Systematic Computer Assisted Proofs of periodic orbits of Hamiltonian systems

The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem.     

Existence of multiple periodic solutions for a natural Hamiltonian system in a potential well

We prove, under a pinching hypothesis, a theorem of existence of at least n periodic orbits for a closed regular energy hypersurface Σ in R2n which is the level set of a natural (i.e. of the type “potential + kinetic energy”) Hamiltonian function and projects onto a potential well in the configuration space.     

Limit cycles and Lie symmetries

Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.     

Continuation of periodic motions of a reversible system in non-structurally stable cases. Application to the N-planet problem

The problem of continuing symmetric periodic solutions of an autonomous or periodic system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not guarantee that the solution can be continued, are considered. Three approaches are proposed to solving the problem: (a) particular consideration of terms that depend on the small parameter and the selection of generating solutions; (b) the selection of a generating system depending on the small parameter; (c) reduction to a quasi-linear system which is then analysed using the first approach. Within the framework of the third approach the existence of a periodic motion is also established that differs from the generating one by a quantity whose order is a fractional power of the small parameter. The theoretical results are used to prove the existence of two families of periodic three-dimensional orbits in the N-planet problem. The orbit of each planet is nearly elliptical and situated in the neighbourhood of its fixed plane; the angle between the planes is arbitrary. The average motions of the planets in these orbits relate to one another as natural numbers (the resonance property), and at instants of time that are multiples of the half-period the planes are either aligned in a straight line—the line of nodes (the first family), or cross the same fixed plane (the second family). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbit are independent.     

The Spin–Orbit Resonances of the Solar System: A Mathematical Treatment Matching Physical Data

In the mathematical framework of a restricted, slightly dissipative spin–orbit model, we prove the existence of periodic orbits for astronomical parameter values corresponding to all satellites of the Solar System observed in exact spin–orbit resonance.     

Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system

Let M be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose M consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the λ-lemma) describing the behavior of trajectories near M. Using this result, trajectories shadowing chains of homoclinic orbits to M are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order µ. As µ → 0, double collisions of small bodies correspond to a symplectic critical manifold M of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem.     

Turning numbers for periodic orbits of disk homeomorphisms

We study braid types of periodic orbits of orientation preserving disk homeomorphisms. If the orbit has period n, we take the closure of the nth power of the corresponding braid and consider linking numbers of the pairs of its components, which we call turning numbers. They are easy to compute and turn out to be very useful in the problem of classification of braid types, especially for small n. This provides us with a simple way of getting useful information about periodic orbits. The method works especially well for disk homeomorphisms that are small perturbations of interval maps.     

Time periodic solutions of compressible Navier-Stokes equations

In this paper, we study the Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n?5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

MAXIMAL SUBGROUPS OF SL(<Emphasis Type="Italic">n</Emphasis>, ℤ)

We establish the existence of maximal subgroups of various different natures in SL(n, ?). In particular, we prove that there are 2^?0 maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n, ?) which is not 2-transitive.     

A Continuous Planar Map With Many Periodic Points

The existence and structure of periodic orbits of the discrete delayed logistic equation x_n=;1=rx_n(1?x_n?1) is studied. In the phase plane (x,y)=(x_n?1,x_n), the corresponding continuous planar map is F_r(x,y)=(y, ry(1?x)). The Birkhoff-Smale theorem for infinitely many periodic points in the neighborhood of a homoclinic point of a differentiable map is appropriately modified for the example; the proof resembles the geometric proof of Birkhoff in two dimensions. Related numerical work is included, and implications of the results in terms of a model for population growth are discussed.     

On the simple isotopy class of a source-sink diffeomorphism on the 3-sphere

The results obtained in this paper are related to the Palis-Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse-Smale systems on a closed smooth manifold M ⁿ . Newhouse and Peixoto showed that such an arc joining flows exists for any n and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For n = 1, this is related to the presence of the Poincaré rotation number, and for n = 2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension n = 3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse-Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.     

On the combinatorial classification of nondegenerate configurations in the plane

We classify nondegenerate plane configurations by attaching, to each such configuration of n points, a periodic sequence of permutations of {1, 2, …, n} which satisfies some simple conditions; this classification turns out to be appropriate for questions involving convexity. In 1881 Perrin stated that every sequence satisfying these conditions was the image of some plane configuration. We show that this statement is incorrect by exhibiting a counterexample, for n = 5, and prove that for n ? 5 every sequence essentially distinct from this one is realized geometrically by giving a complete classification of configurations in these cases; there is 1 combinatorial equivalence class for n = 3, 2 for n = 4, and 19 for n = 5. We develop some basic notions of the geometry of “allowable sequences” in the course of proving this classification theorem. Finally, we state some results and an open problem on the realizability question in the general case.     

Choreographies in the <Emphasis Type="Italic">n</Emphasis>-vortex Problem

We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach.     

Horseshoes for the nearly symmetric heavy top

We prove the existence of horseshoes in the nearly symmetric heavy top. This problem was previously addressed but treated inappropriately due to a singularity of the equations of motion. We introduce an (artificial) inclined plane to remove this singularity and use a Melnikov-type approach to show that there exist transverse homoclinic orbits to periodic orbits on four-dimensional level sets. The price we pay for removing the singularity is that the Hamiltonian system becomes a three-degree-of-freedom system with an additional first integral, unlike the two-degree-of-freedom formulation in the classical treatment. We therefore have to analyze three-dimensional stable and unstable manifolds of periodic orbits in a six-dimensional phase space. A new Melnikov-type technique is developed for this situation. Numerical evidence for the existence of transverse homoclinic orbits on a four-dimensional level set is also given.     

Global structure for a class of dynamical systems

In this paper, we shall consider the global structure of positive bounded systems on the plane which have m singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have at least m−1 connecting orbits; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set. Each of other orbits tends to a singular point as t→+∞, and approaches to the infinity as t→−∞.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F(x,y)=(f(x),g(x,y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite ω-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive.