Types Of Chaotic Behavior Research Articles

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

<p style='text-indent:20px;'>The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> and the detuning <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the <inline-formula><tex-math id="M4">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane is organized by two points of codimension-two bifurcations——a <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}_2 $\end{document}</tex-math></inline-formula>-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the <inline-formula><tex-math id="M6">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.</p>

Safety Behavior Analysis of a Delayed Control System

Time delays in systems are becoming important phenomena now-a-days in regards to its safety issues. A continuous delayed system proposed by A. Uçar is considered for this work. Detailed works are concentrated on finding behavior of this system of continuous delayed system with respect to different system parameters. Self-written code is used to observe the behavior of the system. Self-written code gives flexibility to see behaviors of the system in more in depth. System behavior is observed for a very large range of parameters and comparison is made with others works. Results indicate that for a certain range of values of parameters the system show predictable behavior but after certain range of parameter values the system goes to unpredictable chaotic behavior. In addition, parametric relation is shown for same type of chaotic behavior. It is expected that this finding will increase understanding of complex phenomena involved in delayed dynamical system when safety is prime importance.

Parametric control of self-sustained and self-modulated optomechanical oscillations

Optomechanical systems are known to exhibit a rich set of complex dynamical features including various types of chaotic behavior and multi-stability. Although this exotic behavior has attracted an intense research interest, the utilization of optomechanical systems in technological applications, in most cases necessitates a complex, yet predictable and controllable, oscillatory response. In fact, the various types of robust oscillations supported by optomechanical systems are nested in either the same or neighboring regions of the parameter space, where chaos exists. In this work we systematically dissect the parameter space of the fundamental optomechanical oscillator in order to identify regions where stable self-sustained and self-modulated oscillations exist, by utilizing bifurcation analysis and advanced numerical continuation techniques. Moreover,in cases where bistability occurs, we study the accessibility of these oscillatory states in terms of initial conditions and their location with respect to well-defined basins of attraction. The results provide specific knowledge for the parameter sets enabling the appropriate oscillatory response for different types of applications.

Routes to Chaos Induced by a Discontinuous Resetting Process in a Hybrid Spiking Neuron Model

Several hybrid spiking neuron models combining continuous spike generation mechanisms and discontinuous resetting processes following spiking have been proposed. The Izhikevich neuron model, for example, can reproduce many spiking patterns. This model clearly possesses various types of bifurcations and routes to chaos under the effect of a state-dependent jump in the resetting process. In this study, we focus further on the relation between chaotic behaviour and the state-dependent jump, approaching the subject by comparing spiking neuron model versions with and without the resetting process. We first adopt a continuous two-dimensional spiking neuron model in which the orbit in the spiking state does not exhibit divergent behaviour. We then insert the resetting process into the model. An evaluation using the Lyapunov exponent with a saltation matrix and a characteristic multiplier of the Poincar’e map reveals that two types of chaotic behaviour (i.e. bursting chaotic spikes and near-period-two chaotic spikes) are induced by the resetting process. In addition, we confirm that this chaotic bursting state is generated from the periodic spiking state because of the slow- and fast-scale dynamics that arise when jumping to the hyperpolarization and depolarization regions, respectively.

Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria

This paper reports various chaotic phenomena that occur in a single-link flexible joint (SLFJ) robot manipulator. Four different cases along with subcases are considered here to show different types of chaotic behaviour in a flexible manipulator dynamics. In the first three cases, a partial state feedback as joint velocity and motor rotor velocity feedback is considered, and the resultant autonomous dynamics is considered for analyses. In the fourth case, the manipulator dynamics is considered as a non-autonomous system. The system has (1) one stable spiral and one saddle-node foci, (2) two saddle-node foci and (3) one marginally stable nature of equilibrium points. We found single- and multi-scroll chaotic orbits in these cases. However, with the motor rotor velocity feedback, the system has two unstable equilibria. One of them has an index-4 spiral repellor. In the non-autonomous case, the SLFJ robot manipulator system has an inverse crisis route to chaos and exhibits (1) transient chaos with a stable limit cycle and (2) chaotic behaviour. In all the four cases, the SLFJ manipulator dynamics exhibits coexistence of chaotic orbits, i.e. multi-stability. The various dynamical behaviours of the system are analysed using available methods like phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincare map, parameter space, bifurcation diagram, 0–1 test and frequency spectrum plot. The MATLAB simulation results support various claims made about the system. These claims are further confirmed and validated by circuit implementation using NI Multisim.

