Hyperchaotic Regime Research Articles

Emergence and evolution of unusual inhomogeneous limit cycles displacing hyperchaos in three quorum-sensing coupled identical ring oscillators

We demonstrate that strongly asymmetric limit cycles can be observed in the system of three identical ring oscillators (3-gene networks known as Repressilators) globally coupled by signal molecule diffusion added to the model in a way like the known bacterial “quorum-sensing” mechanism. These cycles are stable over a wide interval of the coupling strengths where they expel the dominant hyperchaotic regime existing in three Repressilators in very large areas of parameters. The bifurcations of the inhomogeneous limit cycle, with a high-amplitude orbit for one oscillator and two low-amplitude identical orbits for the other two, are traced. Bifurcation analysis reveals an unusual cascade of bifurcations ended in the appearance of a new limit cycle with splitted (slightly nonidentical) low-amplitude orbits. Both cycles lose stability giving birth inhomogeneous chaos in the small parameter interval. Hyperchaos dominates in the parameter plane around the “island” with inhomogeneous limit cycles, and this accounts for very long hyper chaotic transients when the system is returning to stable asymmetric cycles after their perturbations. In turn, it is the cycles that contribute the asymmetric and often rather long pieces in hyperchaotic trajectories. The presented cycles differ from the known asymmetric attractors: inhomogeneous limit cycles born from “oscillation death” and the cycles observing in “smallest chimeras”.

Complex Dynamics of Rössler–Nikolov–Clodong O Hyperchaotic System: Analysis and Computations

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The analysis and numerical simulations show that chaos–hyperchaos transition in RNC-O system is coupled to change in the equilibria type as one large hyperchaotic attractor occurs. Moreover, we show that for this system, a zero-Hopf bifurcation is not possible. We also consider the cases when the divergence of the system is a constant and detected two families of exact solutions.

ADVANCED COMPUTING TOPOLOGICAL AND DYNAMICAL INVARIANTS OF RELATIVISTIC BACKWARD-WAVE TUBE TIME SERIES IN CHAOTIC AND HYPERCHAOTIC REGIMES

The advanced results of computing the dynamical and topological invariants (correlation dimensions values, embedding, Kaplan-York dimensions, Lyapunov’s exponents, Kolmogorov entropy etc) of the dynamics time series of the relativistic backward-wave tube with accounting for dissipation and space charge field and other effects are presented for chaotic and hyperchaotic regimes. It is solved a system of equations for unidimensional relativistic electron phase and field unidimensional complex amplitude. The data obtained make more exact earlier presented preliminary data for dynamical and topological invariants of the relativistic backward-wave tube dynamics in chaotic regimes and allow to describe a scenario of transition to chaos in temporal dynamics.

Appearance of chaos and hyperchaos in evolving pendulum network.

The study of deterministic chaos continues to be one of the important problems in the field of nonlinear dynamics. Interest in the study of chaos exists both in low-dimensional dynamical systems and in large ensembles of coupled oscillators. In this paper, we study the emergence of chaos in chains of locally coupled identical pendulums with constant torque. The study of the scenarios of the emergence (disappearance) and properties of chaos is done as a result of changes in (i) the individual properties of elements due to the influence of dissipation in this problem and (ii) the properties of the entire ensemble under consideration, determined by the number of interacting elements and the strength of the connection between them. It is shown that an increase of dissipation in an ensemble with a fixed coupling force and a number of elements can lead to the appearance of chaos as a result of a cascade of period-doubling bifurcations of periodic rotational motions or as a result of invariant tori destruction bifurcations. Chaos and hyperchaos can occur in an ensemble by adding or excluding one or more elements. Moreover, chaos arises hard since in this case, the control parameter is discrete. The influence of the coupling strength on the occurrence of chaos is specific. The appearance of chaos occurs with small and intermediate coupling and is caused by the overlap of the existence of various out-of-phase rotational mode regions. The boundaries of these areas are determined analytically and confirmed in a numerical experiment. Chaotic regimes in the chain do not exist if the coupling strength is strong enough. The dimension of an observed hyperchaotic regime strongly depends on the number of coupled elements.

Chaos and Hyperchaos in a Ka-Band Gyrotron

We present the first experimental proof of hyperchaotic regimes in a gyrotron. Based on experimental data analysis, we calculate the spectrum of Lyapunov exponents and estimate the correlation dimension of the attractors in non-stationary regimes of oscillation of a Ka-band gyrotron. We prove the existence of chaotic and hyperchaotic behavior characterized by one and two positive Lyapunov exponents, respectively. Estimations of dimensions of the obtained attractors show the values higher than 4 in the developed chaos regimes.

Random matrix ensembles in hyperchaotic classical dissipative dynamic systems

We study the statistical fluctuations of Lyapunov exponents in the discrete version of the non-integrable perturbed sine-Gordon equation, the dissipative AC- and DC-driven Frenkel–Kontorova model. Our analysis shows that the fluctuations of the exponent spacings in the strictly overdamped limit, which is nonchaotic, conform to an uncorrelated Poisson distribution. By studying the spatiotemporal dynamics, we relate the emergence of the Poissonian statistics to Middleton’s no-passing rule. Next, by scanning values of the DC drive and the particle mass, we identify several parameter regions for which this one-dimensional model exhibits hyperchaotic behavior. Furthermore, in the hyperchaotic regime where roughly fifty percent of the exponents are positive, the fluctuations exhibit features of the correlated universal statistics of the Gaussian orthogonal ensemble (GOE). Due to the dissipative nature of the dynamics, we find that the match between the Lyapunov spectrum statistics and the universal statistics of GOE is not complete. Finally, we present evidence supporting the existence of the Tracy–Widom distribution in the fluctuation statistics of the largest Lyapunov exponent.

Emergence and control of complex behaviors in driven systems of interacting qubits with dissipation

Progress in the creation of large-scale, artificial quantum coherent structures demands the investigation of their nonequilibrium dynamics when strong interactions, even between remote parts, are non-perturbative. Analysis of multiparticle quantum correlations in a large system in the presence of decoherence and external driving is especially topical. Still, the scaling behavior of dynamics and related emergent phenomena are not yet well understood. We investigate how the dynamics of a driven system of several quantum elements (e.g., qubits or Rydberg atoms) changes with increasing number of elements. Surprisingly, a two-element system exhibits chaotic behaviors. For larger system sizes, a highly stochastic, far from equilibrium, hyperchaotic regime emerges. Its complexity systematically scales with the size of the system, proportionally to the number of elements. Finally, we demonstrate that these chaotic dynamics can be efficiently controlled by a periodic driving field. The insights provided by our results indicate the possibility of a reduced description for the behavior of a large quantum system in terms of the transitions between its qualitatively different dynamical regimes. These transitions are controlled by a relatively small number of parameters, which may prove useful in the design, characterization, and control of large artificial quantum structures.

Quantifying the Transition from Spiral Waves to Spiral Wave Chimeras in a Lattice of Self-sustained Oscillators

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the networks are the van der Pol oscillator and the FitzHugh-Nagumo neuron. Both models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity which enables us to evaluate the sensitivity of each individual oscillator at finite time. Spi-ral waves are observed in both lattices when the interaction between elements have the local character. The dynamics of all the elements is regular. There are no high-sensitive regions. We have discovered that when the coupling becomes nonlocal, the features of the systems significantly changes. The oscillation regime of the spiral wave center element switches to chaotic one. Besides this, a region with high sensitivity occurs around this oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. Formation of this cluster is accompanied by the sharp increase in values of the maximal Lyapunov exponent to the positive region. Furthermore, we explore that the system can even switch to hyperchaotic regime, when several Lyapunov exponents becomes positive.

Multiscroll chaotic attractors in Optical Injected Semiconductor Laser Driven by a Resonant Tunneling Diode Current

This paper shows a complex nonlinear dynamics of an optically injected laser diode (LD) directly modulated by electrical current from a resonant tunneling diode (RTD) circuit. The LD model evolves a new control parameter called effective gain coefficient (EGC) and can switch between a free-running laser and a perturbed laser. A bifurcation diagram analysis sustained Lyapunov exponents’ spectra reveals chaotic and quasi-periodic dynamics for restricted ranges of the frequency fin and by EGC variations, including the generation of the hyperchaotic regime and multiscroll chaotic attractors through a cascade of periodic oscillations to the chaos. Moreover, an implementation of this new coupled system also shows that when an AC signal Vacsin⁡2πfint is added to the DC voltage bias, the LD outputs show several optical signals such as periodic, coherence collapse and chaotic regimes with different strange complex attractors which are devoted to more improve secure optical telecommunications.

Characterizing the complexity of time series networks of dynamical systems: A simplicial approach.

We analyze the time series obtained from different dynamical regimes of evolving maps and flows by constructing their equivalent time series networks, using the visibility algorithm. The regimes analyzed include periodic, chaotic, and hyperchaotic regimes, as well as intermittent regimes and regimes at the edge of chaos. We use the methods of algebraic topology, in particular, simplicial complexes, to define simplicial characterizers, which can analyze the simplicial structure of the networks at both the global and local levels. The simplicial characterizers bring out the hierarchical levels of complexity at various topological levels. These hierarchical levels of complexity find the skeleton of the local dynamics embedded in the network, which influence the global dynamical properties of the system and also permit the identification of dominant motifs. We also analyze the same networks using conventional network characterizers such as average path lengths and clustering coefficients. We see that the simplicial characterizers are capable of distinguishing between different dynamical regimes and can pick up subtle differences in dynamical behavior, whereas the usual characterizers provide a coarser characterization. However, the two taken in conjunction can provide information about the dynamical behavior of the time series, as well as the correlations in the evolving system. Our methods can, therefore, provide powerful tools for the analysis of dynamical systems.

Variety of spatio-temporal regimes in a 2D lattice of coupled bistable FitzHugh-Nagumo oscillators. Formation mechanisms of spiral and double-well chimeras

We consider the dynamics of a 2D lattice of nonlocally coupled FitzHugh-Nagumo oscillators in the regime of bistability. Special attention is paid to the study of influence both of the type of boundary conditions and of the features of the oscillator interaction on the formation of wave regimes and spatiotemporal structures. It is shown that shifting the effective values of the control parameters of partial oscillators under the influence of coupling significantly changes their dynamics and can lead to switching of the dynamical regimes of the whole system. We also calculate the local indexes of sensitivity and the Lyapunov exponents to quantify the observed regimes. A spiral chimera regime is found in the lattice under study for the case of zero flux boundary conditions. It is revealed that the system is in the hyperchaotic regime for this chimera state. We also explain the mechanisms of emergence of the double-well and spiral chimera states which are realized in the lattice.

Hyperchaos and multistability in the model of two interacting microbubble contrast agents.

We study nonlinear dynamics of two coupled contrast agents that are micrometer size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here, we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system, various types of complex dynamics can occur, namely, we observe periodic, quasiperiodic, chaotic, and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of torus destruction. For the onset of hyperchaotic dynamics, we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the dynamics of two bubbles can be essentially multistable, i.e., various combinations of the coexistence of the above mentioned attractors are possible in this model. These cases include the coexistence of a hyperchaotic regime with an attractor of any other remaining type. Thus, the model of two coupled gas bubbles provides a new example of physically relevant system with multistable hyperchaos.

Characteristics of Chaotic Regimes in a Space-distributed Gyroklystron Model with Delayed Feedback

Within the framework of the nonstationary model with nonfixed field structure, we investigate the model of a 3-mm band gyroklystron with delayed feedback. It is shown that both chaotic and hyperchaotic generation regimes are possible in this system. The chaotic regime due to a Feigenbaum period-doubling cascade is characterized by one positive Lyapunov exponent. Further transition to hyperchaos is determined by the appearance of another positive exponent in the Lyapunov spectrum. The correlation dimension of the corresponding attractors reaches values of about 3.5.

