EFFECTIVE COMPUTATION OF THE POINCARÉ MAP FOR THE ANALYSIS OF NONLINEAR DYNAMIC CIRCUITS/SYSTEMS USING RUNGE-KUTTA TRIPLES

1994 ◽  
Vol 04 (01) ◽  
pp. 93-98 ◽  
Author(s):  
L. FINGER ◽  
H. UHLMANN
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Dynamic ◽  
Poincaré Map ◽  
Global Analysis ◽  
Interpolation Polynomial ◽  
Poincare Map ◽  
Dynamic Circuits ◽  
Dense Output ◽  
Computer Aided ◽  
Runge Kutta

An enhancement of the classical Runge—Kutta technique for numerical simulations is presented for the computer-aided global analysis of nonlinear dynamic circuits/systems. With Runge—Kutta triples a remarkable saving of calculation time can be achieved by using an interpolation polynomial for dense output. The Runge—Kutta triples are applied to calculate the Poincaré map for autonomous models/systems.

Nonlinear Rolling Motion of a Statically Biased Ship Under the Effect of External and Parametric Excitation

1993 ◽  
Author(s):  
Isaac Esparza ◽  
Jeffrey Falzarano
Keyword(s):  
Steady State ◽  
Parametric Excitation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Global Analysis ◽  
Wave Train ◽  
Periodic Wave ◽  
Rolling Motion ◽  
Poincare Map ◽  
Safe Basin

Abstract In this work, global analysis of ship rolling motion as effected by parametric excitation is studied. The parametric excitation results from the roll restoring moment variation as a wave train passes. In addition to the parametric excitation, an external periodic wave excitation and steady wind bias are also included in the analysis. The roll motion is the most critical motion for a ship because of the possibility of capsizing. The boundaries in the Poincaré map which separate initial conditions which eventually evolve to bounded steady state solutions and those which lead to unbounded capsizing motion are studied. The changes in these boundaries or manifolds as effected by changes in the ship and environmental conditions are analyzed. The region in the Poincaré map which lead to bounded steady state motions is called the safe basin. The size of this safe basin is a measure of the vessel’s resistance to capsizing.

Symmetry Breaking and Symmetry Increasing in a Vibro-Impact with Symmetric Two-Sided Constraints

2011 ◽  
Vol 66-68 ◽  
pp. 229-234
Author(s):  
Yuan Yue ◽  
Jian Huab Xie
Keyword(s):  
Fixed Point ◽  
Symmetry Breaking ◽  
Numerical Simulations ◽  
Poincaré Map ◽  
Jacobi Matrix ◽  
Degree Of Freedom ◽  
Poincare Map ◽  
Impact System ◽  
Existence Conditions ◽  
Three Degree Of Freedom

A three-degree-of-freedom vibro-impact system with symmetric two-sided constraints is considered. Existence conditions of the symmetric period -2 motion are given, and the symmetric period n-2 motion of the system is deduced analytically. The six dimensional Poincaré map is established, and the Jacobi matrix of the symmetrixc fixed point is obtained. By the numerical simulations, we show that symmetry breaking and symmetry increasing exists in the vibro-impact system with symmetric two-sided constraints.

scholarly journals Dynamical Properties of a Herbivore-Plankton Impulsive Semidynamic System with Eating Behavior

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Yufei Wang ◽  
Huidong Cheng ◽  
Qingjian Li
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Detailed Analysis ◽  
Eating Behavior ◽  
Poincaré Map ◽  
Effective Control ◽  
Poincare Map ◽  
Phase Concentration ◽  
Dynamical Properties ◽  
The Relationship

In this paper, an impulsive semidynamic system of the relationship between plankton and herbivore is established, and the Poincaré map method is used to extend the new properties of the model. We define the Poincaré map of the impulsive point series in phase concentration and analyze the characteristics. A comprehensive and detailed analysis of the periodic solution is performed. In addition, the numerical simulations illustrate the correctness of our arguments. The results show that plankton and herbivore can survive stably under effective control.

Characterizing Nonlinear Dynamic Systems With Periodicity Ratio and Statistic Hypothesis

2011 ◽  
Author(s):  
Lu Han ◽  
Liming Dai
Keyword(s):  
Nonlinear Systems ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Poincaré Map ◽  
Numerical Results ◽  
Present Approach ◽  
Nonlinear Dynamic Systems ◽  
Statistical Hypothesis ◽  
Poincare Map ◽  
Nonlinear Characteristics

By introducing a statistical hypothesis to Periodicity Ratio, the efficiency and accuracy of diagnosing the nonlinear characteristics of dynamic systems are improved. Overlapping points in a Poincare map are verified on a statistically sound basis. The characteristics of nonlinear systems are investigated by using the present approach. The numerical results generated by the approach are compared with that of the conventional approaches.

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.

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.

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