Chaotic Strange Attractor Research Articles

Multi-rotor spacecraft attitude control by triggering chaotic modes on strange chaotic attractors

Multi-rotor spacecraft attitude control by triggering chaotic modes on strange chaotic attractors

Symmetric multi-scroll attractors in magnetized Hopfield neural network under pulse controlled memristor and pulse current stimulation

In this paper, a new method for generating symmetric multi-scroll attractors is proposed based on a magnetized Hopfield neural network (HNN). Firstly, the mathematical model of magnetized HNN by introducing an original memristor to describe the effect of electromagnetic radiation is established, and its dynamic characteristics are analyzed. Then, utilizing two types of multi-level-logic pulse functions, the complex attractor structure changes of the magnetized HNN under external pulse controlled memristor, pulse current stimulation and their combined effects are discussed respectively. The numerical simulation results show that the magnetized HNN under pulse controlled memristor or/and pulse current stimulation can produce multi-scroll chaotic and multi-scroll periodic strange attractors with symmetric structure. The number of attractors can be changed by setting the different series of multi-level-logic pulse functions. Finally, the magnetized HNN circuit under pulse controlled memristor or/and pulse current stimulation is realized by using discrete analog electronic devices. The experimental results verify the validity of the numerical simulation results.

Dynamical study of a novel 4D hyperchaotic system: An integer and fractional order analysis

In this article, a new nonlinear four-dimensional hyperchaotic model is presented. The dynamical aspects of the complex system are analyzed covering equilibrium points, linear stability, dissipation, bifurcations, Lyapunov exponent, phase portraits, Poincaré mapping, attractor projection, sensitivity and time series analysis. To analyze hidden attractors, the proposed system is investigated through nonlocal operator in Caputo sense. The existence of solution of the system in fractional sense is studied by fixed point theory. The stability of fractional order system is demonstrated via Matignon stability criteria. The fractional order system is numerically studied via newly developed numerical method which is based on Newton polynomial interpolation. The evolution of the attractors are depicted with different fractional orders. For few fractional orders, some hidden strange chaotic attractors are observed through graphs. Theoretical and numerical studies demonstrate that this model has complex dynamics with some stimulating physical characteristics. To verify and validate the results, we implement Field Programmable Analog Arrays (FPAA).

Bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals

This study focuses on the stability and local bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals analytically, and numerically. The analytical results are obtained using thenormal form technique and numerical results are obtained using the numerical continuation method. For this model, a number of bifurcations are studied, including the transcritical (pitchfork) and flip bifurcations, the Neimark–Sacker (NS) bifurcations, and the strong resonance bifurcations. We especially determine the dynamical behaviors of the model for higher iterations up to fourth-order. Numerical simulation is employed to present a closed invariant curve emerging about an NS point, and its breaking down to several closed invariant curves and eventuality giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Dynamical Behaviors of an SIR Epidemic Model with Discrete Time

Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot

Abstract This paper presents a new chaotic system that has four attractors including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial condition, Lyapunov spectrum, measure of strangeness, basin of attraction including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation are rigorously treated. It is found by numerical computations that the system has a far-reaching composite basin of attraction, which is important for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding values of electronic components such as the circuital resistances while ensuring the dynamic ranges are well contained. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to innate properties of the system such as sensitivity on initial condition and topological mixing, the robot’s path becomes unpredictable and guaranteed to scan the workspace, respectively.

Synchronization of oscillators with hyperbolic chaotic phases

Synchronization in a population of oscillators with hyperbolic chaotic phases is studied for two models. One is based on the Kuramoto dynamics of the phase oscillators and on the Bernoulli map applied to these phases. This system possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a map for the evolution of the complex order parameter. Beyond a critical coupling strength, this model demonstrates bistability synchrony-disorder. Another model is based on the coupled autonomous oscillators with hyperbolic chaotic strange attractors of Smale-Williams type. Here a disordered asynchronous state at small coupling strengths, and a completely synchronous state at large couplings are observed. Intermediate regimes are characterized by different levels of complexity of the global order parameter dynamics.

Interesting Features of Three-Dimensional Discrete Lotka-Volterra Dynamics

Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.

