Regular and Chaotic Dynamics最新文献

We study bifurcation scenarios leading to the so-called Shilnikov singular attractorsas a result of breakdown of closed invariant curves in the Chialvo map, which is a two-dimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariantcurve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relatesto the immediately occurring big invariant curve after disappearance of a stable fixed pointat the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenariosof breakdown of the invariant curve and chaos development leading to the emergence ofShilnikov singular attractors. Additionally, we study geometrical peculiarities ofthese attractors, such as structures of rotating patterns of orbits inside the Shilnikovsingular funnel and present a two-parametric analysis with Lyapunov exponents and minimala distance between the chaotic attractor and the unstable focus.

We study chaotic dynamics and bifurcations in reversible and time-periodic perturbations of the classical Duffing equation in the case when this equation has a homoclinic figure-8 to the saddle zero equilibrium.We consider the perturbation to be nonconservative and show that the phenomenon of mixed dynamics occurs in the corresponding Poincaré map (T) over the period.However, for small perturbations, the dynamics is predominantly dissipative: here, almost all orbits of (T) from the interior of the homoclinic-8 (perhaps, except for orbits from small resonant zones) tend to a stable fixed point (sink) at forward iterations and to an unstable fixed point (source) at backward iterations.We show that, when the amplitude of perturbation increases, mixed dynamics, as the phenomenon of the intersection of the attractor with the repeller, becomes quite noticeable and even prevalent. In the paper, we propose two different bifurcation scenarios that lead to such mixed dynamics. In the first scenario, it emerges as a result of the phenomenon known as the attractor-repeller collision. In the second scenario, mixed dynamics arises immediately after a symmetric pair of simple saddle-node bifurcations that, due to reversibility, occur simultaneously with the sink and the source.

Within the framework of the two-layer quasi-geostrophic model on a rotating plane, the motions for a special case of three point vortices with intensities (big{(}kappa_{1}^{1},kappa_{2}^{1},kappa_{2}^{2}big{)}=(4,1,-2)) are considered (here, the subscript denotes the layer number: 1 is the upper layer, 2 is the lower layer, and the superscript denotes the vortex number in the layer). Thus, it is assumed that one cyclonic vortex is located in the upper layer, and two vortices, cyclonic and anticyclonic, are located in the lower layer. It is shown that in the general case each vortex performs periodic motions in such a way that every half-period the vortex structure takesa collinear state. In this case, over time, the trajectory of each vortex completely fillsa certain ring region around the vorticity center. However, among the continuum ofthese quasi-ordered trajectories, one can always find a family of closed periodic solutions,both purely circular (preserving the collinear structure) and more complex, so-called(N)-modal ((N)-symmetric) stationary solutions. In this paper, these solutions are constructed and their main properties are described.

In this paper, we prove the theorem given in the title and a similar theorem for generalized pseudo-Anosov homeomorphisms of the Klein bottle with only one “needle” singularity. We reformulate both statements in terms of the invariant foliation singularity type and orientability of foliations. The method we use to prove the theorems implies using the construction of band surface. Every band surface is represented with its combinatorial description, called the configuration. Applying a series of Rauzy transformations to all possible configurations in the cases considered, we show that the necessary conditions imposed on the invariant foliations are violated whence the results follow.

The purpose of this study is to investigate a simple model for the half-center oscillator (HCO), which consists of two nonidentical phase oscillators ((phi)-neurons) with mutual chemical excitatory couplings. In the absence of couplings, each element is in an excitable state. Using one- and two-parameter bifurcation analysis, we determine regions of stability of various synchronous temporal patterns typical for HCO and describe in detail bifurcation transitions between them. The proposed simple HCO model allows one to reproduce the main effects observed in biologically plausible HCO models, and can be easily implemented in locomotor units in neuromorphic robotics.

The problem of motion of a heavy gyrostat with a fixed point under the action of gyroscopic forces, corresponding to the classical Hess case in the problem of motion of a heavy rigid body with a fixed point, is considered. We derive that the problem of motion of a gyrostat is reduced to solving the second-order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions under which the general solution of the corresponding second-order linear differential equation is expressed in terms of Liouvillian functions and, therefore, it can be presented in explicit form. We prove that under the obtained conditions the equations of motion can be integrated by quadratures.

In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows bymeans of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.

We analyze the stability of the straight-line motion of the bicycle depending onthe mass-geometric parameters of the bicycle and its translational velocity.We construct a region in phase space which corresponds to initial conditions under whichthe bicycle tends asymptotically to straight-line motion. To investigate thebifurcations of the periodic solutions of the system, we construct a chart ofdynamical regimes on the plane of two parameters and a three-dimensional Poincaré map.We analyze the possibility of acceleration or deceleration of the bicyclewhen the angular velocity of the rotor periodically changesin time.

We provide a new expansion of the Fourier coefficient of the perturbing function of thePCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formulafor the coefficient in the region of application of KAM theory (i. e., small value of eccentricity and semimajoraxis. See, e. g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes ((m,k)inmathbb{Z}^{2}) and the presence of common zeros as functions of actions between coefficients relative to modes ((m,k)),((2m,2k)) and ((m,k)),((2m,2k)),((3m,3k)).Thanks to the previous expansion, this numerical analysis is done up to order (60) in the powerof eccentricity and semimajor axis. This is thefirst step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called “non-torus” set from (O(1-sqrt{varepsilon})) (implied by standard KAM theory) to (O(1-varepsilon|logvarepsilon|^{c})) for some (c>0).

Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a (SE(2)) Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors.I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term “Nose function” to refer to their expression (N(theta)=big{(}I_{1}cos^{2}theta+I_{3}sin^{2}theta+mz_{C}^{2}(theta)big{)}^{1/2}) where (theta) is the nutation and (z_{C}(theta)) is the center of mass height. (N(theta)) appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles (phi text{(yaw)}, psi text{ roll and } theta). The reduction to 1 DoF is done in two stages: first, reduction by the group (SE(2)={(x,y,phi)}) to (T^{*}S^{2}) with almost symplectic 2-form (Omega_{NH}=dp_{theta}wedge dtheta+dp_{psi}wedge dpsi+Jcdot K). The semibasic term is (Jcdot K=-p_{psi}(dlogbig{(}N(theta)big{)}wedge dpsi). It follows that (Omega_{NH}) is conformally symplectic in the sense that (dleft(frac{1}{N}Omega_{NH}right)=0.)The conserved quantity due to the (S^{1}) symmetry about the body axis is (ell=N(theta)sin^{2}thetadot{psi}), yielding the desired reduction to ((theta,p_{theta})). Further simplification results by taking the new time (dt=sqrt{B(theta)}dtau, text{ with } B=I_{1}+m|CP|^{2}) where (P=(x,y)) is the point of contact.One gets finally (H=frac{1}{2}tilde{p}^{2}_{theta}+V(theta),V(theta)=ell^{2}/2sin^{2}theta+mgz_{C}(theta) text{ with } tilde{p}_{theta}=p_{theta}/sqrt{B}) and usual symplectic form (dtilde{p}_{theta}wedge dtheta). The moments of inertia (I_{1},I_{3}) reappear in the reconstruction.As an example, very basic observations are presented for the torus.A detailed study wasjust finished by A. Kilin and E. Pivovarova in [3].