecewise adiabatic invariance of the tubes, with jumps occurring when the branches penetrate each other.

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Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

Nonlinear Processes in Geophysics ◽

10.5194/npg-5-69-1998 ◽

1998 ◽

Vol 5 (2) ◽

pp. 69-74 ◽

Cited By ~ 11

Author(s):

M. G. Brown

Keyword(s):

Phase Space ◽

Time Dependence ◽

Hamiltonian Systems ◽

Fluid Flows ◽

Space Structure ◽

Degree Of Freedom ◽

Periodic Functions ◽

Phase Space Structure ◽

Fractal Properties ◽

Incompressible Fluid Flows

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

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