Non-symmetric Solutions Research Articles

Reversible Hamiltonian Liapunov center theorem

We study the existence of periodic solutions in the neighbourhood ofsymmetric (partially) elliptic equilibria in purely reversible Hamiltonian vectorfields. These are Hamiltonian vector fields with an involutory reversing symmetry$R$. We contrast the cases where $R$ acts symplectically and anti-symplectically.In case $R$ acts anti-symplectically, generically purely imaginary eigenvaluesare isolated, and the equilibrium is contained in a local two-dimensional invariantmanifold containing symmetric periodic solutions encircling the equilibriumpoint.In case $R$ acts symplectically, generically purely imaginary eigenvaluesare doubly degenerate, and the equilibrium is contained in two two-dimensional invariantmanifolds containing nonsymmetric periodic solutions encirclingthe equilibrium point. In addition, there exists a three-dimensional invariantsurface containing a two-parameter family of symmetric periodic solutions.

Indeterminate moment problems related to birth and death processes with quartic rates

We consider indeterminate symmetric moment problems related to birth and death processes with quartic rates. The Nevanlinna matrices are presented and the entire functions B and D are expressed in terms of the trigonometric functions of order 4. Several examples of symmetric and nonsymmetric solutions are given.

Symmetric and nonsymmetric solutions for an elliptic equation on [formula omitted

We obtain, for N⩾4, the existence of a positive solution as well as a nonradial solution, which changes the sign of the equation − Δu+V(z)u = f(z,u) in R N u ∈ H 1( R N), where the function f is a superlinear nonlinearity and the function V is a potential that is bounded below by a positive constant. As a consequence, we complete for the case V( z)≡1 and f( z, u)≡ f( u) the well-known work of Bartsh and Willen (J. Funct. Anal. 117 (1993) 261.) because there it does not consider the case N=5.

Symmetry-breaking in periodic gravity waves with weak surface tension and gravity-capillary waves on deep water

The method developed by Debiane and Kharif for the calculation of symmetric gravity-capillary waves on infinite depth is extended to the general case of non-symmetric solutions. We have calculated non-symmetric steady periodic gravity-capillary waves on deep water. It is found that they appear via bifurcations from a family of symmetric waves. On the other hand we found that the symmetry-breaking bifurcation of periodic steady class 1 gravity wave on deep water is possible when it approaches the limiting profile, if it is very weakly influenced by surface tension effects. To cite this article: R. Aider, M. Debiane, C. R. Mecanique 332 (2004).

Asymptotic and periodic motion around collinear equilibria in Chermnykh's problem

In this paper we study the asymptotic motion to the collinear equilibrium points of Chermnykh's problem. More specifically, we give three kinds of non-symmetric doubly-asymptotic solutions emanating from L 1 and L 3 . We also show that these solutions are closely connected to the families of periodic orbits generated around these equilibria.

Chaos in the Kuramoto–Sivashinsky equations—a computer-assisted proof

In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczyński (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto–Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions.

A construction of infinitely many singular weak solutions to the equations of nonlinear elasticity

Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.

Nonsymmetric equal sacrifice solutions for claim problem

A claim (allocation) problem is the problem of distributing a given amount of a divisible resource among agents with unequal claims on the resource. The main result of the paper is the following representation of a strictly monotonic, consistent, path independent, and individually unbounded solution of the claim problem. There exists a family of strictly increasing continuous utility functions of agents such that all agents have equal differences between values of their utility functions at the solution point and at the claim point. These solutions of the claim problem are called equal sacrifice solutions with respect to the family of utility functions. Young [Journal of Economic Theory 44 (1988) 321] obtained a similar representation under an additional anonymity condition. Due to this result, we get the solution of goal programming problems with convex feasible sets satisfying natural axioms. The solution of the goal programming problem is the point minimizing the measure of proximity to the goal point. Here the measure of proximity is defined as the sum of antiderivatives of differences between utility functions. Under additional positive homogeneity condition, we get maximal weighted entropy solution.

Dynamical mean-field solution of coupled quantum wells: a bifurcation analysis.

The time evolution of a discrete model of three quantum wells with a localized mean-field electrostatic interaction has been analyzed making use of numerical simulation and bifurcation techniques. The discrete Schrödinger equation can be written as a classical Hamiltonian system with two constants of motion. The frequency spectrum and the Lyapunov exponents show that the system is chaotic as its continuum counterpart. The organizing centers of the dynamical behavior are bifurcations of rotating periodic solutions whose simple structure allows a thorough analytical investigation as the conserved quantities are varied. The global structure of the periodic behavior is organized via subharmonic bifurcations at which tori of nonsymmetric periodic solutions are born. We have found another kind of bifurcation when two pairs of characteristic multipliers split from the unit circle. The chaotic behavior is related to the nonintegrability of the system.

Nonsymmetric solutions for some variational problems

Nonsymmetric solutions for some variational problems

Global solutions of partitioning problems*

In this paper, a special kind of optimization problems is considered. The total cost function for a homogeneous group of participants– each with the same convex-concave cost function – is to be minimized. The particular structure of this problem makes it possible to find global minimizers in dependence on the linking parameter. The symmeric partition is a stationary solution and the optimal one for small parameter values. Beyond a critical parameter value one obtains certain non-symmetric solutions. Two geometrical applications are discussed. They are related to perimeter and volume partitions

On the “genetic memory” of chaotic attractor of the barotropic ocean model

The structure of attractor of barotropic ocean model is studied in this paper. Theorems of the existence of the attractor for the finite dimensional approximation of this model are proved as well as its convergence to the attractor of the model itself. Some properties of stationary solutions of this model and their stability are discussed. The structure of the attractor is partially explained by the sequence of bifurcations the system is subjected to by variations of leading parameters. The principal feature of the studied system is the existence of two “almost invariant” basins of chaotic attractor with very rare transitions between them. This is related to the rise of a couple of non-symmetric stable stationary solutions in the model with symmetric forcing. The “memory” of chaos appears also in the presence of maxima in the spectrum of energy. These maxima correspond either to the principal frequency of the limit cycle which arose in the Hopf bifurcation, or to the frequencies of the Feigenbaum phenomenon.

