Abstract: We consider the Hénon-Heiles Hamiltonian system
H=(px2 + py2)/2 + (x2 + y2)/2 + x2 y - y3/3at the critical energy level. We present a computer assisted proof of the existence of a rich symbolic dynamic structure and of infinitely many periodic, homoclinic and heteroclinic orbits.
Abstract: The n-center problem is a Lagrangian system on a 3-dimensional manifold with the potential energy V having n Newtonian singularities. We describe a global version of the KS regularization of singularities. The regularized configuration space turns out to be 4 or 5-dimensional manifold. As an application, using the results of Gromov and Paternain, we show that the n-center problem in R3 has positive topological entropy for n>2 and energy E > sup V. For the n-center problem in S3 the same holds for n>4. These are purely topological phenomena independent of the concrete form of the Lagrangian.
Abstract: We describe some geometric features of a mechanism for detecting global instability in a priori-unstable nearly integrable Hamiltonian systems.
The mechanism presented is based on decomposing the motion in two types of different dynamics, one called inner that takes place inside a normally hyperbolic invariant manifold, where a lot of regular objects (i.e., invariant tori) live, and another one called outer, that takes into account the asymptotic motions to the hyperbolic manifold. The combination of both types of dynamics gives rise to chaotic dynamics and instability.
This mechanism has been applied to several problems. For instance, it was first applied to the existence of orbits of unbounded energy in generic geodesic flows with a (periodic or quasi-periodic) time-dependent potential. More recently, it has been applied to overcome the large gaps problem in Arnold diffusion.
Abstract: In the modern theory of dynamical systems, after the classical work of Fermi, Pasta and Ulam, much emphasis was given to the point of view of Boltzmann and Jeans, concerning systems of physical interest which can be mathematically described as consisting of weakly coupled harmonic oscillators. Indeed, according to such authors, the observed lack of equipartition of energy among the various degrees of freedom could be explained by taking into account the time actually needed for the relaxation to equilibrium, while the existence of extremely long relaxation times has by now become familiar in the theory of dynamical systems after the work of Nekhoroshev. On the other hand, quantitative estimates were lacking. In the present paper an attempt is made at considering realistic models for the simplest case of interest, namely that of diatomic molecules, in order to make a comparison with the experimental data.
Abstract: This talk will present a brief survey and some new developments on the duality and triality theories in general nonsmooth/nonconvex, nonconservative Hamilton systems. By using the canonical dual transformation method developed by the speaker recently, many nonsmooth problems can be transformed into smooth dual problems, and certain nonconvex partial differential systems can be converted into the so-called Differential-Algebraic equations (DAEs). The triality theory reveals some interesting extremality properties and intrinsic symmetry in nonconvex dynamical systems. Based on this theory, an alternative algorithms is proposed for solving nonsmooth/nonconvex dynamical problems.
Application is illustrated by a semi-linear, nonconvex parametric variational problem. Some interesting phenomena, i.e. meta-chaos and post-chaos are discovered in nonconvex/nonconservative Hamilton systems with double-well potential. Dual feed-back control against chaos is discussed.
Abstract: We present a topological refinement of Easton's method of windows. In our approach, two windows are correctly aligned with respect to a given diffeomorphism provided that the boundary of each window decomposes into naturally defined exit and entry components, and the image of some horizontal in the first window can be homotopically deformed onto a horizontal in the second window through a map with nonzero local Brouwer degree. This method does not require any transverse intersections of horizontals and verticals under iteration, so it can be used to study pathological cases in which heteroclinic connections with topological crossings are present. The main result in this line is that one can see through a sequence of correctly aligned windows. We apply this technique to investigate a class of Hamiltonian systems possessing invariant tori whose stable and unstable manifolds cross topologically along solution curves, relative to some Poincaré sections. We show that symbolic dynamics and small Arnold diffusion can be detected in this case. Our method addresses situations in which one infers the existence of homoclinic and heteroclic orbits by solving for the critical points of a Melnikov potential, without necessarily checking their non-degeneracy.
Abstract: Two-dimensional incompressible fluid flows are described by 1DOF time-dependent Hamiltonian systems. In realistic applications, these ystems are often far from integrable, admit general time dependence, and are only known for finite times in the form of experimental or numerical data. For all these reasons, finding hyperbolic structures that cause chaotic or turbulent mixing is such flows remains a great challenge. In this talk we survey recent analytic results on locating finite-time hyperbolic invariant manifolds in two-dimensional fluid flows. We also show how these results can be used to locate Lagrangian coherent structures in geophysical turbulence.
Abstract: In this work we consider the motion of an infinitesimal particle near the (geometrically defined) equilateral points of the real Earth-Moon system. We use, as real system, the one provided by the JPL ephemeris: the ephemeris give the positions of the main bodies of the solar system (Earth, Moon, Sun and planets) so it is not difficult to write the vector field for the motion of a small particle under the attraction of those bodies. Numerical integrations of this vector field show that trajectories with initial conditions in a vicinity of the equilateral points escape after a short time.