Different types of chimera states: an interplay between spatial and dynamical chaos.

We discuss the occurrence of chimera states in networks of nonlocally coupled bistable oscillators, in which individual subsystems are characterized by the coexistence of regular (a fixed point or a limit cycle) and chaotic attractors. By analyzing the dependence of the network dynamics on the range and strength of coupling, we identify parameter regions for various chimera states, which are characterized by different types of chaotic behavior at the incoherent interval. Besides previously observed chimeras with space-temporal and spatial chaos in the incoherent intervals we observe another type of chimera state in which the incoherent interval is characterized by a central interval with standard space-temporal chaos and two narrow side intervals with spatial chaos. Our findings for the maps as well as for time-continuous van der Pol-Duffing's oscillators reveal that this type of chimera states represents characteristic spatiotemporal patterns at the transition from coherence to incoherence.

Complex Earthquake Cycle Simulations Using a Two-Degree-of-Freedom Spring-Block Model with a Rate- and State-Friction Law

Numerical simulations of complex earthquake cycles are conducted using a two-degree-of-freedom spring-block model with a rate- and state-friction law, which has been supported by laboratory experiments. The model consisted of two blocks coupled to each other and connected by elastic springs to a constant-velocity, moving driver. By widely and systematically varying the model parameters, various slip patterns were obtained, including the periodic recurrence of seismic and aseismic slip events, and several types of chaotic behaviour. The transition in the slip pattern from periodic to chaotic is examined using bifurcation diagrams. The model system exhibits typical period-doubling sequences for some parameter ranges, and attains chaotic motion. Simple relationships are found in iteration maps of the recurrence intervals of simulated earthquakes, suggesting that the simulated slip behaviour is deterministic chaos. Time evolutions of the cumulative slip distance in chaotic slip patterns are well approximated by a time-predictable model. In some cases, both seismic and aseismic slip events occur at a block, and aseismic slip events complicate the earthquake recurrence patterns.

Three Types of Chaos in the Forced Nonlinear Schrödinger Equation

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schrödinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation exhibits analogous types of chaotic phenomena. Thus, by adjusting the forcing frequency, the behavior near the plane wave solution may be set to any one of the three different types of chaos for any periodic box length.

QUALITATIVE BEHAVIOR AND STABILITY OF SOLUTIONS OF A HYBRID SYSTEM

In this paper we consider a certain two-dimensional system of delay differential equations with piecewise constant arguments. We find conditions under which this system has constant solutions; i.e. equilibrium points, and their behavior and stability properties. We also find conditions under which certain of these solutions have a type of chaotic behavior. This paper contains results for more general systems than were dealt with in a previous paper [9].

Chaotic population dynamics and biology of the top-predator

We study how the dynamics of a food chain depends on the biology of the top-predator. We consider two model food chains with specialist and generalist top-predators. Both types of food chains display same type of chaotic behavior, short-term recurrent chaos; but the generating mechanisms are drastically different. Food chains with specialist top-predators are dictated by exogenous stochastic factors. On the contrary, the dynamics of those with the generalist top-predator is governed by deterministic changes in system parameters. The study also suggests that robust chaos would be a rarity.

Quasi-periodically forced nonlinear Helmholtz oscillators

In this paper we study a model that describes the dynamics of the tidal elevation within an almost-enclosed short basin that is connected to a tidal sea by a narrow strait. This model has the form of a forced nonlinear Helmholtz oscillator. The forcing is prescribed by the tide at sea and has a quasi-periodic character of a special nature, since the difference between the frequencies of the lunar and solar components of the forcing tide are very small. We focus on the interactions between the nonlinearity in the oscillator caused by the geometry of the basin, and the external forcing tide. The behavior of small amplitude solutions of the weakly (and quasi-periodically) forced Helmholtz oscillator is studied by a combination of the averaging method and the Melnikov method. We construct three different Melnikov functions. These Melnikov functions enable us to distinguish between six structurally different types of chaotic behavior.

