Nonchaotic Attractors Research Articles

Dynamic of Cytokine Storm in Human Inflammatory Response of Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV)-Induced Disease

Infection of Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV) in the human body is characterized by the detection of large amounts of the virus by cytokine as the response to inflammation. In the human body, there are two types of inflammatory responses by a cytokine called pro-inflammatory and anti-inflammatory cytokine. A non-dimensionalized mathematical model of the inflammatory response system has been constructed in the form of a system of differential equations, which consists of two variables and five parameters. The rate of pro-inflammatory and anti-inflammatory cytokine natural production are two important parameters that correlate with a chronic SARS-CoV infection called a cytokine storm. Varying these parameter values, some non-chaotic torus attractors were found in some numerical simulation with the initial value around the equilibrium point obtained by fixing parameter values at the Generalized Hopf bifurcation point. These phenomena show that in cytokine storm condition, fluctuation of pro-inflammatory and anti-inflammatory cytokines form some patterns which are complex, so that they are difficult to be predicted. Mesenchymal stem cells (MSC), chloroquine (CQ), and hydroxychloroquine (HCQ), which have an effect on pro-inflammatory and anti-inflammatory cytokine production, can be used to reduce the cytokine storm.

Coexistence of Strange Nonchaotic Attractors in a Quasiperiodically Forced Dynamical Map

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters.

Susceptibility of transient chimera states.

Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.

Route to Chaos and Bistability Analysis of Quasi-Periodically Excited Three-Leg Supporter with Shape Memory Alloy

In this paper, the effect of quasi-periodic excitation on a three-leg supporter configured with shape memory alloy is investigated. We derived the equation of motion for the system using the supporter configuration and polynomial constitutive model of the shape memory alloys (SMAs) based on Falk model. Two sets of parameters and symmetric initial conditions are used to analyze the system. The system responded with a chaotic attractor and a strange nonchaotic attractor. Coexistence of these attractors is studied and discussed with corresponding phase portrait, bifurcation plot, and cross section of basin of attraction. We confirm the quasi-periodic excitation results with generation of strange nonchaotic attractors as discussed in the literature. The special properties like symmetricity and bistability are revealed and the parameter ranges of existence of such behaviors are discussed. The system is analyzed for different phases and the existence of bistability in martensite phase and transition phase is explained. While the system enters into austenite phase, the bistability behavior vanishes. The results provide insight knowledge into dynamical response of a quasi-periodically excited SMA leg support system and will be useful for design improvements and controller design.

Route to logical strange nonchaotic attractors with single periodic force and noise.

Strange nonchaotic attractors (SNAs) have been identified and studied in the literature exclusively in quasiperiodically driven nonlinear dynamical systems. It is an interesting question to ask whether they can be identified with other types of forcings as well, which still remains an open problem. Here, we show that robust SNAs can be created by a small amount of noise in periodically driven nonlinear dynamical systems by a single force. The robustness of these attractors is tested by perturbing the system with logical signals, leading to the emulation of different logical elements in the SNA regions.

The existence of strange nonchaotic attractors in the quasiperiodically forced Ricker family.

In this paper, the Ricker family (a population model) with quasiperiodic excitation is considered. The existence of strange nonchaotic attractors (SNAs) is analyzed in a co-dimension-2 parameter space by both theoretical and numerical methods. We prove that SNAs exist in a positive measure parameter set. The SNAs are nowhere differentiable (i.e., strange). We use numerical methods to identify the existence of SNAs in a larger parameter set. The nonchaotic property of SNAs is verified by evaluating the Lyapunov exponents, while the strange property is characterized by phase sensitivity and rational approximations. We also find that there is a transition region in a parameter plane in which SNAs alternate with chaotic attractors.

A New Route to Strange Nonchaotic Attractors in an Interval Map

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that the smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus. The nonsmooth points on the torus increase more and more with the change of control parameter. Finally, the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.

Long-range memory effects in a magnetized Hindmarsh-Rose neural network

We consider a model network of diffusively coupled Hindmarsh-Rose neurons to study both analytically and numerically, long-range memory effects on the modulational instability phenomenon, chaotic, synchronous and chimera states within the network. The multiple scale method is used to reduce the generic model into a discrete nonlinear Schrödinger equation. The latter is explored in the linear stability analysis and the instability criterion along with the critical amplitude are derived. The analytical results predict that strong local coupling, high electromagnetic induction and strong long-range interactions may support the formation of highly localized excitations in neural networks. Through numerical simulations, the largest Lyapunov exponents are computed for studying chaos, the synchronization factor and the strong of incoherence are recorded for studying, respectively synchronous and chimera states in the network. We find the appropriate domains of space parameters where these rich activities could be observed. As a result, quasi-periodic synchronous patterns, chaotic chimera and synchronous states, strange chaotic and non-chaotic attractors are found to be the main features of membrane potential coupled with memristive current during long-range memory activities of neural networks. Our results suggest that a combination of long-range activity and memory effects in neural networks may produces a rich variety of membrane potential patterns which are involved in information processing, odors recognition and discrimination and various diseases in the brain.

