Bifurcations of equilibria and periodic solutions in a nonlinear autonomous system with symmetry

1993 ◽  
Vol 76 (7) ◽  
pp. 1-14 ◽  
Author(s):  
Yuuji Katsuta ◽  
Hiroshi Kawakami
Keyword(s):  
Periodic Solutions ◽  
Autonomous System ◽  
Nonlinear Autonomous System

scholarly journals Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tiantian Ma ◽  
Zaihong Wang
Keyword(s):  
Periodic Solutions ◽  
Autonomous System ◽  
Positive Periodic Solutions ◽  
Duffing Equations ◽  
Time Map ◽  
Twist Theorem ◽  
The Given ◽  
Infinitely Many Periodic Solutions

We study the periodic solutions of Duffing equations with singularitiesx′′+g(x)=p(t). By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided thatgsatisfies the singular condition and the time map related to autonomous systemx′′+g(x)=0tends to zero.

ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS

1996 ◽  
Vol 06 (11) ◽  
pp. 2015-2030 ◽  
Author(s):  
A. KLÍČ ◽  
P. POKORNÝ
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Autonomous System ◽  
Vector Fields ◽  
Numerical Study ◽  
Method Of Averaging ◽  
Controlling Of Chaos

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.

scholarly journals Counting equilibria of large complex systems by instability index

2021 ◽  
Vol 118 (34) ◽  
pp. e2023719118 ◽  
Author(s):  
Gérard Ben Arous ◽  
Yan V. Fyodorov ◽  
Boris A. Khoruzhenko
Keyword(s):  
Autonomous System ◽  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Interaction Strength ◽  
Absolute Instability ◽  
Instability Index ◽  
Random Interactions ◽  
Stable Equilibria ◽  
The Mean ◽  
Nonlinear Autonomous System

We consider a nonlinear autonomous system of N≫1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

scholarly journals A MULTIPLICITY THEOREM FOR A PERTURBED SECOND-ORDER NON-AUTONOMOUS SYSTEM

2006 ◽  
Vol 49 (2) ◽  
pp. 267-275 ◽  
Author(s):  
Francesca Faraci ◽  
Antonio Iannizzotto
Keyword(s):  
Variational Principle ◽  
Periodic Solutions ◽  
Autonomous System ◽  
Second Order ◽  
Mountain Pass ◽  
Connected Components ◽  
Multiplicity Result ◽  
Global Minima ◽  
Multiplicity Theorem

AbstractIn this paper we establish a multiplicity result for a second-order non-autonomous system. Using a variational principle of Ricceri we prove that if the set of global minima of a certain function has at least $k$ connected components, then our problem has at least $k$ periodic solutions. Moreover, the existence of one more solution is investigated through a mountain-pass-like argument.

scholarly journals Infinite-Scroll Attractor Generated by the Complex Pendulum Model

2013 ◽  
Vol 2013 ◽  
pp. 1-3
Author(s):  
Sachin Bhalekar
Keyword(s):  
Autonomous System ◽  
Saddle Points ◽  
Pendulum Equation ◽  
Pendulum Model ◽  
Nonlinear Autonomous System ◽  
Complex Valued

We report the finding of the simple nonlinear autonomous system exhibiting infinite-scroll attractor. The system is generated from the pendulum equation with complex-valued function. The proposed system is having infinitely many saddle points of index two which are responsible for the infinite-scroll attractor.

Word problems related to periodic solutions of a non-autonomous system

1990 ◽  
Vol 108 (1) ◽  
pp. 127-151 ◽  
Author(s):  
James Devlin
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Word Problems ◽  
Autonomous System ◽  
Homogeneous Polynomials ◽  
Image Position ◽  
Abstract Formulation

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the formwhere p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the formwhere P and Q are polynomials, and their possible configurations.

scholarly journals Stability Analysis of Complex-Valued Nonlinear Differential System

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tao Fang ◽  
Jitao Sun
Keyword(s):  
Stability Analysis ◽  
Differential System ◽  
Comparison Principle ◽  
Autonomous System ◽  
New Method ◽  
Stability Criteria ◽  
General Complex ◽  
The Stability ◽  
Nonlinear Autonomous System ◽  
Complex Valued

This paper studies the stability of complex-valued nonlinear differential system. The stability criteria of complex-valued nonlinear autonomous system are established. For the general complex-valued nonlinear non-autonomous system, the comparison principle in the context of complex fields is given. Those derived stability criteria not only provide a new method to analyze complex-valued differential system, but also greatly reduce the complexity of analysis and computation.

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