THE STABILITY OF SOME KINDS OF GENERALIZED HOMOCLINIC LOOPS IN PLANAR PIECEWISE SMOOTH SYSTEMS

2013 ◽  
Vol 23 (02) ◽  
pp. 1350027 ◽  
Author(s):  
FENG LIANG ◽  
MAOAN HAN
Keyword(s):  
Poincaré Map ◽  
Piecewise Smooth ◽  
Poincare Map ◽  
Piecewise Smooth Systems ◽  
The Stability

In this paper, we present some kinds of generalized homoclinic loops in planar piecewise smooth systems. By establishing Poincaré map, we study the stability of generalized homoclinic loops for each case. Some criteria are given for their stability, respectively. As applications, several concrete piecewise systems are considered.

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.

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.

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.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

Analysis of a Mass-Spring-Relay System With Periodic Forcing

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

Abstract The 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. A Poincare map is constructed to simplify the mathematical analysis. The stability of the xed points of the Poincare 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 xed point. The global dynamics of the system is investigated, showing discontinuity induced bifurcations of the xed points.

Hierarchical Implicit Feedback Structure in Passive Dynamic Walking

2008 ◽  
Vol 20 (4) ◽  
pp. 559-566 ◽  
Author(s):  
Yasuhiro Sugimoto ◽  
◽  
Koichi Osuka
Keyword(s):  
Poincaré Map ◽  
Implicit Feedback ◽  
Poincare Map ◽  
Dynamic Walking ◽  
Passive Dynamic Walking ◽  
Bifurcation Phenomenon ◽  
Feedback Structure ◽  
The Stability ◽  
Interesting Structure

Using a linearized analytical Poincaré map, we analyzed the stability of Passive Dynamic Walking (PDW), focusing on bifurcation phenomenon in PDW. Although it is well-known, bifurcation of the walking period has not been well studied. Based on our previous research, we derive an analytical Poincaré map for 2-period walking to discuss PDW stability of PDW with this map. In addition, we point out that there is a similar interesting structure in this Poincaré map.

Stability analysis of thermoacoustic interactions in vortex shedding combustors using Poincaré map

2016 ◽  
Vol 801 ◽  
pp. 597-622 ◽  
Author(s):  
Balasubramanian Singaravelu ◽  
Sathesh Mariappan
Keyword(s):  
Heat Release ◽  
Heat Release Rate ◽  
Vortex Shedding ◽  
Release Rate ◽  
Poincaré Map ◽  
Vortex Breakdown ◽  
Acoustic Mode ◽  
Experimental Investigations ◽  
Poincare Map ◽  
The Stability

Vortex separation and breakdown are an important source of unsteady heat release rate fluctuations in thermoacoustic systems. The coupling between the acoustic field and the energy released by vortex breakdown can cause combustion instability. The objective of this work is to perform linear stability analyses and to quantify the stability of thermoacoustic interactions where vortex breakdown is the dominant cause of heat release rate fluctuations. The dynamics of the system is modelled as a kicked oscillator, where the energy released by vortex breakdown is represented as the kick. Assuming small fluctuations, periodic and low values of kick, a Poincaré map is derived analytically. The stability of the system is determined from the eigenvalues associated with the Poincaré map. The results allow us to identify the region where vortex shedding or the acoustic mode is dominant. Previous experimental investigations report the transition between the two modes for variation in the flow Mach number. Similar transitions are observed in the present study.

Periodic Solution Bifurcation and Spiking Dynamics of Impacting Predator–Prey Dynamical Model

2018 ◽  
Vol 28 (12) ◽  
pp. 1850147 ◽  
Author(s):  
Sanyi Tang ◽  
Xuewen Tan ◽  
Jin Yang ◽  
Juhua Liang
Keyword(s):  
Periodic Solution ◽  
Poincaré Map ◽  
Complex Dynamics ◽  
Poincare Map ◽  
Predator Prey ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
The Stability ◽  
Nonmonotonic Functional Response ◽  
Solution Bifurcation

A planar predator–prey impacting system model with a nonmonotonic functional response function is proposed and analyzed. The existence and stability of a boundary order-1 periodic solution were investigated and the threshold conditions for a transcritical bifurcation and stable switching were obtained, and also the definition and properties of the Poincaré map are discussed. The main results indicate that multiple discontinuous points of the Poincaré map could induce the coexistence of multiple order-1 periodic solutions. Numerical analyses reveal the complex dynamics of the model including periodic adding and halving bifurcations, which could result in multiple active phases, among them rapid spiking and quiescence phases which can switch from one to another and consequently create complex bursting patterns. The main results reveal that it is beneficial to restore the stability and balance of a ecosystem for species with group defence by moderately reducing population densities and the group defence capacity.

scholarly journals STABILITY ANALYSIS OF PARALLEL CONNECTED DC-DC CONVERTERS

2020 ◽  
Vol 4 (2) ◽  
pp. 35-39
Author(s):  
Abdulmajed O. Elbkosh
Keyword(s):  
Poincaré Map ◽  
Monodromy Matrix ◽  
The Other ◽  
Nonlinear Behaviour ◽  
Matrix Approach ◽  
Poincare Map ◽  
Bifurcation And Chaos ◽  
The Matrix ◽  
The Stability ◽  
Map Approach

Parallel controlled DC-DC converters are nonlinear and non-smooth systems, they show various nonlinear behaviour including smooth, non-smooth bifurcation, and chaos when they work outer their design conditions. Usually, the Poincaré map approach is the most common method for studying the stability of those nonlinear systems. Stability is indicated using the eigenvalues of the Jacobian of the map computed at the fixed point. The other method is the monodromy matrix approach, where the stability can be concluded by computed the eigenvalues of the matrix. In this paper, the nonlinear dynamics of parallel connected DC-DC converters are investigated. It is shown that the concept of the monodromy matrix can be applied to determine the stability of the system as well as the Poincare map approach.

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