Symmetric Periodic Solutions Research Articles

Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems [5]. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them [5]. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar idea in [5], based on the Hill-type formula, we will build up the Krein-type trace formula for the first order ODEs, which can be seen as a non-self-adjoint version of the case of Hamiltonian system.

Periodic motions of a gyrostat satellite with a large gyrostatic moment about the centre of mass

An axisymmetric gyrostat satellite, whose centre of mass moves in a fixed circular orbit about an attractive centre is considered. The satellite moves about the centre of mass under the action of the gravitational moment. Under the assumption that the gyrostatic moment of the satellite is large and is directed along its axis of symmetry, the periodic motions of this axis are investigated in a small vicinity of a fixed direction, lying in the orbital plane in absolute space. Such motions are described by a system of fourth-order differential equations with periodic coefficients that contains a large parameter. A theorem for the existence and uniqueness of a symmetric periodic solution of this system that has the required properties is proved using methods for constructing periodic solutions of differential equations with a large parameter.

New families of symmetric periodic solutions of the spatial anisotropic Manev problem

Applying perturbation theory and symmetry conditions, we prove the existence of new families of periodic orbits for the 3-dimensional anisotropic Manev problem, which depends on three parameters.

Symmetric Periodic Orbits and Schubart Orbits in The Charged Collinear Three-Body Problem

Using analytic continuation of periodic orbits and symmetries we construct symmetric periodic solutions in the charged collinear three-body problem for the case where the outer particles are sufficiently small and have small charge. The results obtained allows to construct Schubart-like periodic solutions for different values of masses and charges in this setting and for the case of equal masses and charges of the outer bodies we have explicit Schubart periodic solutions.

The periodic motions of a class of symmetric superlinear Hill’s impact equations

We are concerned with the existence, multiplicity and distribution of the even and periodic solutions of a class of symmetric superlinear Hill's equations with impact. We transform the impact phase-plane into a whole phase-plane by a coordinate transformation and, we prove the existence of infinite numbers of symmetric harmonic and subharmonic solutions of forced impactors as well as the densely distribution of symmetric subharmonic solutions of the unforced equations under some superlinear assumption about time-mapping. We also develop a lemma on sufficient condition for symmetric periodic solutions to symmetric second order equations.

Symmetric periodic solutions for a class of differential delay equations with distributed delay

We consider the nonlinear distributed delay equation x 0 (t) = f Z t d t 1 g(x(s)) ds , d2 (0, 1), where g and f are smooth, bounded, and odd and satisfy a positive and a negative feedback condition, respectively. Using elementary fixed point theory we prove the ex- istence of a nontrivial periodic solution of period 2 + 2d satisfying certain symmetries, given certain growth conditions on f and g near zero.

Symmetric periodic solutions of the Hill’s problem. I

The planar circular Hill’s problem is considered, as well as its limiting integrable variant called the Henon problem, for which the original Hill’s problem is a singular perturbation. Among solutions to the Henon problem there are a countable number of generating solutions-arcs that are uniquely determined by the condition of successive passage through the origin of coordinates—singular point of equations of motion of the Hill’s problem. Using the generating solutions-arcs as “letters” of a certain “alphabet”, one can compose, according to some rules, the “words”: generating solutions of families of periodic orbits of the Hill’s problem. The sequence of letters in a word determines the order of orbit transfer from one invariant manifold to another, while the set of all properly specified words determine the system’s symbolic dynamics.

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i −1 of linear Hamiltonian systems. As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity. This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption. We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity which was proved by Fei, Kim and Wang in 2001.

Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses

The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with equal masses, or a uniformly rotating planar 2N+1 body relative equilibrium with a large central mass [18]. We study the question of stability or instability for symmetric periodic solutions of the equal mass $2N$-body problem without perturbation methods. The hip-hop family is a family of $\mathbb{Z}_2$-symmetric action minimizing solutions, investigated by [7,23], and is shown to be generically hyperbolic on its reduced energy-momentum surface. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing hip-hop orbit to develop conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.

The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

We consider the existence of the periodic solutions in the neighbourhood of equilibria for C∞ equivariant Hamiltonian vector fields. If the equivariant symmetry S acts antisymplectically and S2 = I, we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two‐dimensional flow‐invariant manifold, consisting of a one‐parameter family of symmetric periodic solutions and two two‐dimensional flow‐invariant manifolds each containing a one‐parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian systems.

Nonlinear coupling and instability in the forced dynamics of a non-shallow arch: theory and experiments

Dynamic instability of a non-shallow circular arch, under harmonic time-depending load, is investigated in this paper both in analytical and experimental ways. The analytical model is a 2-d.o.f. reduced model obtained by using a Galerkin projection of a mono-dimensional curved polar continuum. The determination of the regions of instability of the symmetric periodic solution and the discussion of the post-critical behavior are carried out, comparing the results with the experimental evidence on a companion laboratory steel prototype. During post-critical evolution, both periodic and non-periodic solutions are obtained varying the excitation control parameters. The theoretical and experimental models are analyzed around the primary external resonance condition of the first symmetric mode, in the case of a nearly 2:1 internal resonance condition between the first symmetric and anti-symmetric modes. When the motion loses regularity, synthetic complexity indicators are used to describe, in quantitative sense, the nonlinear response.

