Regular and Chaotic Dynamics最新文献

It is known that there is a duality between the Davey – Stewartson type coupled systems and a class of integrable two-dimensional Toda type lattices. More precisely, the coupled systems are generalized symmetries for the lattices and the lattices can be interpreted as dressing chains for the systems. In our recent study we have found a novel lattice which is apparently not related to the known ones by Miura type transformation. In this article we describe higher symmetries to this lattice and derive a new coupled system of DS type.

We discuss some families of integrable and superintegrable systems in (n)-dimensional Euclidean space which are invariant under (mgeqslant n-2) rotations. The invariant Hamiltonian (H=sum p_{i}^{2}+V(q)) is integrable with (n-2) integrals of motion (M_{alpha}) and an additional integral ofmotion (G), which are first- and fourth-order polynomials in momenta, respectively.

This paper describes an approach to constructing the asymptotics of Gaussian beams, based on the theory of the canonical Maslov operator and the study of the dynamics and singularities of the corresponding Lagrangian manifolds in the phase space. As an example, we construct global asymptotics of Laguerre – Gauss beams, which are solutions of the Helmholtz equation in the paraxial approximation. Depending on the type of the beam and the emerging singularity on the Lagrangian manifold, asymptotics are expressed in terms of the Airy function or the Bessel function. One of the advantages of the described approach is that we can abandon the paraxial approximation and construct global asymptotics in terms of special functions also for solutions of the original Helmholtz equation, which is illustrated by an example.

We consider the effect of an external periodic force on chimera states in the phase oscillator model proposed in [Phys. Rev. Lett, v. 101, 00319007 (2008)].Using the Ott – Antonsen reduction, the dynamical equations for the global order parametercharacterizing the degree of synchronization are constructed. The frequency locking by an external periodic force region isconstructed. The possibility of stable chimeras synchronization and unstable chimerasstabilization is established. The instability development of the chimera states leads to the appearance of breather chimeras or complete synchronization.

In [1], a stable heteroclinic cycle was proposed as a mathematical image of switching activity. Due to the stability of the heteroclinic cycle, the sequential activity of the elements of such a network is not limited in time. In this paper, it is proposed to use an unstable heteroclinic cycle as a mathematical image of switching activity. We propose two dynamical systems based on the generalized Lotka – Volterra model of three excitable elements interacting through excitatory couplings. It is shown that in the space of coupling parameters there is a region such that, when coupling parameters in this region are chosen, the phase space of systems contains unstable heteroclinic cycles containing three or six saddles and heteroclinic trajectories connecting them.Depending on the initial conditions, the phase trajectory will sequentially visit theneighborhood of saddle equilibria (possibly more than once). The described behavior isproposed to be used to simulate time-limited switching activity in neural ensembles.Different transients are determined by different initial conditions. The passage of thephase point of the system near the saddle equilibria included in the heteroclinic cycle isproposed to be interpreted as activation of the corresponding element.

We provide a detailed bifurcation analysis in a three-dimensional system describing interaction between tumor cells, healthy tissue cells, and cells of the immune system. As is well known from previous studies, the most interesting dynamical regimes in this model are associated with the spiral chaos arising due to the Shilnikov homoclinic loop to a saddle-focus equilibrium [1, 2, 3]. We explain how this equilibrium appears and how it gives rise to Shilnikov attractors.The main part of this work is devoted to the study of codimension-two bifurcations which,as we show, are the organizing centers in the system. In particular, we describe bifurcationunfoldings for an equilibrium state when: (1) it has a pair of zero eigenvalues(Bogdanov – Takens bifurcation) and (2) zero and a pair of purely imaginary eigenvalues(zero-Hopf bifurcation). It is shown how these bifurcations are related to the emergenceof the observed chaotic attractors.

In this paper we consider an (Omega)-stable 3-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic 2-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a 3-torus, by Smale’s surgery. Each such attractor has bunches of degree 1 and 2. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with k bunches and k isolated periodic points.

We present some results on the absence of a wide class of invariant measures for dynamical systems possessing attractors.We then consider a generalization of the classical nonholonomic Suslov problem which shows how previous investigations of existence ofinvariant measures for nonholonomicsystems should necessarily be extended beyond the class of measures with strictly positive (C^{1}) densitiesif one wishes to determine dynamical obstructions to the presence of attractors.

The second-order differential equation (ddot{x}+axdot{x}+bx^{3}=0) with (a,binmathbb{R}) has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits according to its parameters (a) and (b). This classification is done in the Poincaré disc in order to control the orbits that escape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincaré disc of the first-order differential system associated to the second-order differential equation. Additionally, we show that this system is always integrable, providing explicitly its first integrals.

In the work we use integral formulas for calculating the monodromy data for the Painlevé-2 equation. The perturbation theory for the auxiliary linear system is constructed and formulas for the variation of the monodromy data are obtained. We also derive a formula for solving the linearized Painlevé-2 equation based on the Fourier-type integral of the squared solutions of the auxiliary linear system of equations.