To do a preliminary study, we will introduce an analytic model that can be written as a quasi-periodic perturbation of the Restricted Three-Body Problem, and that tries to model the effect on the Sun and the eccentricity of the Moon. Then, we will compute some families of normally elliptic 3-D invariant tori near the triangular points, that give rise to regions of effective stability. By means of numerical integrations, we will show that they seem to persist in the real system, at least for time spans of 1000 years.
Abstract: We shall present a geometric proof of two important theorems by J.Mather about diffusing orbits of Hamiltonian systems. One (connecting) says that for an area-preserving twist map inside of region of instability there are trajectories connecting any two Aubry-Mather sets. Area-preserving twist maps naturally arise in Hamiltonian systems in two-degrees of freedom. Another (accelerating) says that for a generic Hamiltonian systems on a two-torus which is time periodic (2 1/2 degrees of freedom) there is a trajectory whose speed gradually accelerates to an arbitrary large speed. Other proofs of the last theorem were given by Bolotin-Treschev and Delshams-de la Llave-Seara.
Abstract: We describe a renormalization group (RG) transformation for partially analytic Hamiltonians in 2n variables, and its relation to a RG transformation for n-tuples of commuting maps. The conjecture is that a transformation of this type has a fixed point with nontrivial scaling properties. We show that, under some technical assumptions, such a fixed point (Hamiltonian) has an invariant torus that is critical, in the sense that its degree of differentiability is positive but less than one. We will also report on progress toward a proof of the above mentioned conjecture.
Abstract: We will review our previous results on the global behavior of two degrees of freedom classical mechanical systems with cubic homogeneous potentials for the simple case of negative energy. Then we will present new results for the case of positive energy. This case is harder than the case of negative energy, because in McGehee like blow up coordinates at infinity the flow is gradient-like only at the infinity manifold. Besides, solutions may reach the origin of configuration plane, which is sent to the unbounded part in blow up coordinates.
Abstract: Near a nonresonant, elliptic fixed point, a symplectic map can be transformed into Birkhoff normal form. In these coordinates, the dynamics is represented entirely by the Lagrangian frequency map which gives the rotation number as a function of the action. The twist matrix, given by the Jacobian of the rotation number, describes the anharmonicity in the system. When the twist is singular the frequency map is in general not locally one-to-one. We will discuss the occurrence of fold and cusp singularities in the frequency map and show that these necessarily occur near third order resonances. We illustrate the results by numerical computations of frequency maps for a quadratic, symplectic map.
Abstract: In this paper we start from the discrete version of linear Hamiltonian systems with periodic coefficients
where Ak and Dk are Hermitian matrices, Ak, Bk, Dk define N-periodic sequences, and l is a complex parameter. For this system a Krein-type theory of the l -zones of strong (robust) stability may be constructed. Within this theory the side l -zones' width may be estimated using the multipliers' traffic rules of Krein while the central stability zone (centered around l = 0) is estimated using the eigenvalues of a certain self-adjoint boundary value problem. In the discrete-time there occur some specific differences with respect to the continuous time case due to the fact that the transition matrix (hence the monodromy matrix also) is not entire with respect to l but rational. During the paper we consider some specific cases (the matrix analogue of the discretized Hill equation, the J-unitary and symplectic systems, real scalar systems) for which the results on the eigenvalues are complete and obtain some simplified estimates of the central stability zones.
yk+1 - yk = l Bk yk + l Dk zk+1 zk+1 - zk = - l Ak yk - l Bk* zk+1
Abstract: We use dynamical systems theory to construct a general phase space version of transition state theory. Special multidimensional separatrices are found which act as impenetrable barriers in phase space between reacting and nonreacting trajectories. The elusive momentum-dependent transition state between reactants and products is thereby characterized. A practical algorithm is presented and applied to a strongly coupled Hamiltonian.
Abstract: In this talk we show the normal form approach as a way of getting an analytical handle on geometric objects, such as normally hyperbolic invariant manifolds (NHIM), and the breakthrough this is in chemistry problems. Computer visualization is used in order to see these objects. In order to illustrate the technique we focus on the Rydberg atom problem. Concretely, we show how to determine analytically the transition state (TS) in this type of chemical reactions. For that we calculate the normal form and transform the original three-degree-of-freedom (3DOF) Hamiltonian to one of 0DOF. In fact, we are able to construct three asymptotic integrals of the original Hamiltonian by inverting the normal form transformation. Moreover, we calculate in the original system the three-dimensional normally hyperbolic invariant manifold (NHIM), its stable and unstable manifolds, as well as the transition state. We compute trajectories that start on the NHIM in the five-dimensional energy surface. Besides, we determine trajectories in either the forward or backward stable and unstable manifolds associated to the NHIM. These trajectories are simply chosen and computed from the normal form vector field. The normal form transformation then allows us to visualize them in the original coordinates. Thus, we have complete control and knowledge of the exact dynamical trajectories near the TS in a 3DOF system. This is the first time this has been demonstrated for a 3DOF chemical or atomic system.