Quasi-periodic bifurcations in reversible systems

Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parameterized by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.     

Imposing symmetry in augmented linear systems

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Degenerate lower dimensional tori in reversible systems

In this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems. The normal variable is two-dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure.     

Boltzmann–Gibbs distribution of fortune and broken time reversible symmetry in econodynamics

Within the framework of stochastic differential equations it is demonstrated that the existence of Boltzmann–Gibbs type distribution in economy is independent of the time reversal symmetry in econodynamics. Both power law and exponential distributions can be accommodated naturally. The demonstration is based on a mathematical structure discovered during a study in gene regulatory network dynamics. Further possible analogy between equilibrium economy and thermodynamics is explored, suggesting that statistical physics methods can indeed play an important role in the study of complex systems.     

The structure of evolutional jumps in reversible systems

A general theory for jump structures in reversible symmetric systems is developed. It is established that, as in systems with dissipation [1–3], the type of jump structure depends on the number of intersections of the dispersion curve and the straight line corresponding to the jump velocity, while jumps for which a structure exists turn out to be at the same time evolutional jumps. The theory is applicable in cases when the dispersive properties of the medium prevail over its dissipative properties.     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.     

Hamiltonian systems with a discrete symmetry

If $\text{?: M → M}$ is an antisymplectic involution of a symplectic manifold M then the fixed set of ? is a Lagrangian submanifold L ? M. Moreover there exist cotangent bundle coordinates in a neighborhood of L in M such that ? in these coordinates maps a covector into its negative. Thus classical examples which have a discrete symmetry such as the restricted three-body problems are locally like a reversible system.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

The symmetry approach on economic systems

This paper studies the symmetry on the dynamic equations of economic systems. Symmetry group transforms a solution of the equations to other solutions. Thus we can understand the transformational relations among these solutions. Symmetry describes the invariance of the dynamic equations. Then the symmetry provides more information for the inner structure of the economic systems. The complexity of the socioeconomic phenomena is discussed by using dynamic systems with symmetry. For the calculation of the symmetry, the prolongation technique is adopted. This method reformulates the basic problem of finding solutions of differential equations in a more geometrical form which is suited to the investigation of symmetry groups. Some examples of dynamic equations by using symmetry analysis are also illustrated.     

Radial symmetry for systems of fractional Laplacian

In this paper, we consider systems of fractional Laplacian equations in Rnwith nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].     

Homoclinic period blow-up in reversible and conservative systems

We show that in conservative systems each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric homoclinic orbits in reversible systems. Our results extend earlier work by Devaney and Henrard, and provide a positive answer to a conjecture of Strömgren. We present a unified approach to both the conservative and the reversible case, based on a technique introduced recently by X.-B. Lin.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday     

Equilibrium fluctuations of Gradient reversible particle systems

Summary The Boltzmann-Gibbs principle is known to be crucial in the study of the fluctuations of interacting particle systems. A new method is proposed in this paper which confirms this principle for models with gradient reversible dynamics in equilibrium. The method is simpler and can be applied to more general models than the conventional one which is developed by Brox et al. To illustrate the idea in more detail, we study the weakly asymmetric simple exclusion process of which the jump rates are slowly varying. As a consequence of the Boltzmann-Gibbs principle, the limit of the density fluctuation fields is identified as a generalized Ornstein-Uhlenbeck process. Finally, the extension to models with long range interactions is briefly discussed.     

Boundedness of solutions for semilinear reversible systems

In this paper we will study the boundedness of all solutions for second-order differential equations

where and satisfies the sublinear growth condition. Since the system in general is non-Hamiltonian, we have to introduce reversibility assumptions to apply the twist theorem for reversible mappings. Under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions.

     

Strong isochronism of two-dimensional reversible cubic systems

We obtain the maximum order of strong isochronism of reversible cubic systems on the plane.     

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria. To cite this article: F. Grabsi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).