Synchronization in driven chaotic systems: Diagnostics and bifurcations

We investigate generic aspects of chaos synchronization in an externally forced Rössler system. By comparing different diagnostic methods, we show the existence of a well-defined cut-off of synchronization associated with the transition from weak to fully developed chaos. Chaotic synchronization is found to be lost at this cut-off only after the last band-merging bifurcation has occurred. Everywhere at the boundary of phase synchronization, one of the Lyapunov exponents becomes equal to zero. Two types of chaotic behavior, differing by the number of vanishing Lyapunov exponents, are observed outside the synchronization region.

Chaos and One-to-Oneness

The term was introduced in a mathematical context by T. Li and J. Yorke [6]. This paper considers chaos in the setting of an interval I of real numbers and a function f that maps I into I. Such a pair (I, f) is called a dynamical system, or simply a system. Given a system (I, f), the function x -*f(f(x)) also maps I into I and is denoted by f[2]. More generally, the function x >f(f(. . . f(f( x) ... ))), where f appears n times, maps I into I, is denoted by f[n], and is called the nt/ iterate off, f[l] being f itself. It is sometimes useful to think of I as the set of states of a physical system. Then f(x) may be thought of as the state of the system that results from an initial state x after one unit of time. Thus, f [n](x) is the state of the system after n time units, given that the initial state was x. A sequence of states x,f(x), f[2](X),...,f[ln](X),... describes the evolution of the system over time. Whether the system (I,f) behaves chaotically is related to the behavior of the sequence of iterates f, f[2],f[3],... f[n] Roughly speaking, in the system means that the iterates of f churn up the points of I. For chaos to occur, this churning up process should not occur separately in disjoint parts of I. A function f is one-to-one if f(x) =f( y) only if x = y. We will prove that, for a one-to-one function f on I, chaotic behavior cannot occur if f is continuous, but that a type of chaotic behavior may occur if f has even one discontinuity. Although it is not the main concern of this paper, we note in passing that continuous functions that are not one-to-one-even seemingly harmless ones-may also lead to chaos.

Parabolic resonances and instabilities.

A parabolic resonance is formed when an integrable two-degrees-of-freedom (d.o.f.) Hamiltonian system possessing a circle of parabolic fixed points is perturbed. It is proved that its occurrence is generic for one parameter families (co-dimension one phenomenon) of near-integrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits a new type of chaotic behavior which includes instabilities in some directions and long trapping times in others. Moreover, in a degenerate case, near a flat parabolic resonance, large scale instabilities appear. A model arising from an atmospherical study is shown to exhibit flat parabolic resonance. This supplies a simple mechanism for the transport of particles with small (i.e. atmospherically relevant) initial velocities from the vicinity of the equator to high latitudes. A modification of the model which allows the development of atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities are clearly observed. (c) 1997 American Institute of Physics.

WAVE PROPAGATION, PATTERN FORMATION AND MEMORY EFFECTS IN LARGE ARRAYS OF INTERCONNECTED CHAOTIC CIRCUITS

We investigate complex dynamic phenomena in arrays composed of interacting chaotic circuits. Such arrays can be thought of as a model of nonlinear phenomena in spatially extended (high-dimensional or infinite-dimensional) systems and active media with potential applications in brain function modelling and signal processing. In this paper we consider a particular structure of the network in which there exist locally double diffusive interactions between the cells. Such a double interaction can be considered as a paradigm and means for understanding very complex interactions existing in real systems where neighboring cells can communicate in various ways. We consider two basic cases where separate cells without coupling exhibit two different types of chaotic behavior. Depending on the connection structure, initial conditions imposed in the cells the array exhibits various kinds of spatially ordered chaotic waves. Patterns of behavior depending on the excitation of the array and the connection structure are studied in some detail. We present results of simulations showing cooperative phenomena. Depending on the connection structure between the cells and their initial states the array can show spatially ordered patterns of behavior. Such orderly spatial patterns resulting in arrays of chaotic elements are referred to as self-organization. We observed travelling target waves and chaotic wavefronts behaving like autowaves. This kind of autowave-like behavior in arrays of chaotic elements is to our knowledge described for the first time.