Multistability in a quasiperiodically forced piecewise smooth dynamical system

Considering a class of quasiperiodically forced piecewise smooth systems, we uncover a dynamic phenomenon in which strange nonchaotic attractors (SNAs) and quasiperiodic attractors coexist in nonsmooth dynamical system, obtaining the domains of attraction of these coexisting attractors in parameter space in order to analyze the global dynamics. The global dynamics analysis demonstrates that SNAs are the transition from quasiperiodic attractors to chaotic attractors. The routes to SNAs, including torus-doubling route, torus fractalization route or, simply, fractal route, and intermittency route, are also investigated. The characteristics of SNAs are described by dynamical invariants such as the Lyapunov exponent, power spectrum, phase sensitivity and rational approximations.

Strange nonchaos in self-excited singing flames

We report the first experimental evidence of a strange nonchaotic attractor (SNA) in the natural dynamics of a self-excited laboratory-scale system. In the previous experimental studies, the birth of a SNA was observed in quasiperiodically forced systems; however, such evidence of a SNA in an autonomous laboratory system is yet to be reported. We discover the presence of a SNA between the attractors of quasiperiodicity and chaos through a fractalization route in a laboratory thermoacoustic system. The observed dynamical transitions from order to chaos via a SNA is confirmed through various nonlinear characterization methods prescribed for the detection of a SNA.

STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

Rich dynamics and anticontrol of extinction in a prey–predator system

This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete systems, it is numerically found that, for some small parameter ranges, the system seemingly presents strange nonchaotic attractors. It is shown both analytically and by numerical simulations that the original system and the anticontrolled system undergo several Neimark-Sacker bifurcations. Beside the classical numerical tools for analyzing chaotic systems, such as phase portraits, time series and power spectral density, the 0-1 test is used to differentiate regular attractors from chaotic attractors.

Observation of Strange Nonchaotic Dynamics in the Frame of State-Controlled Cellular Neural Network-Based Oscillator

Abstract In this paper, the strange nonchaotic dynamics of a quasi-periodically driven state-controlled cellular neural network (SC-CNN) based on a simple chaotic circuit is investigated using hardware experiments and numerical simulations. We report here two different routes to strange nonchaotic attractors (SNAs) taken by this SC-CNN based circuit system. These routes were confirmed using rational approximation (RA) theory, finite time Lyapunov exponents, spectrum of the largest Lyapunov exponents and their variance, and phase sensitivity exponent. It is observed that the results from both computer simulations as well as laboratory experiments have spectacular resemblance.

New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation

A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a new addition to the rich nonlinear motions of the Duffing system including equlibria, periodic responses, quasi-periodic oscillations and chaos. In the current slow-fast Duffing system, we find three new attractors in the form of periodic-chaotic motions. These are called the fixed-point chaotic attractor, the fixed-point strange nonchaotic attractor, and the critical behavior with the maximum Lyapunov exponent fluctuating around zero. The system periodically switches between one attractor with a fixed single-well potential and the other with time-varying two-well potentials in every period of excitation. This behavior is apparently the mechanism to generate the periodic-chaotic motion.

Strange nonchaotic attractors in a nonsmooth dynamical system

Strange nonchaotic attractors (SNAs) have fractal geometric structure, but are nonchaotic in the dynamical sense. Since Grebogi et al. discovered SNA in 1984, it has become one of the important topics in nonlinear dynamics. Now, the study of SNAs has been mainly confined to smooth dynamics with quasiperiodic excitation or random excitation. In this paper, we consider a class of single-degree-of-freedom gear dynamical system with quasiperiodic forcing. We show that the gear transmission system can be modeled as a three-dimensional piecewise linear system, which belongs to a typical class of nonsmooth system. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Dynamics of a neuron exposed to integer- and fractional-order discontinuous external magnetic flux

We propose a modified Fitzhugh-Nagumo neuron (MFNN) model. Based on this model, an integer-order MFNN system (case A) and a fractional-order MFNN system (case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractional-order magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity.

Электронное устройство, реализующее странный нехаотический аттрактор Ханта–Отта

Электронное устройство, реализующее странный нехаотический аттрактор Ханта–Отта

Strange nonchaotic attractors in oscillators sharing nonlinearity

We identified the plausibility of the existence of strange nonchaotic attractors (SNAs) in a model of two sinusoidal driven LCR dissipative oscillators sharing a common piecewise nonlinearity. The detected strange nonchaos has been assayed through numerical studies encompassing phase space analysis, Poincaré cross sections, Lyapunov exponents, etc., The strange nonchaotic nature of attractors was examined by recurrence quantification measure. Phase synchronization features of the attractors in the SNA regime have also been included. Experimental evidences have been furnished by means of a real-time electronic circuit to reinforce the observed results. It is expected that the outcomes of the present study could facilitate the understanding and the observation of SNAs in nonlinearly coupled networks, including small-world, scale-free networks and neuronal networks.

Transient chaos generates small chimeras

While the chimera states themselves are usually believed to be chaotic transients, the involvement of chaos behind their self-organization is not properly distinguished or studied. In this work, we demonstrate that small chimeras in the local flux dynamics of an array of magnetically coupled superconducting quantum interference devices (SQUIDs) driven by an external field are born through transiently chaotic dynamics. We deduce analytic expressions for small chimeras and synchronous states which correspond to nonchaotic attractors in the model. We also numerically study the bifurcations underlying the multistability responsible for their generation. Transient chaos manifests itself in the short term flux oscillations with erratically fluctuating amplitudes, exponential escape time distribution and irregular dependence of the escape time to initial conditions. We classify the small chimera states in terms of the position of the non-synchronized member and numerically construct their basin of attraction. The basin is shown to possess an interesting structure consisting of both ordered and fractal parts, which again can be attributed to transient chaos.