scholarly journals On a method to compute periodic solutions of the general autonomous system

1960 ◽  
Vol 24 (2) ◽  
pp. 189-196 ◽  
Author(s):  
Minoru Urabe
Keyword(s):  
Periodic Solutions ◽  
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 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.

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.

Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals

1977 ◽  
Vol 77 (1-2) ◽  
pp. 163-175 ◽  
Author(s):  
In-Ding Hsü ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Periodic Solutions ◽  
Stability Criterion ◽  
Autonomous System ◽  
Third Order ◽  
Non Linear ◽  
Existence And Stability ◽  
Linear Autonomous System ◽  
Non Linear System

SynopsisA 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

Periodic solutions of a class of Hamiltonian systems with singularities

1990 ◽  
Vol 114 (1-2) ◽  
pp. 1-13 ◽  
Author(s):  
Antonio Ambrosetti ◽  
Ivar Ekeland
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Autonomous System ◽  
Time Dependent ◽  
Unperturbed Problem ◽  
Time Periodic

SynopsisThis paper deals with a class of time-periodic Hamiltonian systems obtained by a time-dependent perturbation from an autonomous system with a singularity at q = 0 in configuration space. It is shown that, T being the period of the perturbation, nondegenerate families of T-periodic orbits in the unperturbed problem branch off into a certain number of T-periodic orbits for the perturbed problem.

Periodic solutions of general autonomous system of liénard type

2004 ◽  
Vol 83 (7) ◽  
pp. 735-746
Author(s):  
Ping Yan ◽  
Jifa Jiang †
Keyword(s):  
Periodic Solutions ◽  
Autonomous System
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