Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons
Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 458-470
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In the present paper we consider and study numerically two systems based on model FitzHugh–Nagumo neurons, where in the presence of periodic modulation of parameters it is possible to implement chaotic dynamics on the attractor in the form of a Smale–Williams solenoid in the stroboscopic Poincaré map. In particular, hyperbolic chaos characterized by structural stability occurs in a single neuron supplemented by a time-delay feedback loop with a quadratic nonlinear element.
hyperbolic chaos, Smale–Williams solenoid, FitzHugh–Nagumo neuron, time-delay system
Kuznetsov S. P., Sedova Y. V., Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 458-470
Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
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Examples of mechanical systems are discussed, where quasi-periodic motions may occur, caused by an irrational ratio of the radii of rotating elements that constitute the system. For the pendulum system with frictional transmission of rotation between the elements, in the conservative and dissipative cases we note the coexistence of an infinite number of stable fixed points, and in the case of the self-oscillating system the presence of many attractors in the form of limit cycles and of quasi-periodic rotational modes is observed. In the case of quasi-periodic dynamics the frequencies of spectral components depend on the parameters, but the ratio of basic incommensurate frequencies remains constant and is determined by the irrational number characterizing the relative size of the elements.
Kuznetsov A. P., Kuznetsov S. P., Sedova Y. V., Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
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We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
Universal two-dimensional map and its radiophysical realization
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471
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We suggest a simple two-dimensional map, parameters of which are the trace and Jacobian of the perturbation matrix of the fixed point. On the parameters plane it demonstrates the main universal bifurcation scenarios: the threshold to chaos via period-doublings, the situation of quasiperiodic oscillations and Arnold tongues. We demonstrate the possibility of implementation of such map in radiophysical device.
maps, bifurcations, phenomena of quasiperiodicity
Kuznetsov A. P., Kuznetsov S. P., Pozdnyakov M. V., Sedova Y. V., Universal two-dimensional map and its radiophysical realization, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471