Polynomial Approximation of Poincaré Maps for Hamiltonian Systems

1996 ◽  
pp. 51-56 ◽  
Author(s):  
Claude Froeschlé ◽  
Elena Lega
Keyword(s):  
Hamiltonian Systems ◽  
Polynomial Approximation ◽  
Poincaré Maps ◽  
Poincare Maps

Bifurcations and chaos in converters. Discontinuous vector fields and singular Poincaré maps

Nonlinearity ◽  
2000 ◽  
Vol 13 (4) ◽  
pp. 1095-1121 ◽  
Author(s):  
Gerard Olivar ◽  
Enric Fossas ◽  
Carles Batlle
Keyword(s):  
Vector Fields ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Discontinuous Vector Fields

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections

2017 ◽  
Vol 18 (5) ◽  
pp. 371-384 ◽  
Author(s):  
Luis Alberto Quezada-Téllez ◽  
Salvador Carrillo-Moreno ◽  
Oscar Rosas-Jaimes ◽  
José Job Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Analysis ◽  
Complex Variables ◽  
Bifurcation Diagrams ◽  
Poincaré Maps ◽  
Five Dimensions ◽  
Strange Nonchaotic Attractor ◽  
The Real ◽  
Poincare Maps ◽  
Extended Complex ◽  
Six Dimensions

AbstractIn this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

Observation and Evolution of Chaos for an Autonomous System

1984 ◽  
Vol 51 (3) ◽  
pp. 664-673 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Elastic Plate ◽  
Physical Interpretation ◽  
Autonomous System ◽  
Phase Plane ◽  
Power Spectra ◽  
Poincaré Maps ◽  
Flowing Fluid ◽  
Poincare Maps ◽  
Time Histories

Time histories, phase plane portraits, power spectra, and Poincare maps are used as descriptors to observe the evolution of chaos in an autonomous system. Although the motions of such a system can be quite complex, these descriptors prove helpful in detecting the essential structure of the motion. Here the principal interest is in phase plane portraits and Poincare maps, their methods of construction, and physical interpretation. The system chosen for study has been previously discussed in the literature, i.e., the flutter of a buckled elastic plate in a flowing fluid.

A Non-Intrusive Experimental Technique for Constructing Poincaré Maps in 3D Chaotically Advected Flows

2001 ◽  
Author(s):  
Fotis Sotiropoulos ◽  
Igor Mezić ◽  
Donald R. Webster
Keyword(s):  
Light Intensity ◽  
Experimental Technique ◽  
Vortex Breakdown ◽  
Laser Sheet ◽  
Three Dimensional ◽  
Chaotic Advection ◽  
Time Averaging ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
The Rich

Abstract We propose and develop the theoretical framework for a new experimental technique for constructing Poincaré maps in three-dimensional flows exhibiting chaotic advection. The technique is non-intrusive and, thus, simple to implement. Planar laser-induced fluorescence (LIF) is employed to collect a sufficiently long sequence of instantaneous light intensity fields on the plane of section of the Poincaré map (defined by the laser sheet). The chains of unmixed (regular) islands in the flow are visualized by time-averaging the instantaneous images and plotting iso-contours of the resulting mean light intensity field. A rigorous theoretical justification for this technique is derived using concepts from ergodic theory. We demonstrate the capabilities of the method by applying it to visualize the rich Lagrangian dynamics within steady vortex breakdown bubbles in a closed cylinder with a rotating bottom. The experimental results are shown to be in excellent agreement with numerical simulations.

Experimental and Computational Studies of Chaotic Stirring in Complex 3D Flows

2002 ◽  
Author(s):  
Fotis Sotiropoulos ◽  
Tahirih C. Lackey ◽  
S. Casey Jones
Keyword(s):  
Swirling Flow ◽  
Experimental Studies ◽  
Computational Studies ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Numerical Studies ◽  
Helical Vortices ◽  
3D Flows ◽  
Confined Flows ◽  
Particle Paths

Recent progress in experimental and computational studies of complex chaotically advected 3D flows is reviewed for the confined swirling flow in a cylindrical container with a rotating bottom and the open flow in a helical static mixer. The concept of Lagrangian averaging along particle paths, whose theoretical foundation stems from ergodic theory, is proposed as a powerful tool for constructing Poincare´ maps in numerical studies of confined flows. The same concept has also been employed to develop the first non-intrusive experimental technique for constructing Poincare´ maps in complex 3D flows. The potential of these ergodic concepts is demonstrated in computational and experimental studies for the confined swirling flow. Numerical computations for the helical mixer flow show that increasing the Reynolds number from Re = 100 to 500 leads to the appearance of unmixed islands in the flow. The mechanism that leads to the formation of such islands is shown to be linked to the growth of coherent helical vortices in the flow.

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