Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems(DPSs). This paper first provides a new class of four-dimensional(4 D) two-zone discontinuous piecewise affine systems(DPASs), and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.     

Symmetries of discontinuous flows and the dual Rankine-Hugoniot conditions in fluid dynamics

It has recently been shown that the maximal kinematical invariance group of polytropic fluids, for smooth subsonic flows, is the semidirect product of SL (2, R) and the static Galilei group G. This result purports to offer a theoretical explanation for an intriguing similarity, that was recently observed, between a supernova explosion and a plasma implosion. In this paper we extend this result to discuss the symmetries of discontinuous flows, which further validates the explanation by taking into account shock waves, which are the driving force behind both the explosion and implosion. This is accomplished by constructing a new set of Rankine-Hugoniot conditions, which follow from Noether’s conservation laws. The new set is dual to the standard Rankine-Hugoniot conditions and is related to them through the SL (2, R) transformations. The entropy condition, that the shock needs to satisfy for physical reasons, is also seen to remain invariant under the transformations.     

Global synchronization for a class of dynamical complex networks

This paper is concerned with the problem of global synchronization for a class of dynamical complex networks composed of general Lur’e systems. Based on the absolute stability theory and the Kalman-Yakubovich-Popov (KYP) lemma, sufficient conditions are established to guarantee global synchronization of dynamical networks with complex topology, directed and weighted couplings. Several global synchronization criteria formulated in the form of linear matrix inequalities (LMIs) or frequency-domain inequalities are also proposed for undirected dynamical networks. In order to obtain global results, no linearization technique is involved through derivation of the synchronization criteria. Numerical examples are provided to demonstrate the effectiveness of the proposed results.     

On the stability of a class of kinetic equations

Stability properties of kinetic model equations (including discrete versions of the master equation and Boltzmann's equation) are derived by means of Lyapunov's direct method. The construction of suitable Lyapunov functions leads to results about the structural stability of the dynamical systems and makes it possible to compose more complicated systems from the given ones, preserving automatically some form of stability.     

Quantum aspects of triangular billiards

The energy spectrum of a particle enclosed in a two-dimensional triangular box can in general not be calculated analytically. Therefore, in order to establish the degree of intergrability, using some well known criteria, this spectrum must be calculated numerically.

For more than 800 triangles these calculations were performed and then used to check a number of conjectures found in the literature. For almost all of them we find counter examples.

Our own suggestion is that in general acute triangles are more integrable than obtuse ones. However, also this rule has its exceptions, which cannot be explained away by some triangles being more rational than others.     

Resonance tunneling in double-well billiards with a point-like scatterer

Coherent tunneling is investigated in rectangular billiards divided into two domains by a classically unclimbable potential barrier. We show that by placing a point-like scatterer inside the billiard, we can control the occurrence and the resonance tunneling rate. The key role of the avoided crossing is stressed.     

Global stability of a susceptible-infected-susceptible epidemic model on networks with individual awareness

Recent research results indicate that individual awareness can play an important influence on epidemic spreading in networks. By local stability analysis, a significant conclusion is that the embedded awareness in an epidemic network can increase its epidemic threshold. In this paper, by using limit theory and dynamical system theory, we further give global stability analysis of a susceptible-infected-susceptible(SIS) epidemic model on networks with awareness. Results show that the obtained epidemic threshold is also a global stability condition for its endemic equilibrium, which implies the embedded awareness can enhance the epidemic threshold globally. Some numerical examples are presented to verify the theoretical results.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

Conditions of stochasticity of two-dimensional billiards

There are two known mechanisms that produce chaos in billiard systems. The first one, discovered by Ya. G. Sinai, is called dispersing, the second, found by the author, is called defocusing. The same mechanisms produce chaos for geodesic flows. Some results on two-dimensional billiards, which indicate that only these two mechanisms can produce chaos in Hamiltonian systems, are discussed.     

Complexity of trajectories in rectangular billiards

To a trajectory of the billiard in a cube we assign its symbolic trajectory-the sequence of numbers of coordinate planes, to which the faces met by the trajectory are parallel. The complexity of the trajectory is the number of different words of lengthn occurring in it. We prove that for generic trajectories the complexity is well defined and calculate it, confirming the conjecture of Arnoux, Mauduit, Shiokawa and Tamura [AMST].The author was supported by DFG.     

Convergence of Hamiltonian systems to billiards

We examine in detail a physically natural and general scheme for gradually deforming a Hamiltonian to its corresponding billiard, as a certain parameter k varies from one to infinity. We apply this limiting process to a class of Hamiltonians with homogeneous potential-energy functions and further investigate the extent to which the limiting billiards inherit properties from the corresponding sequences of Hamiltonians. The results are mixed. Using theorems of Yoshida for the case of two degrees of freedom, we prove a general theorem establishing the "inheritability" of stability properties of certain orbits. This result follows naturally from the convergence of the traces of appropriate monodromy matrices to the billiard analog. However, in spite of the close analogy between the concepts of integrability for Hamiltonian systems and billiards, integrability properties of Hamiltonians in a sequence are not necessarily inherited by the limiting billiard, as we show by example. In addition to rigorous results, we include numerical examples of certain interesting cases, along with computer simulations. (c) 1998 American Institute of Physics.     

Global stability analysis of fluid flows using sum-of-squares

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems.We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.     

Three unequal masses on a ring and soft triangular billiards

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.