Nikov Chaos Research Articles

Grazing, Homoclinic Orbits and Chaos in a Single-Loop Feedback System with a Discontinuous Function

Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil’nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron.

Grazing Bifurcations, Homoclinic Orbits and Chaos in a Ring Network of Linear Neuron-Like Elements with a Single Nonmonotonic Piecewise Constant Output Function

The bifurcations and chaos of a ring of three unidirectionally coupled neuron-like elements are examined as a minimal chaotic neural network. The output function of one neuron is nonmonotonic and piecewise constant while those of the other two neurons are linear. Two kinds of nonmonotonic output functions are considered and it is shown that periodic solutions undergo grazing bifurcations owing to discontinuity in the nonmonotonic functions. Chaotic attractors are created directly through a grazing bifurcation and homoclinic orbits based at pseudo steady states are generated. It is shown that homoclinic/heteroclinic orbits satisfying the condition of Shil’nikov chaos are caused by overshoot in the nonmonotonic functions.

Bifurcation analysis for Shil'nikov Chaos Electro-dissolution of Copper.

This paper is devoted to study the local bifurcations and stability of three dimensional systems that representing a Shil'nikov chaos during copper electro-dissolution. The local stability analysis of equilibrium points has been studied. It is shown that transcritical bifurcation can appears in the system. Also, the existence of Hopf bifurcation of the system around the equilibrium points is studied when the parameter passes through the critical value. Normal form theory is used to study bifurcating periodic solutions.

Tunable bimodal explorations of space from memory-driven deterministic dynamics.

We present a wave-memory-driven system that exhibits intermittent switching between two propulsion modes in free space. The model is based on a pointlike particle emitting periodically cylindrical standing waves. Submitted to a force related to the local wave-field gradient, the particle is propelled, while the wave field stores positional information on the particle trajectory. For long memory, the linear motion is unstable and we observe erratic switches between two propulsive modes: linear motion and diffusive motion. We show that the bimodal propulsion and the stochastic aspect of the dynamics at long time are generated by a Shil'nikov chaos. The memory of the system controls the fraction of time spent in each phase. The resulting bimodal dynamics shows analogies with intermittent search strategies usually observed in living systems of much higher complexity.

Bifurcations and chaos in convection taking non-Fourier heat-flux

In this Letter, we report the influences of thermal time-lag on the onset of convection, its bifurcations and chaos of a horizontal layer of Boussinesq fluid heated underneath taking non-Fourier Cattaneo–Christov hyperbolic model for heat propagation. A five-dimensional nonlinear system is obtained for a low-order Galerkin expansion, and it reduces to Lorenz system for Cattaneo number tending to zero. The linear stability agreed with existing results that depend on Cattaneo number C. It also gives a threshold Cattaneo number, CT, above which only oscillatory solutions can persist. The oscillatory solutions branch terminates at the subcritical steady branch with a heteroclinic loop connecting a pair of saddle points for subcritical steady-state solutions. For subcritical onset of convection two stable solutions coexist, that is, hysteresis phenomenon occurs at this stage. The steady solution undergoes a Hopf bifurcation and is of subcritical type for small value of C, while it becomes supercritical for moderate Cattaneo number. The system goes through period-doubling/noisy period-doubling transition to chaos depending on the control parameters. There after the system exhibits Shil'nikov chaos via homoclinic explosion. The complexity of spiral strange attractor is analyzed using fractal dimension and return map.

Shil’nikov chaos in the 4D Lorenz–Stenflo system modeling the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere

The Lorenz–Stenflo system serves as a model of the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere. In the present paper, we study the Shil’nikov chaos which arises in the 4D Lorenz–Stenflo system. The analytical and numerical results constitute an application of the Shil’nikov theorems to a 4D system (whereas most results present in the literature deal with applying the Shil’nikov theorems to 3D systems), which allows for the study of chaos along homoclinic and heteroclinic orbits arising as solutions to the Lorenz–Stenflo system. We verify the observed chaos via competitive modes analysis—a diagnostic for chaotic systems. We give an analytical test, completely in terms of the model parameters, for the Smale horseshoe chaos near homoclinic orbits of the origin, as well as for the case of specific heteroclinic orbits. Numerical results are shown for other cases in which the general analytical method becomes too complicated to apply. These results can be extended to more complicated higher-dimensional systems governing plasmas, and, in particular, may be used to shed light on period-doubling and Smale horseshoe chaos that arises in such models.

Shil’nikov chaos and mixed-mode oscillation in Chua circuit

We report experimental observations of Shil'nikov-type homoclinic chaos and mixed-mode oscillations in asymmetry-induced Chua's oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcation. The asymmetry is introduced in the Chua circuit by forcing a dc voltage. Then by tuning a control parameter, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of mixed-mode oscillations interspersed by chaotic states. We provide experimental evidences that the asymmetry effect can also be induced in the oscillatory Chua circuit when it is coupled with another one in a rest state. The coupling strength then controls the strength of asymmetry and thereby reproduces all the features of Shil'nikov chaos.

