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.

scholarly journals STUDYING THE BASIN OF CONVERGENCE OF METHODS FOR COMPUTING PERIODIC ORBITS

2011 ◽  
Vol 21 (08) ◽  
pp. 2079-2106 ◽  
Author(s):  
MICHAEL G. EPITROPAKIS ◽  
MICHAEL N. VRAHATIS
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Newton's Method ◽  
Poincaré Map ◽  
Specific Problem ◽  
Poincare Map ◽  
Robust Methods ◽  
Surface Of Section ◽  
Cubic Equation ◽  
Nonlinear Mappings

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

A HORSESHOE IN A CELLULAR NEURAL NETWORK OF FOUR-DIMENSIONAL AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 17 (09) ◽  
pp. 3211-3218 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Poincaré Map ◽  
Cellular Neural Network ◽  
Poincare Map ◽  
Invariant Subset ◽  
The Neural Network

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

Poincaré Map-Based Continuation of Periodic Orbits in Dynamic Discontinuous and Hysteretic Systems

1999 ◽  
Author(s):  
Walter Lacarbonara ◽  
Fabrizio Vestroni ◽  
Danilo Capecchi
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Poincaré Map ◽  
Period Doubling ◽  
Path Following ◽  
Return Time ◽  
Poincare Map ◽  
Response Curves ◽  
Hysteretic Systems

Abstract A numerical algorithm is proposed to compute variation of periodic solutions and their codimension-one bifurcations in discontinuous and hysteretic systems in the relevant control parameter space. For dynamic systems with discontinuities and hysteresis, some components of the associated vector fields are nondifferentiable. Therefore, one cannot resort on classical numerical tools based on the evaluation of the Jacobian of the vector field for path-following analyses. Using the pertinent state space, periodic orbits are sought as the fixed points of a Poincaré map based on an appropriate return time. The Jacobian of the map is computed numerically via either a forward or a central finite-difference scheme and a Newton-Raphson procedure is used to determine the fixed points. The continuation scheme is a pseudo-arclength algorithm based on arclength parameterization. The eigenvalues of the Jacobian of the map — Floquet multipliers — are computed to ascertain the stability of the periodic orbits and the associated bifurcations. The procedure is used to construct frequency-response curves of a bilinear, a Masing-type, and a Bouc-Wen oscillator in the primary and superharmonic frequency ranges for various excitation levels. The proposed numerical strategy proves to be very effective in capturing a rich class of solutions and bifurcations — including jump phenomena, pitchfork (symmetry-breaking), and period-doubling.

ON ENTROPY OF CHUA'S CIRCUITS

2005 ◽  
Vol 15 (05) ◽  
pp. 1823-1828 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Computer Simulation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Topological Entropy ◽  
Poincaré Map ◽  
Chua’S Circuit ◽  
Poincare Map ◽  
Chua's Circuit

In this paper we revisit the well-known Chua's circuit and give a discussion on entropy of this circuit. We present a formula for the topological entropy of a Chua's circuit in terms of the Poincaré map derived from the ordinary differential equations of this Chua's circuit by computer simulation arguments.

HORSESHOES IN A NEW SWITCHING CIRCUIT VIA WIEN-BRIDGE OSCILLATOR

2005 ◽  
Vol 15 (07) ◽  
pp. 2271-2275 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Cross Section ◽  
Poincaré Map ◽  
Modern Theory ◽  
Poincare Map ◽  
Switching Circuit ◽  
Shift Map ◽  
Computer Aided ◽  
Topological Horseshoes

In this paper we revisit a switching circuit designed by the authors and present a theoretical analysis on the existence of chaos in this circuit. For the ordinary differential equations describing this circuit, we give a computer-aided proof in terms of cross-section and Poincare map, by applying a modern theory of topological horseshoes theory to the obtained Poincare map, that this map is semiconjugate to the two-shift map. This implies that the corresponding differential equations exhibit chaos.

scholarly journals Analysis of a mass-spring-relay system with periodic forcing

2021 ◽  
Author(s):  
János Lelkes ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Fixed Points ◽  
Poincaré Map ◽  
Pitchfork Bifurcation ◽  
Global Dynamics ◽  
Poincare Map ◽  
Periodic Forcing ◽  
Relay System ◽  
Saddle Type ◽  
The Stability ◽  
Mass Spring

AbstractThe dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

THE NONLINEAR MATHIEU EQUATION

1994 ◽  
Vol 04 (01) ◽  
pp. 71-86 ◽  
Author(s):  
J.W. NORRIS
Keyword(s):  
Vector Field ◽  
Perturbation Method ◽  
Small Amplitude ◽  
Vector Fields ◽  
Poincaré Map ◽  
Mathieu Equation ◽  
Poincare Map

The purpose of this paper is to classify the different sequences of bifurcation that can occur for small amplitude solutions to the nonlinear Mathieu equation near to the Mathieu regions of instability. We do this by using the Lindstedt-Poincare perturbation method to construct a vector field which interpolates the successive iterations of the Poincare map. These vector fields are then analysed to determine the sequence of bifurcations.

Steady-state modelling method for matrix-reactance frequency converter with boost topology

2011 ◽  
Vol 60 (2) ◽  
pp. 137-148
Author(s):  
Igor Korotyeyev ◽  
Beata Zięba
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Differential Equations ◽  
Numerical Method ◽  
Frequency Converter ◽  
Double Fourier Series ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Modelling Method ◽  
Three Phase

Steady-state modelling method for matrix-reactance frequency converter with boost topologyThis paper presents a method intended for calculation of steady-state processes in AC/AC three-phase converters that are described by nonstationary periodical differential equations. The method is based on the extension of nonstationary differential equations and the use of Galerkin's method. The results of calculations are presented in the form of a double Fourier series. As an example, a three-phase matrix-reactance frequency converter (MRFC) with boost topology is considered and the results of computation are compared with a numerical method.

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