Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

[+] Author and Article Information
Albert C. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805aluo@siue.edu

J. Comput. Nonlinear Dynam 1(2), 135-142 (Jul 11, 2005) (8 pages) doi:10.1115/1.2162868 History: Received April 17, 2005; Revised July 11, 2005

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Copyright © 2006 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Maximum equipotential curves for given energies h={−0.75,−0.5,⋯,1.0}. The equilibrium points (±2,±2) with the minimum potential energy Vmin=−1.0 (filled circles). The equilibrium point (0,0) is connected by the curve with maximum equipotential energy with h=0.

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Figure 2

Maximum and minimum energy spectrums varying with displacement x1(h=0) on the sub-Poincaré mapping surface Σ0Ξ21 in (a) the right potential well and (b) the left-potential well (α=β=γ=ε=1, c=0). The white-filled, shaded, and gray-filled areas are quasi-periodic, resonant layer, and chaotic motion, respectively. The dashed vertical lines are the boundaries separated from one motion to another. The dark solid vertical line is the boundary for nonmotion.

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Figure 3

The domains on the potential well for chaotic and periodic motions: (a) overall view and (b) detailed view (α=β=γ=ε=1). The shaded areas are for chaotic motion. The white areas inside the boundary are for quasi-periodic motion (QP) and resonance layer (RL). NR represent the new resonance.

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Figure 4

The Poincaré surfaces of chaotic motions on the homoclinic surface with the chosen initial condition (x1,x2;y1,y2)≈(±0.86,0.0;±0.6827,0.0): (a) ΣhΞ21 on (x1,y1), (b) ΠhP on (x1,x2), and (c) ΣhΞ2 on (x1,y1,x2), (α=β=γ=ε=1).

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Figure 5

The motion on the subspace (x1,y1,x2) of the homoclinic surface (h=0): (a) chaotic motion with the initial condition (x1,x2;y1,y2)≈(−0.87,0.2,0.2871,0.0) and (b) quasi-periodic motion with the initial condition (x1,x2;y1,y2)≈(−0.82,0.2,0.2799,0.0), (α=β=γ=ε=1).

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Figure 6

Chaotic motion on the homoclinic surface (h=0) projected on: (a) phase plane (x1,y1), (b) phase plane (x2,y2), (c) displacement subspace (or potential domain), and (d) velocity subspace (or kinetic energy domain). The initial condition is (x1,x2;y1,y2)≈(−0.87,0.2,0.2871,0.0), (α=β=γ=ε=1).

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Figure 7

Quasi-periodic motion on the homoclinic surface (h=0) projected on: (a) phase plane (x1,y1), (b) phase plane (x2,y2), (c) displacement subspace (or potential domain), and (d) velocity subspace (or kinetic energy domain). The initial condition is (x1,x2;y1,y2)≈(−0.82,0.2,0.2799,0.0), (α=β=γ=ε=1).



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