Shil’nikov chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode

We report experimental observations of potential oscillations in electrocatalytic oxidation of formic acid with bismuth ion on the Pt ring electrode. Potential oscillations were investigated under galvanostatic conditions. The presence of bismuth ion enables to get the low and stable potential values at larger current range. Periodic and Shil’nikov chaotic potential oscillations were observed at fixed current. To extract information from the oscillation pattern, we used time series, reconstructed attractor, next-maximum map, pattern formation of interfacial potential above ring electrode.

Sil'nikov chaos of the Liu system

A new chaotic attractor is discovered for the Liu system. The homoclinic and heteroclinic orbits in the Liu system have been found by using the undetermined coefficient method. It analytically demonstrates that there exists one heteroclinic orbit of the Sil'nikov type that connects two nontrivial equilibrium points, and therefore Smale horseshoes and the horseshoe chaos occur for this system via the Sil'nikov criterion. In addition, there also exists one homoclinic orbit joined to the origin. The convergence of the series expansions of these two types of orbits is proved.

Feedback loops for chaos in activator-inhibitor systems

Previous investigations have revealed that special constellations of feedback loops in a network can give rise to saddle-node and Hopf bifurcations and can induce particular bifurcation diagrams including the occurrence of various codimension-two points. To elucidate the role of feedback loops in the generation of more complex dynamics, a minimal prototype for these networks will be taken as purely periodic starting model which will be extended by an additional species in different feedback loops. The dynamics of the resulting systems will be analyzed numerically for the occurrence of chaotic attractors. Especially, the consequences of codimension-two bifurcations and the role of homoclinic orbits in view of the emergence of Shil'nikov chaos will be discussed.

Analytical search for bifurcation surfaces in parameter space

The method of resultants can be used to compute Hopf bifurcations in ODE systems. We discuss this method from the applicant’s point of view. Furthermore, we show by theory and examples that the method may be extended to cover other bifurcation situations as well. Among them are the real Hopf situation, which plays an important role in the transition to Shil’nikov chaos as well as some higher codimension bifurcations, such as Takens–Bogdanov, Gavrilov–Guckenheimer, and double Hopf bifurcations. The method yields an analytical test function that can be solved analytically or by computer algebra systems. In contrast to common analytical techniques based on eigenvalue computation (which can only be applied to systems of size N≤4), the method is applicable for systems of intermediate size ( N<10). We illustrate the power of the method by discussing examples from different disciplines of science: a Lorenz-like oscillator, two coupled oscillators, and a five-species food chain.

Information encoding in homoclinic chaotic systems.

We present a simple method for real-time encoding of information in the interspike intervals of a homoclinic chaotic system. The method has been experimentally tested on a CO2 laser with feedback displaying Sil'nikov chaos and synchronized with an external pulsed signal. Information is encoded by the length of the temporal intervals between consecutive pulses of the external signal. This length is varied each time a new pulse is generated.

Analysis of spontaneous electrochemical oscillations by the wavelet transformation method

A new way of analyzing current oscillations that accompany anodic metal dissolution is presented. A wavelet transformation is used for separation and spectral analysis of singular courses—elementary components of electrochemical oscillation recordings. A method of obtaining oscillation energy distribution analogous to the spectral power density function is presented. It is shown that the chaotic component (Shil'nikov chaos) can be separated from the deterministic component of the described process.

Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: šil'nikov's chaos and the devil's staircase

We study the motion of non-diffusive, passive particles within steady, three-dimensional vortex breakdown bubbles in a closed cylindrical container with a rotating bottom. The velocity fields are obtained by solving numerically the three-dimensional Navier–Stokes equations. We clarify the relationship between the manifold structure of axisymmetric (ideal) vortex breakdown bubbles and those of the three-dimensional real-life (laboratory) flow fields, which exhibit chaotic particle paths. We show that the upstream and downstream fixed hyperbolic points in the former are transformed into spiral-out and spiral-in saddles, respectively, in the latter. Material elements passing repeatedly through the two saddle foci undergo intense stretching and folding, leading to the growth of infinitely many Smale horseshoes and sensitive dependence on initial conditions via the mechanism discovered by šil'nikov (1965). Chaotic šil'nikov orbits spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then downward near the surface, wrapping around the toroidal region in the interior of the bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent with what we would generically anticipate for a mildly perturbed, volume-preserving, three-dimensional dynamical system (MacKay 1994; Mezić &amp; Wiggins 1994a). Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. We calculate residence times of upstream-originating non-diffusive particles and show that when mapped to initial release locations the resulting maps exhibit fractal properties. We argue that there exists a Cantor set of initial conditions that leads to arbitrarily long residence times within the breakdown region. We also show that the emptying of the bubble does not take place in a continuous manner but rather in a sequence of discrete bursting events during which clusters of particles exit the bubble at once. A remarkable finding in this regard is that the rate at which an initial population of particles exits the breakdown region is described by the devil's staircase distribution, a fractal curve that has been already shown to describe a number of other chaotic physical systems.

