HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

Heteroclinic Orbit, Forced Lorenz System, and Chaos

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Dibakar Ghosh ◽  
Anirban Ray ◽  
A. Roy Chowdhury
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lorenz System ◽  
Heteroclinic Orbit ◽  
Convergent Series ◽  
Heteroclinic Orbits ◽  
Onset Of Chaos ◽  
Uniformly Convergent ◽  
Analytic Expression

Forced Lorenz system, important in modeling of monsoonlike phenomena, is analyzed for the existence of heteroclinic orbit. This is done in the light of the suggested new mechanism for the onset of chaos by Magnitskii and Sidorov (2006, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations,” Diff. Eq., 39, pp. 1593–1602), where heteroclinic orbits plays important and dominant roles. The analysis is performed based on the theory laid down by Shilnikov. An analytic expression in the form of uniformly convergent series is obtained. The same orbit is also obtained numerically by a technique enunciated by Magnitskii and Sidorov, reproducing the necessary important features.

scholarly journals ON AN ELLIPTIC SYSTEM OF P(X)-KIRCHHOFF-TYPE UNDER NEUMANN BOUNDARY CONDITION

2012 ◽  
Vol 17 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Zehra Yucedag ◽  
Mustafa Avci ◽  
Rabil Mashiyev
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Principle ◽  
Variational Method ◽  
Elliptic System ◽  
Neumann Boundary Condition ◽  
Existence Of Solutions ◽  
Neumann Boundary ◽  
Kirchhoff Type ◽  
Direct Variational Method

In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution.

scholarly journals A Class of Variable-Order Fractional p · -Kirchhoff-Type Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yong Wu ◽  
Zhenhua Qiao ◽  
Mohamed Karim Hamdani ◽  
Bingyu Kou ◽  
Libo Yang
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Variational Method ◽  
Elliptic System ◽  
Type Systems ◽  
Ekeland Variational Principle ◽  
Variable Order ◽  
Variable Exponents ◽  
Kirchhoff Type ◽  
Direct Variational Method

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional p x -operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

Complex Dynamics of a Four-Dimensional Circuit System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Haijun Wang ◽  
Hongdan Fan ◽  
Jun Pan
Keyword(s):  
Complex Dynamics ◽  
Numerical Technique ◽  
Heteroclinic Orbit ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
Heteroclinic Cycle ◽  
Stable Node ◽  
Geometrical Structures ◽  
Heteroclinic Cycles ◽  
Forming Mechanism

Combining qualitative analysis and numerical technique, the present work revisits a four-dimensional circuit system in [Ma et al., 2016] and mainly reveals some of its rich dynamics not yet investigated: pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, globally exponentially attractive set, invariant algebraic surface and heteroclinic orbit. The main contributions of the work are summarized as follows: Firstly, it is proved that there exists a globally exponentially attractive set with three different exponential rates by constructing a suitable Lyapunov function. Secondly, the existence of a pair of heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, numerical simulations not only are consistent with theoretical results, but also illustrate potential existence of hidden attractors in its Lorenz-type subsystem, singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small [Formula: see text], i.e. hyperchaotic attractors and nearby pseudo singularly degenerate heteroclinic cycles, i.e. a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity, or the true ones consisting of normally hyperbolic saddle-foci (or saddle-nodes) and stable node-foci, giving some kind of forming mechanism of hyperchaos.

The weak Ekeland variational principle and fixed points

2014 ◽  
Vol 102 ◽  
pp. 91-96 ◽  
Author(s):  
Gerald Beer ◽  
Asen L. Dontchev
Keyword(s):  
Variational Principle ◽  
Fixed Points ◽  
Ekeland Variational Principle

On Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Heteroclinic Orbit ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Undetermined Coefficient ◽  
Global Connections ◽  
Coefficient Method ◽  
T System ◽  
Undetermined Coefficient Method

In the referenced paper, the authors use the undetermined coefficient method to analytically construct homoclinic and heteroclinic orbits in the T system. Unfortunately their method is not valid because they assume odd functions for the first component of the homoclinic and the heteroclinic orbit whereas these Shil'nikov global connections do not exhibit symmetry.

scholarly journals Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150665 ◽  
Author(s):  
P. Ponte Castañeda
Keyword(s):  
Composite Materials ◽  
Variational Principle ◽  
Variational Method ◽  
Single Crystal ◽  
Second Order ◽  
Duality Gaps ◽  
Nonlinear Composite ◽  
Constitutive Response ◽  
Correction Terms ◽  
Macroscopic Response

A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.

Extensions of the mild-slope equation

1995 ◽  
Vol 300 ◽  
pp. 367-382 ◽  
Author(s):  
D. Porter ◽  
D. J. Staziker
Keyword(s):  
Free Surface ◽  
Variational Principle ◽  
Variational Method ◽  
Water Wave ◽  
Jump Condition ◽  
Jump Conditions ◽  
Free Surface Elevation ◽  
Mild Slope Equation ◽  
Bed Slope ◽  
Smooth Approximations

The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.

Variational Method for Nonconservative Problems

1970 ◽  
Vol 37 (1) ◽  
pp. 133-136 ◽  
Author(s):  
R. N. Dubey
Keyword(s):  
Boundary Condition ◽  
Variational Principle ◽  
Variational Method ◽  
Boundary Surface ◽  
Adjoint System ◽  
Work Condition ◽  
Nonconservative Systems ◽  
Admissible Velocity ◽  
Geometric Boundary ◽  
Geometric Boundary Condition

It is shown that the Ritz variational method can be applied to nonconservative problems provided the admissible velocity satisfies not only the geometric boundary condition but also a work condition on the part of the boundary surface where traction and traction rate are prescribed. It is further found that the modified variational principle can also be applied to nonconservative systems for which it is possible to construct an adjoint system. The principle is expected to be useful for application to a wide class of problems, including those encountered in connection with hydrodynamic and hydromagnetic stability investigation.

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