scholarly journals On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

2015 ◽  
Vol 29 (1) ◽  
pp. 7-17
Author(s):  
Roman Srzednicki
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Conley Index ◽  
Computer Assisted ◽  
Small Step ◽  
Index Pair ◽  
Time Periodic ◽  
Autonomous Ordinary Differential Equation

AbstractWe announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

The G-invariant implicit function theorem in infinite dimensions

1982 ◽  
Vol 92 (1-2) ◽  
pp. 13-30 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Infinite Dimensions ◽  
Basic Theorem

SynopsisOur basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.

Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

2018 ◽  
Vol 2020 (23) ◽  
pp. 9440-9470
Author(s):  
Jian Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Existence Result ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.

A combinatorial procedure for finding isolating neighbourhoods and index pairs

1997 ◽  
Vol 127 (5) ◽  
pp. 1075-1088 ◽  
Author(s):  
Andrzej Szymczak
Keyword(s):  
Periodic Orbits ◽  
Conley Index ◽  
Finite Union ◽  
Computer Assisted ◽  
Invariant Subset ◽  
Index Pair ◽  
Index Pairs

SynopsisWe present a purely combinatorial procedure for finding an isolating neighbourhood and an index pair contained in a given set, being a finite union of cubes in Rs. It is applied for a computer-assisted computation of the Conley index of an isolated invariant subset of the Hénon attractor. As a corollary, it is shown that the Hénon attractor contains periodic orbits of all principal periods except for 3 and 5.

ON THE STRUCTURE OF THE SOLUTIONS OF A TWO-PARAMETER FAMILY OF THREE-DIMENSIONAL ORDINARY DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 1287-1298 ◽  
Author(s):  
SERKAN T. IMPRAM ◽  
RUSSELL JOHNSON ◽  
RAFFAELLA PAVANI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Wide Range ◽  
Two Parameters ◽  
Two Parameter ◽  
Autonomous Ordinary Differential Equation

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

scholarly journals Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Yongxiang Li ◽  
Lanjun Guo
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Existence Result ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Superlinear Growth ◽  
Existence Of Periodic Solutions

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation u′′(t)=ft,ut,u′t, t∈R, where the nonlinearity f:R3→R is continuous and f(t,x,y) is 2π-periodic in t. Under certain inequality conditions that f(t,x,y) may be superlinear growth on (x,y), an existence result of odd 2π-periodic solutions is obtained via Leray-Schauder fixed point theorem.

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