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

2005 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a 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 energy incremental energy approach. The Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The chaotic and quasiperiodic 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 systems. The resonant-periodic motions 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.

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

2005 ◽  
Vol 1 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Hamiltonian System ◽  
Two Degree Of Freedom

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.

Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

2003 ◽  
Vol 70 (5) ◽  
pp. 732-738
Author(s):  
K. Yagasaki
Keyword(s):  
Differential Equation ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Degree Of Freedom ◽  
Differential Galois Theory ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Infinite Degree ◽  
Simulation Results

We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

scholarly journals On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies

2021 ◽  
Vol 17 (1) ◽  
pp. 77-102
Author(s):  
O. V. Kholostova ◽  
Keyword(s):  
Hamiltonian System ◽  
Normal Form ◽  
Small Parameter ◽  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Periodic Motions ◽  
Half Integer ◽  
Stability And Instability ◽  
Trivial Equilibrium ◽  
Two Parameters

This paper examines the motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small; when it has zero value, the system is autonomous. Consideration is given to a set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of the linearized equations of perturbed motion are identical and are integer or half-integer numbers (the case of multiple parametric resonance). It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form in terms through degree four in perturbations and up to various degrees in a small parameter (systems of first, second and third approximations). The structure of the regions of stability and instability of trivial equilibrium is investigated, and solutions are obtained to the problems of the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described. As an application, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.

Stability and Bifurcation for the Equispaced, Periodic Motion of a Horizontal Impact Damper

2001 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Impact Damper ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Good Agreement

Abstract Stability and bifurcation conditions for the asymmetric, periodic motion of a horizontal impact damper under a periodic excitation are developed through four mappings for two switch-planes relative to discontinuities. Period-doubling bifurcation for equispaced motion does not occur, but the asymmetric period-1 motions change to the asymmetric, period-2 ones through a period doubling bifurcation. A numerical prediction for equispaced to chaotic motions is completed. The numerical and analytical predictions of the periodic motion are in very good agreement. The asymmetric, periodic motions are also simulated.

Global Chaos in a Periodically Forced, Linear System With a Dead-Zone Restoring Force

2003 ◽  
Author(s):  
Albert C. J. Luo ◽  
Santhosh Menon
Keyword(s):  
Linear System ◽  
Periodic Motion ◽  
Dead Zone ◽  
Poincaré Mapping ◽  
Periodic Motions ◽  
Restoring Force ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Poincare Mapping ◽  
The Stability

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are developed through the switching planes pertaining to the two constraints. The global periodic motions based on the Poincare mapping are determined, and the analysis for the stability and bifurcation of periodic motion is carried out. From the global periodic motions, the global chaos in such a system is investigated numerically. The bifurcation scenario with varying parameters was presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – II

2021 ◽  
Vol 31 (12) ◽  
pp. 2150188
Author(s):  
Matthaios Katsanikas ◽  
Stephen Wiggins
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbit ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Dimensional Space ◽  
New Method ◽  
Normally Hyperbolic Invariant Manifold ◽  
Dividing Surface ◽  
Dimensional Object ◽  
Three Degrees Of Freedom

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [ Katsanikas & Wiggins, 2021 ]. Similar to that method, for an [Formula: see text] degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a [Formula: see text] dimensional geometrical object in the energy surface of a [Formula: see text] dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.

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