Nonlinear dynamics of laser systems with elements of a chaos: Advanced computational code

A general, uniform chaos-geometric computational approach to analysis, modelling and prediction of the non-linear dynamics of quantum and laser systems (laser and quantum generators system etc) with elements of the deterministic chaos is briefly presented. The approach is based on using the advanced generalized techniques such as the wavelet analysis, multi-fractal formalism, mutual information approach, correlation integral analysis, false nearest neighbour algorithm, the Lyapunov’s exponents analysis, and surrogate data method, prediction models etc There are firstly presented the numerical data on the topological and dynamical invariants (in particular, the correlation, embedding, Kaplan-York dimensions, the Lyapunov’s exponents, Kolmogorov’s entropy and other parameters) for laser system (the semiconductor GaAs/GaAlAs laser with a retarded feedback) dynamics in a chaotic and hyperchaotic regimes.

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Emergence of amplitude death scenario in a network of oscillators under repulsive delay interaction

We report the existence of amplitude death in a network of identical oscillators under repulsive mean coupling. Amplitude death appears in a globally coupled network of identical oscillators with instantaneous repulsive mean coupling only when the number of oscillators is more than two. We further investigate that, amplitude death may emerge even in two coupled oscillators as well as network of oscillators if we introduce delay time in the repulsive mean coupling. We have analytically derived the region of amplitude death island and find out how strength of delay controls the death regime in two coupled or a large network of coupled oscillators. We have verified our results on network of delayed Mackey–Glass systems where parameters are set in hyperchaotic regime. We have also tested our coupling approach in two paradigmatic limit cycle oscillators: Stuart–Landau and Van der Pol oscillators.

Quasiperiodic routes to chaos in confined two-dimensional differential convection.

The complete cascade of bifurcations from steady to chaotic convection, as the Rayleigh number is varied, is considered numerically inside an air-filled differentially heated cavity. The system is assumed to be two-dimensional and is invariant under a generalized reflection about the center of the cavity. In the neighborhood of several codimension-two points, two main routes emerge, characterized by different symmetries of the first oscillatory eigenstate. Along these two competing routes, different sequences of bifurcations and symmetry breakings lead from the steady base flow to the hyperchaotic regime. Several families of two- and three-frequency tori have been identified via the computation of the leading Lyapunov exponents. Modal structures extracted from time series reveal the occurrence of slow internal oscillations in the center of the cavity and faster wall modes confined to vertical boundary layers. Further quasiperiodicity windows have been detected on each route. The different regimes eventually disappear in a boundary crisis in favor of a single, globally symmetric, hyperchaotic regime.

Route to hyperchaos in Rayleigh-Bénard convection

Transition to hyperchaotic regimes in Rayleigh-Bénard convection in a square periodicity cell is studied by three-dimensional numerical simulations. By fixing the Prandtl number at and varying the Rayleigh number as a control parameter, a bifurcation diagram is constructed where a route to hyperchaos involving quasiperiodic regimes with two and three incommensurate frequencies, multistability, chaotic intermittent attractors and a sequence of boundary and interior crises is shown. The three largest Lyapunov exponents exhibit a linear scaling with the Rayleigh number and are positive in the final hyperchaotic attractor. Thus, a route to weak turbulence is found.

Diagnostics of the regime of hyperchaotic dynamics from sequences of threshold-crossing time intervals

We consider a method of hyperchaotic regime diagnostics in a system with self-sustained oscillations by monitoring point processes representing sequences of time intervals between the moments at which the system response signal crosses a threshold level. The possibility of determining two positive Lyapunov exponents from a single point process of short duration is demonstrated.

Characterization of the chaos-hyperchaos transition based on return times.

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.