Symmetric Periodic Solutions of Parabolic Problems With Discontinuous Hysteresis

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model. We construct all periodic solutions with exactly two switching on the period and study their stability. Coexistence of several periodic solutions with different stability properties is proved to be possible. A mechanism of appearance and disappearance of periodic solutions is investigated.

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related with the energy of the primaries.

Non-smooth bifurcation analysis of a piecewise linear chaotic circuit

The dynamics of a nonlinear capacitor circuit is investigated in this paper. The symmetric periodic solution and the chaotic attractor can be observed in numerical simulations. Furthermore, the generalized Jacobian matrix at the non-smooth boundaries is introduced to explore the non-smooth bifurcation mechanism for the periodic solutions. Discontinuous bifurcation in the combination of the Hopf bifurcation and the turning point bifurcation occurs at the non-smooth boundaries. Here, the Hopf bifurcation may result in a new frequency, which leads to periodic oscillation. With the variation of the parameter, the periodic symmetric solution oscillates more quickly, which can also be explained through non-smooth bifurcation, and the conclusion accord well with the numerical results.

10.1007/s11208-008-2007-7

We consider the planar circular restricted three-body problem. It is described by an autonomous Hamiltonian system with two degrees of freedom and one parameter μ ∈ [0, 1/2], which is the mass ratio of the two massive bodies. Periodic solutions of this problem form two-parameter families. We propose methods of computation of symmetric periodic solutions for all values of the parameter μ. Each solution has a period and two traces, namely, the plane and the vertical one. Two characteristics of a family, i.e., its intersection with the symmetry plane, are plotted in the four coordinate systems used in the investigations: two global and two local ones related to the two massive bodies. We also describe generating families, i.e., the limits of families as μ → 0, which are known almost explicitly. As examples, we consider the family c, which begins at the fixed collinear point L 1, and the family i, which begins with direct circular orbits of an infinitely small radius around the primary P 1 of a bigger mass. We give a complete description of the generating families c and i for μ = 0.

Global bifurcations from the center of mass in the Sitnikov problem

The Sitnikov problem is a restricted three body problem where the eccentricity of the primaries acts as a parameter. We find families of symmetric periodic solutions bifurcating from the equilibrium at the center of mass. These families admit a global continuation up to excentricity $e=1$. The same techniques are applicable to the families obtained by continuation from the circular problem ($e=0$). They lead to a refinement of a result obtained by J. Llibre and R. Ortega.

Nonholonomic double-bracket equations and the Gauss thermostat

In this Rapid Communication we consider certain equations that arise from imposing a constant kinetic-energy constraint on a one-dimensional set of oscillators. This is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with Gauss's law of least constraint. Dynamics of this sort are of interest in nonequilibrium molecular dynamics. We show that under certain choices of external potential these equations give rise to a generalization of the so-called double-bracket equations which are of interest in studying gradient flows and integrable systems such as the Toda lattice. In the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained.

A new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem

We show that there exists a new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem. In such a solution, the infinitesimal body is confined to the vicinity of a primary and moves on a nearly circular orbit. This orbit is almost perpendicular to the orbital plane of the primaries, where the line of symmetry of the orbit lies. The existence is shown by applying a corollary of Arenstorf’s fixed point theorem to a periodicity equation system of the problem. And this existence doesn’t require any restriction on the mass ratio of the primaries, nor on the eccentricity of their relative elliptic orbit. Potential relevance of this new class of periodic solutions to real celestial body systems and the follow-up studies in this respect are also discussed.

Closed families of periodic solutions of a restricted three-body problem

The planar restricted three-body problem has an infinite number of families of symmetric periodic solutions (SPSs). The natural SPS families include certain families which are self-closed with respect to small variations in a parameter. These families remain closed for any admissible variations in the mass parameter μ. However, there are closed SRS families of another type, which exist only in bounded intervals of μ and are formed via self-bifurcations of some SPS families. This type of SPS families is poorly understude. This work describes the initial stage (4 bifurcations) of a bifurcation cascade of the natural family i and points out other closed SPS families known to date.

Family h of periodic solutions of the restricted problem for big μ

The results of the computation of the family h of symmetric periodic solutions of the circular planar restricted three-body problem for μ = 0.3, 0.4, and 0.5 are presented. This family begins with retrograde circular orbits around a massive body. Associated with each value of μ are the table of critical orbits, the orbit pictures, the graphs of characteristics of the family in four coordinate systems, and the graphs of the period and traces (planar and vertical). Regularities on the family and its evolution as μ increased were observed.