Regular and Chaotic Dynamics最新文献

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplecticmap, the so-called rationalstandard map, at moderate-to-large values of theamplitude parameter is addressed. After reviewing the model, a discussion concerning an analyticaldetermination of the maximum Lyapunov exponent is provided together with thorough numerical experiments.The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values.Correlations among phases lead to departures from the expected estimates.In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included.Finally, an experimental relationship between the Lyapunov and instability times is shown,while an analytical one applies when correlations are irrelevant, which is the case, in general,for large values of the amplitude parameter.

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.

An example of an analytic system of differential equations in (mathbb{R}^{6}) with an equilibriumformally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories.These tori do not fill all phase space. Though the “gap” between these tori has zero measure, this set is everywhere dense in (mathbb{R}^{6}) and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré – Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.

For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent.In 2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics on closed triangulated surfaces.In 2004, Luo developed a theory of discrete Yamabe flow for closed triangulated surfaces.He investigated the formation of singularities and convergence to a metric of constant curvature.

In this note we develop the theory of a naïve discrete Ricci flow and its modification — the so-called weighted Ricci flow. We prove that this flow has a rich family of first integrals and is equivalent to a certain modification of Luo’s discrete Yamabe flow.We investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weightedconstant curvature.

In this note we introduce the V-shaped action functional with delay in a symplectization,which is an intermediate action functional between the Rabinowitz action functionaland the V-shaped action functional. It lives on the same space as theV-shaped action functional, but its gradient flow equation is a delay equationas in the case of the Rabinowitz action functional. We show that there is a smooth interpolationbetween the V-shaped action functional and the V-shaped action functional with delayduring which the critical points and its actions are fixed. Moreover, we prove that thereis a bijection between gradient flow lines of the V-shaped action functional with delayand the ones of the Rabinowitz action functional.

We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We will use for this purpose the digital line topology on the set ({mathbb{Z}}) of relative integers, also called the Khalimsky topology. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the annulus by using the Lifting Theorem and the Intermediate Value Theorem. More precisely, we will interpretate Birkhoff theory about annular invariant open sets in this formalism. Then we give a proof of Mather’s theorem stating the existence of crossing orbits in a Birkhoff region of instability. Finally we will give a proof of Poincaré – Birkhoff theorem in a particular case, that includes the case where the map is a composition of positive twist maps.

We introduce Smale A-homeomorphisms that include regular, semichaotic, chaotic, andsuperchaotic homeomorphisms of a topological (n)-manifold (M^{n}), (ngeqslant 2). Smale A-homeomorphisms contain axiom A diffeomorphisms (in short, A-diffeomorphisms) provided that (M^{n}) admits a smooth structure. Regular A-homeomorphisms contain all Morse – Smale diffeomorphisms, while semichaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Superchaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. The reason to consider Smale A-homeomorphisms instead of A-diffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform hyperbolicity and pseudo-hyperbolicity, a subject which has already seen an immense number of applications.

We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms (in particular, for all Morse – Smale diffeomorphisms). We applythese necessary and sufficient conditions for structurally stable surface diffeomorphismswith an arbitrary number of expanding attractors. We also use these conditions to obtain acomplete classification of Morse – Smale diffeomorphisms on projective-like manifolds.

This paper presents the stability of resonant rotation of a symmetric gyrostat under third- and fourth-order resonances, whose center of mass moves in an elliptic orbit in a central Newtonian gravitational field. The resonant rotation is a special planar periodic motion of the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space during two orbital revolutions of its center of mass. The equations of motion ofthe gyrostat are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed motion, the unstable region of the resonant rotation and the region of stability in the first-order approximation are determined in the dimensionless parameter plane. In addition, the third- and fourth-order resonances are obtained in the linear stability region and further nonlinear stability analysis is performed in the third- and fourth-order resonant cases.

This paper investigates the dynamics of a toy known as the chatter ring.Specifically, it examines the mechanism by which the small ring rotates around the large ring,the mechanism by whichthe force from the large ring provides torque to the small ring, andwhether the motion of the small ring is the same as that of a hula hoop.The dynamics of a chatter ring has been investigated in previous work [13, 14, 15];however, a detailed analysis has not yet been performed.Thus, to understand the mechanisms described above,the equations of motion and constraint conditionsare obtained, and an analysis of the motion is performed.To simplify the problem, a model consisting ofa straight rod and a washer ring is analyzed under the no-slip condition.The motion of a washer has two modes: the one point of contact (1PC) mode andtwo points of contact (2PC) mode.The motion of the small ring of the chatter ring is similarto that of a washer in the 2PC mode,whereas the motion of a hula hoop is similar to thatof a washer in the 1PC mode.The analysis indicates that the motion of a washer with two points of contactis equivalent to free fall motion. However, in practice, the velocity reaches a constantvalue through energy dissipation.The washer rotates around an axis that passes through the two points of contact.The components of the forces exerted by the rod at the points of contact that are normal to the plane of the washerprovide rotational torque acting at the center of mass.The components of the forces parallel to the horizontal planeare centripetal forces, whichinduce the circular motion of the center of mass.

The chaotic behaviour of dynamical systems can be suppressed if we couple them in some way. In order to do that, the coupling strengths must assume particular values. We illustrate it for the situation that leads to a fixed point behaviour, using two types of couplings corresponding either to a diffusive interaction or a migrative one. For both of them, we present strategies that easily calculate coupling strengths that suppress the chaotic behaviour. We analyse the particular situation of these couplings that consists in a symmetric one and we propose a strategy that provides the suppression of the chaotic evolution of a population.