Spatial chaos in weakly dispersive and viscous media: A nonperturbative theory of the driven KdV-Burgers equation

The asymptotic travelling wave solution of the KdV-Burgers equation driven by the long scale periodic driver is constructed. The solution represents a shock-train in which the quasi-periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions. It is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes. Two types of chaotic behavior are considered. The first type is a weak chaos, where only a small chaotic deviation from the periodic solution occurs. The second type corresponds to the developed chaos where the solution “ignores” the driver period and represents a random sequence of uncorrelated shocks. In the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor, while in the case of developed chaos it moves on a repeller. Both solutions depend on a parameter indicating the reference shock position in the shock-train. The structure of a one dimensional set to which this parameter belongs is investigated. This set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos. The capacity dimension of this set is calculated.

Nonlinear-time-series analysis of chaotic laser dynamics.

We report on the analysis of experiments on a neodymium-doped yttrium aluminum garnet laser with an intracavity frequency-doubling crystal. Three modes of the laser were excited in differing polarization configurations. The total intensity of infrared light was observed and then analyzed using methods of nonlinear-time-series analysis. We present clear evidence using global false nearest neighbors that when all polarizations are parallel, the intensity is chaotic with two positive Lyapunov exponents and the system can be embedded in dimension 7. The noise level in this operating condition, which we call type I chaos, is small. When one of the polarizations is perpendicular to the others, the intensity is again chaotic with positive Lyapunov exponents, but there is substantial noise in the signal of high dimensional origin, and no finite embedding dimension appears possible. We call this type II chaos. We suggest that the origin of this phenomenon is the intrinsic quantum noise associated with the generation of green light, which is 25 times more intense in the type II operating configuration than in the first. In past experiments with this system we have found that the type I chaos can be controlled to unstable periodic orbits while type II cannot. In each type of chaotic laser operation we use local false nearest neighbors to demonstrate that the local dimension of the dynamics is 7. This means seven differential equations can capture the full dynamics of these regimes of the laser. We evaluate local and global false nearest neighbors to support our conclusions and determine the Lyapunov spectrum of each type of chaotic behavior. The predictability of type II chaos is shown to be much less than that of type I, and we make local polynomial models in reconstructed-state space to demonstrate that we can predict rather well for type I chaos. Finally we suggest a fairly standard model for the interaction of the infrared light with the nonlinear frequency doubling medium and with a two-level of the active medium. \textcopyright{} 1996 The American Physical Society.

Chaotic waves and spatio-temporal patterns in large arrays of doubly-coupled Chua's circuits

We investigate complex dynamic phenomena in arrays composed of interacting chaotic circuits. Such arrays can be thought of as a model of nonlinear phenomena in spatially extended (high-dimensional or infinite-dimensional) systems and active media with potential applications in signal processing. In this paper, we consider a particular structure of the network in which there exists double diffusive interactions between the cells. Such a double interaction can be considered as a paradigm and means for understanding very complex interactions existing in real systems where separate cells can communicate in various ways. We consider two basic cases where separate cells without coupling exhibit two different types of chaotic behavior. Depending on the connection structure, initial conditions imposed in the cells, the array exhibits various kinds of spatially ordered chaotic waves. Patterns of behavior depending on the excitation of the array and the connection structure are studied in this paper. Chua's circuits are taken as standard chaotic cells.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Chattering and related behaviour in impact oscillators

One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation

The dynamical behavior of a large one-dimensional system obeying the cubic complex Ginzburg-Landau equation is studied numerically as a function of parameters near a supercritical bifurcation. Two types of chaotic behavior can be distinguished beyond the Benjamin-Feir instability, a phase turbulence regime with a conserved phase winding number and no phase dislocations (space-time defects), and a defect regime with a nonzero density of defects. The transition between the two can either be continuous or discontinuous (hysteretic), depending on parameters. The spatial decay of the phase correlation function is inferred to be exponential in both regimes, with a sharp decrease of the correlation length upon entering the defect phase. The temporal decay of correlations is exponential in the defect regime.