Narrow Results

This is the first of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, the results of an experimental analysis are presented together with an intuitive geometrical qualitative model of the phenomenon.

This is the second of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, a detailed mathematical analysis of the qualitative resonance phenomenon is presented, confirming the intuitions given by the geometrical model discussed in Part I.

Various forms of chaotic synchronization have been proposed as ways of realizing associative memories and/or pattern recognizers. To exploit this kind of synchronization phenomena in temporal pattern recognition, a chaotic dynamical system representing the class of signals that are to be recognized must be established. As shown recently [De Feo, 2003], this system can be determined by means of identification techniques where chaos emerges by itself to model the diversity of nearly periodic signals. However, the emerging chaotic behavior is subharmonic, i.e. period doubling-like, and therefore, as explained in [De Feo, 2004a, 2004b], it is not suitable for a synchronization-based pattern recognition technique. Nevertheless, as shown here, bifurcation theory and continuation techniques can be combined to modify a subharmonic chaotic system and drive it to homoclinic conditions; obtaining in this way a model suitable for synchronization-based pattern recognition.

In this paper we analyze the qualitative behavior of a fixed offshore oil platform under sea action perturbation (flows and tides), which is modeled by an inverted rigid pendulum with an elastic articulation at sea surface. We study their bifurcation properties provided by analytical framework and investigate their response to small oscillations, considered a more realistic case taking into consideration the platform features as flexibility and saturation.

Our goal is to establish analytic conditions for chaotic oscillations occurrence in this model using the Poincaré–Mel'nikov's criterion for separatrix splitting in transverse orbits, considering external periodic force.

Starting from historical researches, we used, like Van der Pol and Le Corbeiller, a cubic function for modeling the current–voltage characteristic of a direct current low-pressure plasma discharge tube, i.e. a neon tube. This led us to propose a new four-dimensional autonomous dynamical system allowing to describe the experimentally observed phenomenon. Then, mathematical analysis and detailed numerical investigations of such a fourth-order torus circuit enabled to highlight bifurcation routes from torus breakdown to homoclinic chaos following the Newhouse–Ruelle–Takens scenario.

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the $x$-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

In this paper, theoretical framework and numerical verification for suppressing homoclinic chaos of a class of vibro-impact oscillators are discussed by adding parametric excitations in the form of $xfcos(\omega \tau +\phi )$ as the control item. The analytical Melnikov method for planar vibro-impact systems is employed to obtain the corresponding thresholds of parameters as sufficient conditions for suppressing chaos. Two typical oscillators are presented to show the effectiveness of theoretical analysis for suppressing homoclinic chaos by tuning the amplitudes, frequencies and phases of the parametric excitations.

The existence of homoclinic orbits is discussed analytically for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, that is, under the same parameter setting, homoclinic orbits and chaos appear simultaneously in the system. In addition, homoclinic chaos can be suppressed to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations.

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.

Robust chaos is defined as the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space since the existence of such windows in the chaotic region implies fragility of the chaos. In this paper, we introduce a new terminology called robustification of chaos, which means creating robust chaos (in the sense of the above definition) in a dynamical system. As a first step, a new chaotification (robustification) method to generate robust chaos in planar maps is presented using simple piecewise smooth feedback to create a border collision bifurcation in the resulting system under some realizable conditions. The results are applied to an elementary example to illustrate the validity of the proposed method.

We review the ideas and experiments that established the onset of laser coherence beyond a suitable threshold. That threshold is the first of a chain of bifurcations in a non linear dynamics, leading eventually to deterministic chaos in lasers. In particular, the so called HC behavior has striking analogies with the electrical activity of neurons. Based on these considerations, we develop a dynamical model of neuron synchronization leading to coherent global perceptions.

Synchronization implies a transitory control of neuron chaos. Depending on the time duration of this control, a cognitive agent has different amounts of awareness.

Combining this with a stream of external inputs, one can point at an optimal use of internal resources, that is called cognitive creativity.

While coherence is associated with long range correlations, complexity arises whenever an array of coupled dynamical systems displays multiple paths of coherence.

What is the relation among the three concepts in the title? While coherence is associated with long range correlations, complexity arises whenever an array of coupled dynamical systems displays multiple paths of coherence. Creativity corresponds to a free selection of a coherence path within a complex nest. As sketched above, it seems dynamically related to chaos control.