Chaos in a mean field model of coupled quantum wells; bifurcations of periodic orbits in a symmetric hamiltonian system

We analyze a discrete model of coupled quantum wells with short-range mean-field interaction in one site. The system evolves according to the time dependent Schrödinger equation with a nonlinear electrostatic term. The simplest vector field that accounts for the chaotic dynamical behaviour present in the continuum case has four degrees of freedom and can be written as a classical hamiltonian system. It is invariant under diagonal rotations in C 4, reversible, autonomous and nonintegrable. The conserved quantities are the energy and the total charge. The organizing centers of the dynamical behaviour are bifurcations of rotating periodic solutions. The global structure of the periodic behaviour is organized via subharmonic bifurcations in which the characteristic multipliers (CM) pass each other on the unit circle and a branch of torus filled with nonsymmetric periodic solutions is born. We have also found another kind of bifurcation in which two pairs of CM depart the unit circle and the symmetric periodic orbit becomes unstable.

Bifurcation of nonsymmetric solutions for some duffing equations

For some symmetric Duffing equation, the existence of bifurcation of nonsymmetric, periodic solutions from symmetric periodic solutions is proved by using the change of index of symmetric, periodic solutions for variation of parameters.

Invariant solutions of two models of evolution of turbulent bursts

We consider two problems related to the b–l and b–ε models of propagation of turbulent bursts. We show that these equations admit some particular exact solutions which reduce to a finite-dimensional dynamical system. This makes it possible to describe a singular effect of finite-time extinction, and in particular, nonsymmetric solutions which do not exhibit symmetrization in the asymptotic behaviour. We show that in the multi-dimensional equation related to the b–l model, the nonsymmetric extinction behaviour is governed by the first-order equation. For the b–ε model with α=β=1 and γ<1, using such particular solutions, we establish that the ω-limit set of all the rescaled extinction orbits is essentially infinite-dimensional.

Steady Nonsymmetrical Gravity Waves with Shear in Water

A new type of steady two‐dimensional inviscid gravity wave with shear is computed numerically. These waves appear at relatively low amplitudes and lack symmetry with respect to any crest or trough. A boundary integral formulation is used to obtain a one‐parameter family of nonsymmetrical solutions through a symmetry‐breaking bifurcation.

Pecularities of acoustic phonon scattering from a planar crystal defect and pseudosurface phonons (Resonant scattering of transverse phonons from a thin defect layer)

A scattering of acoustical phonons from a planar crystal defect is studied. A resonance reflection of the transverse phonons from the planar defect is discovered and conditions of such a reflection are investigated. It is shown that the resonance conditions are connected with existence of longitudinal phonons localized at the planar defect and travelling along it. The problem is analyzed basing both on the lattice dynamics and on the theory of elasticity. A non-symmetrical stationary solution which consists of (1) a homogeneous transverse wave propagating only in one elastic semispace and (2) longitudinal localized waves in the both elastic semispaces is found. The condition of the resonance reflection of the transverse phonons from the defect layer coincides with a condition under which the non-symmetrical stationary vibration exists. The influence of properties of the models considered on the resonance frequencies is discussed.

Small viscosity behavior of a homogeneous, quasi-geostrophic, ocean circulation model

Insensitivity to turbulent closure in the wind-driven nonlinear Stommel-Munk model is addressed theoretically and numerically. The QG energy equation is used to show that, with the assumption that the maximum velocities occur at inertial length scales or smaller, a Sverdrup interior is consistent with the small Rossby number assumption only when the frictional parameters exceed critical values. For frictional parameters smaller than these values, valid solutions must decrease the energy source. This is possible for non-Sverdrup solutions since the energy source is dependent on the solution. The steady-state behavior of the model was investigated via a pseudo-arclength continuation algorithm. Dependence on the boundary layer Reynolds number, Re, was investigated by varying the eddy viscosity for e xed wind forcing. A tendency to decrease the energy source was found for solutions that are nonsymmetric about the center latitude. Antisymmetric solutions displayed the opposite behavior and diverged more quickly with increasing Re. The robustness of the results to dynamic boundary condition, symmetry and strength of wind stress, time dependence and bottom friction were tested. Some aspects of the nonsymmetric solutions appeared insensitive to Re.

Numerical analysis of radially nonsymmetric blow-up solutions of a nonlinear parabolic problem

The process of combustion for a nonlinear heat conducting medium with a nonlinear volume source is considered. Blow-up self-similar solutions which describe the evolution of radially nonsymmetric waves — with complex symmetry and spiral waves — are realized numerically. Their asymptotic behaviour is analysed and their metastability is established. To solve the self-similar nonlinear problem the continuous analog of the Newton method and the finite element method are used. The semidiscrete Galerkin finite element method and an explicit difference scheme are used to solve the nonlinear parabolic problem. Special adaptive grids, consistent with the structure of the self-similar solutions, are utilized.

On the blow-up time convergence of semidiscretizations of reaction-diffusion equations

Semidiscretizations of reaction-diffusion equations are studied and special attention is devoted to symmetric solutions. Also nonsymmetric solutions are considered when the reaction term is such that f(0) = 0. Sufficient conditions for blow-up in such discretizations are established and upper bounds of the blow-up time, which depend on the maximum norm of the initial conditions, are provided. Convergence of the blow-up times of the semidiscrete problems to the theoretical one is proved.