Canonical Coordinates Defined on a Curved Poincaré Section and a Relation to Micro-Canonical Averages in Nonlinear Hamiltonian Dynamical System

Abstract Understanding of dynamical behavior in the parameter-state space plays a vital role in the optimal design and motion control of mechanical governor systems. By combining the GPU parallel computing technique with two determinate indicators, namely, the Lyapunov exponents and Poincaré section, this paper presents a detailed study on the two-parameter dynamics of a mechanical governor system with different time delays. By identifying different system responses in two-parameter plane, it is shown that the complexity of evolutionary process can increase significantly with the increase of time delay. The path-following strategy and the time domain collocation method are used to explore the details of the evolutionary process. An interesting phenomenon is found in the dynamical behavior of the delayed governor system, which can cause the inconsistency between the number of intersection points of certain periodic response on Poincaré section and the actual period characteristic. For example, the commonly exhibited period-1 orbit may have two or more intersection points on the Poincaré section instead of one point. Furthermore, the variations of basins of attraction are also discussed in the plane of initial history conditions to demonstrate the observed multistability phenomena and chaotic transitions.

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Smale spaces via inverse limits

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.19 ◽

2013 ◽

Vol 34 (6) ◽

pp. 2066-2092 ◽

Cited By ~ 6

Author(s):

SUSANA WIELER

Keyword(s):

Dynamical System ◽

Inverse Limits ◽

Canonical Coordinates ◽

Chaotic Dynamical System ◽

Smale Space ◽

Smale Spaces ◽

Special Case ◽

Basic Sets

AbstractA Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom $A$ systems are a key class of examples. We consider the special case of irreducible Smale spaces with zero-dimensional contracting directions, and characterize these as stationary inverse limits satisfying certain conditions.

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The onset of chaos in vortex sheet flow

Journal of Fluid Mechanics ◽

10.1017/s0022112001007066 ◽

2002 ◽

Vol 454 ◽

pp. 47-69 ◽

Cited By ~ 29

Author(s):

ROBERT KRASNY ◽

MONIKA NITSCHE

Keyword(s):

Vortex Ring ◽

Vortex Core ◽

Axisymmetric Flow ◽

Period Doubling ◽

Vortex Sheet ◽

Small Scale ◽

Poincaré Section ◽

Poincare Section ◽

Sheet Flow ◽

Onset Of Chaos

Regularized point-vortex simulations are presented for vortex sheet motion in planar and axisymmetric flow. The sheet forms a vortex pair in the planar case and a vortex ring in the axisymmetric case. Initially the sheet rolls up into a smooth spiral, but irregular small-scale features develop later in time: gaps and folds appear in the spiral core and a thin wake is shed behind the vortex ring. These features are due to the onset of chaos in the vortex sheet flow. Numerical evidence and qualitative theoretical arguments are presented to support this conclusion. Past the initial transient the flow enters a quasi-steady state in which the vortex core undergoes a small-amplitude oscillation about a steady mean. The oscillation is a time-dependent variation in the elliptic deformation of the core vorticity contours; it is nearly time-periodic, but over long times it exhibits period-doubling and transitions between rotation and nutation. A spectral analysis is performed to determine the fundamental oscillation frequency and this is used to construct a Poincaré section of the vortex sheet flow. The resulting section displays the generic features of a chaotic Hamiltonian system, resonance bands and a heteroclinic tangle, and these features are well-correlated with the irregular features in the shape of the vortex sheet. The Poincaré section also has KAM curves bounding regions of integrable dynamics in which the sheet rolls up smoothly. The chaos seen here is induced by a self-sustained oscillation in the vortex core rather than external forcing. Several well-known vortex models are cited to justify and interpret the results.

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Aspects of laser Lorenz dynamics

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000001033x ◽

1994 ◽

Vol 36 (2) ◽

pp. 152-174 ◽

Cited By ~ 1

Author(s):

P. B. Chapman

Keyword(s):

Hopf Bifurcation ◽

Poincaré Section ◽

Poincare Section ◽

Lorenz Equations

AbstractThe laser Lorenz equations are studied by reducing them to a form suitable for application of an extension of a method developed by Kuzmak. The method generates a flow in a Poincaré section from which it is inferred that a certain Hopf bifurcation is always subcritical.

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COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021762 ◽

2008 ◽

Vol 18 (08) ◽

pp. 2391-2396 ◽

Cited By ~ 3

Author(s):

QUAN YUAN ◽

XIAO-SONG YANG

Keyword(s):

Poincaré Map ◽

Chaotic Behavior ◽

Computer Assisted ◽

Poincaré Section ◽

Poincare Map ◽

Poincare Section ◽

Numerical Studies ◽

Topological Horseshoes

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

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Constructing the time independent Hamiltonian from a time dependent one

Open Physics ◽

10.2478/s11534-007-0024-7 ◽

2007 ◽

Vol 5 (3) ◽

Cited By ~ 1

Author(s):

Michał Dobrski

Keyword(s):

Dynamical System ◽

Harmonic Oscillator ◽

Time Dependent ◽

Damped Harmonic Oscillator ◽

Hamiltonian Dynamical System ◽

Equivalent Time ◽

New Time

AbstractIn this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.

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Horseshoe Chaos in a Simple Memristive Circuit

Journal of Applied Mathematics ◽

10.1155/2014/546091 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Lei Wang ◽

XiaoSong Yang ◽

WenJie Hu ◽

Quan Yuan

Keyword(s):

Stability Analysis ◽

Poincaré Map ◽

Circuit Model ◽

Computer Assisted ◽

Poincaré Section ◽

Poincare Map ◽

Poincare Section ◽

Topological Horseshoe ◽

Memristive Circuit ◽

The Stability

A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

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