Chaotic Motions of a Parametrically Excited Pendulum With Non-Exponentially Small Parameters

2005 ◽  
Author(s):  
Chunqing Lu
Keyword(s):  
Poincaré Map ◽  
Poincare Map ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Small Parameters ◽  
Exponentially Small

The paper mathematically proves that a pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than using the Poincare’ map. It provides more information about when the chaos occurs.

Properties of cross-well chaos in an impacting system

1994 ◽  
Vol 347 (1683) ◽  
pp. 391-410 ◽  
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Poincare Map ◽  
Chaotic Motions ◽  
Higher Harmonics ◽  
Potential Wells ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
Phase Space Transport ◽  
Basin Boundaries

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

2007 ◽  
Vol 17 (03) ◽  
pp. 837-850 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Coupled Systems ◽  
Poincare Map ◽  
Circuit Implementation ◽  
Coupling Structure ◽  
Bifurcation Phenomena ◽  
Junctional Coupling ◽  
Current Coupling

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Path Following Control and Analysis of Snake Robots Based on the Poincaré Map

2013 ◽  
pp. 89-101
Author(s):  
Pål Liljebäck ◽  
Kristin Y. Pettersen ◽  
Øyvind Stavdahl ◽  
Jan Tommy Gravdahl
Keyword(s):  
Poincaré Map ◽  
Path Following ◽  
Poincare Map ◽  
Path Following Control ◽  
Snake Robots

On the Chaotic Dynamics Associated with the Center Manifold Equations of Double-Diffusive Convection Near a Codimension-Four Bifurcation Point at Moderate Thermal Rayleigh Number

2018 ◽  
Vol 28 (08) ◽  
pp. 1850094 ◽  
Author(s):  
Justin Eilertsen ◽  
Jerry Magnan
Keyword(s):  
Bifurcation Point ◽  
Center Manifold ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Thermosolutal Convection ◽  
Poincare Map ◽  
Lorenz Equations ◽  
Frequency Locking ◽  
Diffusive Convection ◽  
Double Diffusive

We analyze the dynamics of the Poincaré map associated with the center manifold equations of double-diffusive thermosolutal convection near a codimension-four bifurcation point when the values of the thermal and solute Rayleigh numbers, [Formula: see text] and [Formula: see text], are comparable. We find that the bifurcation sequence of the Poincaré map is analogous to that of the (continuous) Lorenz equations. Chaotic solutions are found, and the emergence of strange attractors is shown to occur via three different routes: (1) a discrete Lorenz-like attractor of the three-dimensional Poincaré map of the four-dimensional center manifold equations that forms as the result of a quasi-periodic homoclinic explosion; (2) chaos that follows quasi-periodic intermittency occurring near saddle-node bifurcations of tori; and, (3) chaos that emerges from the destruction of a 2-torus, preceded by frequency locking.

Computing the Number of Fixed Joints on Poincare Map in Nonlinear Mathieu Equation

1993 ◽  
Author(s):  
Hisato Fujisaka ◽  
Chikara Sato
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Numerical Method ◽  
Topological Degree ◽  
Poincaré Map ◽  
Mathieu Equation ◽  
Time Varying ◽  
Poincare Map ◽  
Newton's Iteration ◽  
Combined Use

Abstract A numerical method is presented to compute the number of fixed points of Poincare maps in ordinary differential equations including time varying equations. The method’s fundamental is to construct a map whose topological degree equals to the number of fixed points of a Poincare map on a given domain of Poincare section. Consequently, the computation procedure is simply computing the topological degree of the map. The combined use of this method and Newton’s iteration gives the locations of all the fixed points in the domain.

Sign in / Sign up

Export Citation Format

Share Document