scholarly journals Aspects of laser Lorenz dynamics

1994 ◽  
Vol 36 (2) ◽  
pp. 152-174 ◽  
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.

scholarly journals Two-Parameter Dynamics of An Autonomous Mechanical Governor System With Time Delay

2021 ◽  
Author(s):  
Shuning Deng ◽  
Jinchen Ji ◽  
Guilin Wen ◽  
Huidong Xu
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Evolutionary Process ◽  
Basins Of Attraction ◽  
Poincaré Section ◽  
Poincare Section ◽  
Interesting Phenomenon ◽  
Computing Technique ◽  
Intersection Points ◽  
Two Parameter

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.

scholarly journals The onset of chaos in vortex sheet flow

2002 ◽  
Vol 454 ◽  
pp. 47-69 ◽  
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.

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

2008 ◽  
Vol 18 (08) ◽  
pp. 2391-2396 ◽  
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.

scholarly journals Horseshoe Chaos in a Simple Memristive Circuit

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.

Nonlinear differential delay equations using the Poincaré section technique

1996 ◽  
Vol 54 (6) ◽  
pp. 6925-6928 ◽  
Author(s):  
Zhao Hong ◽  
Zhang Feizhou ◽  
Yan Jie ◽  
Wang Yinghai
Keyword(s):  
Delay Equations ◽  
Poincaré Section ◽  
Poincare Section ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Section Technique

scholarly journals Dynamical Monte Carlo Simulations of 3-D Galactic Systems in Axisymmetric and Triaxial Potentials

2017 ◽  
Vol 34 ◽  
Author(s):  
Ali Taani ◽  
Juan C. Vallejo
Keyword(s):  
Phase Space ◽  
Invariant Tori ◽  
Dynamical Behavior ◽  
Dark Halo ◽  
Short Axis ◽  
Poincaré Section ◽  
Invariant Curves ◽  
Poincare Section ◽  
Dynamical Behaviors ◽  
Stable Periodic

AbstractWe describe the dynamical behavior of isolated old ( ⩾ 1Gyr) objects-like Neutron Stars (NSs). These objects are evolved under smooth, time-independent, gravitational potentials, axisymmetric and with a triaxial dark halo. We analysed the geometry of the dynamics and applied the Poincaré section for comparing the influence of different birth velocities. The inspection of the maximal asymptotic Lyapunov (λ) exponent shows that dynamical behaviors of the selected orbits are nearly the same as the regular orbits with 2-DOF, both in axisymmetric and triaxial when (ϕ, qz)= (0,0). Conversely, a few chaotic trajectories are found with a rotated triaxial halo when (ϕ, qz)= (90, 1.5). The tube orbits preserve direction of their circulation around either the long or short axis as appeared in the triaxial potential, even when every initial condition leads to different orientations. The Poincaré section shows that there are 2-D invariant tori and invariant curves (islands) around stable periodic orbits that bound to the surface of 3-D tori. The regularity of several prototypical orbits offer the means to identify the phase-space regions with localized motions and to determine their environment in different models, because they can occupy significant parts of phase-space depending on the potential. This is of particular importance in Galactic Dynamics.

Local dimension of ergodic measures for two-dimensional Lorenz transformations

2000 ◽  
Vol 20 (3) ◽  
pp. 911-923 ◽  
Author(s):  
THOMAS STEINBERGER
Keyword(s):  
Probability Measures ◽  
Two Dimensional ◽  
Poincaré Section ◽  
Poincare Section ◽  
Local Dimension ◽  
Lorenz Equation ◽  
Characteristic Exponents ◽  
Ergodic Measures

A class of transformations on $[0,1]^2$, which includes transformations obtained by a Poincaré section of the Lorenz equation, is considered. We prove a formula which connects local dimension, entropy and characteristic exponents of ergodic invariant probability measures.

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