Homoclinic orbits and chaos in a multimode laser

We show experimentally that under certain conditions the chaotic intensity dynamics of an optically pumped NH3 bidirectional ring laser could be well described in terms of Shil'nikov homoclinic orbits and chaos. We found that the mechanism that resulted in this kind of dynamics of the laser is the competition between effects caused by the mode interaction between the forward and the backward modes of the laser and by the intrinsic single-mode dynamics of the interacting modes.

Dynamic Patterns: The Self-Organization of Brain and Behavior

July 01 1996 Dynamic Patterns: The Self-Organization of Brain and Behavior In Special Collection: CogNet J. A. ScottKelso, Cambridge, MA: The MIT Press, 1995, Hardbound, 334 pages, $49.95. ISBN 0-262-11200-0 A. J. van Opstal A. J. van Opstal Department of Medical Physics and Biophysics, University of Nijmegen, Nijmegen, The Netherlands. Search for other works by this author on: This Site Google Scholar Author and Article Information A. J. van Opstal Department of Medical Physics and Biophysics, University of Nijmegen, Nijmegen, The Netherlands. Online ISSN: 1530-8898 Print ISSN: 0898-929X © 1996 by the Massachusetts Institute of Technology1996 Journal of Cognitive Neuroscience (1996) 8 (4): 385–386. https://doi.org/10.1162/jocn.1996.8.4.385 Cite Icon Cite Permissions Share Icon Share Facebook Twitter LinkedIn MailTo Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Search Site Citation A. J. van Opstal; Dynamic Patterns: The Self-Organization of Brain and Behavior. J Cogn Neurosci 1996; 8 (4): 385–386. doi: https://doi.org/10.1162/jocn.1996.8.4.385 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll JournalsJournal of Cognitive Neuroscience Search Advanced Search This content is only available as a PDF. © 1996 by the Massachusetts Institute of Technology1996 Article PDF first page preview Close Modal You do not currently have access to this content.

Self-organizing dynamics of the human brain: Critical instabilities and Sil'nikov chaos.

Using a sensorimotor coordination task in conjunction with an array of SQUIDs (Superconducting QUantum Interference Devices) we demonstrate critical instabilities in human brain activity patterns. Analysis of the dominant spatial pattern of the brain and its time-varying amplitude displays a task-dependent geometry characteristic of Sil'nikov-like chaos, which changes qualitatively at the transition. (c) 1995 American Institute of Physics.

Shil’nikov case of antiphase dynamics in a multimode laser

We describe Shil’nikov chaos in the antiphase dynamics of a multimode laser with intracavity second-harmonic generation. A theoretical model of a laser operating with a sparse-mode spectrum has been studied using multimode rate equations, which include the effects of spatial gratings and birefringence of the intracavity optical elements. We use the methods of dynamical systems theory to prove the existence of Shil’nikov behavior. Experimental observations of Shil’nikov chaos were carried out with a miniature lithium-neodymium-tetraphosphate (LNP) laser with an intracavity KTP doubler. Numerical modeling is in good agreement with experimental results.

Poincaré versus Boltzmann in Šil'nikov phenomena

By suitable adjustment of the control parameters in a CO 2 laser with feedback we show experimental evidence of Šil'nikov chaos, characterized by intensity pulses almost equal in shape, but irregularly separated in time. The times of return to a Poincaré section are statistically spread, however their iteration map is one-dimensional and in close agreement with that arising from Šil'nikov theory. Thus, the iteration map of the time intervals becomes the most appropriate indicator of this chaos. The residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics. This appears as an inestricable mixture of deterministic unpredictability as introduced by Poincaré and stochastic fluctuations as considered in Boltzmann statistical mechanics.

Rate Processes in Low Dimensional Chaotic Systems with Many Attractors

AbstractKramers' paper and its successive elaborations up to the last decade have considered the transition rate between two stable situations. In the language of system dynamics, we say that the process consists in the transition between the basins of attraction of two fixed point attractors. – Here I present four situations of transitions in non equilibrium systems explored experimentally, and which await a formal treatment in terms of a generalized Kramers transition rate theory.I) Since 1967, with reference to Q‐switched lasers, we have studied the decay out of an unstable state, exploring the passage time statistics in its first and higher order moments.II) In the '80s we have been exploring situations of low dimensional chaos where two or more strange attractors coexist. The jumps from one to another are either noise induced jumps, or they are induced by a change in a control parameter which lowers the energy barrier separating the two attractors (crises).III) Investigating a dynamics with competing unstable points we have shown evidence of Shil'nikov chaos. The identification is done in terms of the iteration map of the successive return times to a given surface of section. This may be considered as an extension of the passage time method to the case of multiple passages.IV) We have given experimental evidence of “chaotic itinerancy”, that is, of jump processes self triggered by the same dynamics. So far, the corresponding understanding is based only on numerical simulations. Since chaotic itinerancy has been observed in optical systems displaying space‐time chaos, a short survey of these phenomena is presented for sake of completeness.