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# Vol. 12, No. 1, 2016

 Matyushkin I. V. Abstract The properties of an $e^{iz}$ map are studied. It is proved that the map has one stable and an infinite number of unstable equilibrium positions. There are an infinite number of repellent twoperiodic cycles. The nonexistence of wandering points is heuristically shown by using MATLAB. The definition of helicity points is given. As for other hyperbolic maps, Cantor bouquets are visualized for the Julia and Mandelbrot sets. Keywords: holomorphic dynamics, fractal, Cantor bouquet, hyperbolic map Citation: Matyushkin I. V., On some properties of an ${\rm exp}(iz)$ map , Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 3-15 DOI:10.20537/nd1601001
 Morozov Y. Abstract We consider a class of symmetric planar Filippov systems. We find the interval of variation of the bifurcation parameter for which there is an unstable limit cycle. There exist stationary points into the domain, which has this cycle as a boundary. The type of points depends on the value of the bifurcation parameter. There is a redistribution of the area, bounded by this cycle, between the attraction domains of stationary points. The results of numerical simulations are presented for the most interesting values of the bifurcation parameter. Keywords: limit cycle, planar system with a discontinuous right-hand side, global bifurcation Citation: Morozov Y., The limit cycle as a result of global bifurcation in a class of symmetric systems with discontinuous right-hand side, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 17-30 DOI:10.20537/nd1601002
 Kostromina O. S. Abstract Small time-periodic perturbations of an asymmetric Duffing – Van der Pol equation with a homoclinic “figure-eight” of a saddle are considered. Using the Melnikov analytical method and numerical simulations, basic bifurcations associated with the presence of a non-rough homoclinic curve in this equation are studied. In the main parameter plane the bifurcation diagram for the Poincaré map is constructed. Depending on the parameters, the boundaries of attraction basins of stable fixed (periodic) points of the direct (inverse) Poincaré map are investigated. It is ascertained that the transition moment of the fractal dimension of attraction basin boundaries of attractors through the unit may be preceded by the moment of occurrence of the first homoclinic tangency of the invariant curves of the saddle fixed point. Keywords: bifurcations, homoclinic Poincaré structures, attraction basins, fractal dimension, sensitive dependence on initial conditions Citation: Kostromina O. S., On the investigation of the bifurcation and chaotic phenomena in the system with a homoclinic “figure-eight”, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 31-52 DOI:10.20537/nd1601003
 Jalnine A. Y. Abstract In the present paper we consider a family of coupled self-oscillatory systems presented by pairs of coupled van der Pol generators and FitzHugh–Nagumo neural models, with the parameters being periodically modulated in anti-phase, so that the subsystems undergo alternate excitation with a successive transmission of the phase of oscillations from one subsystem to another. It is shown that, due to the choice of the parameter modulation and coupling methods, one can observe a whole spectrum of robust chaotic dynamical regimes, taking the form ranging from quasiharmonic ones (with a chaotically floating phase) to the well-defined neural oscillations, which represent a sequence of amplitude bursts, in which the phase dynamics of oscillatory spikes is described by a chaotic mapping of Bernoulli type. It is also shown that 4D maps arising in a stroboscopic Poincaré section of the model flow systems universally possess a hyperbolic strange attractor of the Smale–Williams type. The results are confirmed by analysis of phase portraits and time series, by numerical calculation of Lyapunov exponents and their parameter dependencies, as well as by direct computation of the distributions of angles between stable and unstable tangent subspaces of chaotic trajectories. Keywords: chaos, hyperbolicity, Smale–Williams attractor, neurons, FitzHugh–Nagumo model Citation: Jalnine A. Y., From quasiharmonic oscillations to neural spikes and bursts: a variety of hyperbolic chaotic regimes based on Smale – Williams attractor, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 53-73 DOI:10.20537/nd1601004
 Markeev A. P. Abstract We study the inertial motion of a material point in a planar domain bounded by two coaxial parabolas. Inside the domain the point moves along a straight line, the collisions with the boundary curves are assumed to be perfectly elastic. There is a two-link periodic trajectory, for which the point alternately collides with the boundary parabolas at their vertices, and in the intervals between collisions it moves along the common axis of the parabolas. We study the nonlinear problem of stability of the two-link trajectory of the point. Keywords: map, canonical transformations, Hamilton system, stability Citation: Markeev A. P., On the stability of the two-link trajectory of the parabolic Birkhoff billiards, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 75-90 DOI:10.20537/nd1601005
 Munitsyn A. I.,  Munitsyna M. A. Abstract An analytical solution of the problem of forced oscillation of the solid parallelepiped on a horizontal base is presented. It is assumed that the slippage between the body and the base is absent, and the base moves harmonically in a horizontal direction. It is also assumed that the height of the box is much larger than the width. The dissipation of impact is taken into account in the framework of Newton’s hypothesis. The forced oscillation modes of parallelepiped corresponding to the main and two subharmonic resonances are found by using the averaging method. The results are shown in the form of amplitude-frequency characteristics. Keywords: supported plane, nonlinear oscillations, averaging method Citation: Munitsyn A. I.,  Munitsyna M. A., Oscillations of a solid parallelepiped on a supported base, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 91-98 DOI:10.20537/nd1601006
 Tenenev V. A.,  Vetchanin  E. V.,  Ilaletdinov L. F. Abstract This paper is concerned with the process of the free fall of a three-bladed screw in a fluid. The investigation is performed within the framework of theories of an ideal fluid and a viscous fluid. For the case of an ideal fluid the stability of uniformly accelerated rotations (the Steklov solutions) is studied. A phenomenological model of viscous forces and torques is derived for investigation of the motion in a viscous fluid. A chart of Lyapunov exponents and bifucation diagrams are computed. It is shown that, depending on the system parameters, quasiperiodic and chaotic regimes of motion are possible. Transition to chaos occurs through cascade of period-doubling bifurcations. Keywords: ideal fluid, viscous fluid, motion of a rigid body, dynamical system, stability of motion, bifurcations, chart of Lyapunov exponents Citation: Tenenev V. A.,  Vetchanin  E. V.,  Ilaletdinov L. F., Chaotic dynamics in the problem of the fall of a screw-shaped body in a fluid, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 99-120 DOI:10.20537/nd1601007
 Kuznetsov S. P. Abstract Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston–Weeks–Hunt–MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation. Keywords: dynamical system, chaos, hyperbolic attractor, Anosov dynamics, rotator, Lyapunov exponent, self-oscillator Citation: Kuznetsov S. P., Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 121-143 DOI:10.20537/nd1601008
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Abstract In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs. Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., On the Hadamard–Hamel problem and the dynamics of wheeled vehicles, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 145-163 DOI:10.20537/nd1601009
 Abstract Citation: VI International Conference “Geometry, Dynamics, Integrable Systems – GDIS 2016”, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp. 165-166

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