Regular and Chaotic Dynamics最新文献

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.

We study the spike activity of two mutually coupled FitzHugh – Nagumo neurons, which is influenced by two-frequency signals. The ratio of frequencies in the external signal corresponds to musical intervals (consonances). It has been discovered that this system can exhibit selective properties for identifying musical intervals. The mechanism of selectivity is shown, which is associated with the influence on the spiking frequency of neurons by intensity of the external signal and nature of the interaction of neurons.

We consider the infinite-dimensional vector of frequencies (omega(mathtt{m})=(sqrt{j^{2}+mathtt{m}})_{jinmathbb{Z}}), (mathtt{m}in[1,2])arising from a linear Klein – Gordon equation on the one-dimensional torus and prove that there exists a positive measure set of masses (mathtt{m}^{prime})s for which (omega(mathtt{m})) satisfies a Diophantine condition similar to the one introduced by Bourgain in [14],in the context of the Schrödinger equation with convolution potential.The main difficulties we have to deal with arethe asymptotically linear nature of the (infinitely many) (omega_{j}^{prime})s and the degeneracy coming from having only one parameter at disposal for their modulation.As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category.

In this paper we consider the persistence of elliptic lower-dimensional invariant tori with prescribed frequencies in Hamiltonian systems with small parameters. Under the Brjuno nondegeneracy condition,if the prescribed frequencies satisfy a Diophantine condition, by the KAM technique we prove that for most of small parameters in the sense of Lebesgue measure, the Hamiltonian systems admit a lower-dimensionalinvariant torus whose frequency vector is a dilation of the prescribed frequencies.

We study the KAM-stability of several single star two-planetnonresonant extrasolar systems. It is likely that the observedexoplanets are the most massive of the system considered. Therefore,their robust stability is a crucial and necessary condition for thelong-term survival of the system when considering potentialadditional exoplanets yet to be seen. Our study is based on theconstruction of a combination of lower-dimensional elliptic and KAMtori, so as to better approximate the dynamics in the framework ofaccurate secular models. For each extrasolar system, we explore theparameter space of both inclinations: the one with respect to theline of sight and the mutual inclination between the planets. Ourapproach shows that remarkable inclinations, resulting inthree-dimensional architectures that are far from being coplanar,can be compatible with the KAM stability of the system. We findthat the highest values of the mutual inclinations are comparable tothose of the few systems for which the said inclinations are determinedby the observations.