Spontaneous symmetry breaking and superposition of states in systems with dynamic chaos

The article discusses one of the typical problems described by the equations of nonlinear dynamics—forced oscillations in a system with W-potential. It focuses, in particular, on chaotic oscillations in the presence of dissipation. In this case the state of the system is described by a chaotic (strange) attractor and can be characterized by the probability density in the phase space. A partial differential equation for the probability density is presented. It is shown that when the parameters change in the system under consideration, symmetry breaking can occur. In this case the superposition principle for the probability density is valid. It is similar to the superposition principle for the quantum mechanical function in the problem of particle motion in the W-potential field.

ИССЛЕДОВАНИЕ ВЛИЯНИЯ СЛУЧАЙНЫХ ВОЗМУЩЕНИЙ НА ДИНАМИКУ СИСТЕМЫ В ЗАДАЧЕ СУСЛОВА

The paper considers the generalized Suslov problem with variable parameters and the influence of random perturbations on the dynamics of the system under consideration. The physical meaning of the Suslov problem is Chaplygin's sleigh, which moves along the inner side of the circle. In the case of a deterministic system, a brief review of the previously obtained results is made, the presence of chaotic dynamics in the system and such effects as the appearance of a strange attractor and noncompact (escaping) trajectories is shown. Moreover, the latter may indicate a possible acceleration in the system. The appearance of chaotic strange attractors occurs due to a cascade of bifurcations of doubling the period. We also consider the dynamics of a perturbed system which arises due to the addition of «white noise» modeled by the Wiener process to one of the equations. Changes in the dynamics of a perturbed system compared to an unperturbed one are studied: chaotization of periodic regimes, the appearance of noncompact trajectories, and the premature destruction of strange attractors. In this paper, phase portraits, maps for the period, graphs of system solutions, and a chart of dynamical regimes are constructed using the Maple software package and the software package «Computer Dynamics: Chaos» (/http://site4.ics.org.ru//chaos_pack).

An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

In this paper, we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that is built on the original approach introduced in [ Lopesino et al., 2015a ], and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete-time system, the Hénon map.

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.

Linear properties of chaotic states of systems described by equations of nonlinear dynamics: Analogy with quantum theory

The paper provides a theoretical exploration of properties of systems described by equations of nonlinear dynamics in a chaotic state. Using the example of a system described by Duffing equations, it is shown that when the state of the system corresponds to a chaotic (strange) attractor, it is possible to determine a function whose meaning corresponds to the probability density. In this case, the resulting equation for the probability density is linear, so that the solution methods developed for linear differential equations, in particular the method of perturbation theory, can be applied to solve the equation in question. This results in a linear dependence of the average values of physical quantities on the parameter that characterizes small perturbations of the system. The numerical experiment confirms this linear relationship.

SENSING AND ANALYSIS OF RADIOACTIVE RADON 222Rn CONCENTRATION CHAOTIC VARIABILITY IN AN ATMOSPHERE ENVIRONMENT

For the first time we present the results of computational analysis and modelling the atmospheric radon 222Rn concentration temporal dynamics using the data of the Chester surface observations of the Environmental Measurements Laboratory (USA Dept. of Energy). A chaotic behaviour has been discovered and in details investigated by using nonlinear methods of the chaos and dynamical systems theories. To reconstruct the corresponding strange chaotic attractor, the time delay and embedding dimension are computed. The former is determined by the methods of autocorrelation function and average mutual information, and the latter is calculated by means of correlation dimension method and algorithm of false nearest neighbours. The topological and dynamical invariants for the observed time series of the Rn concentrations are computed..

Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods

Two classes of differentiation and integral operators have been successfully used in modeling real world problems. On the one hand, researchers used the concept of non-local differentiation to capture the heterogeneity of nature linked to heavy-tailed, and many other, nonlocalities’ dependency. On the other hand, they used a local operator with power law setting to capture processes with similarities. Both operators were used for different purposes; however, we shall note that there exist in nature chaotic processes that exhibit both kinds of behavior. Thus, neither the nonlocal operator nor the differential operator with power law setting can capture such processes. In connection to this, a new class of differential operators has been recently introduced. The differential operator has two orders: the first is the fractional order and the second is the fractal dimension. To open new doors to capturing more chaotic behaviors, in this paper, the Thomas cyclically symmetric attractor, the Shilnikov attractor, the Rossler attractor, the Langford attractor and the King Cobra attractor under the domain of newly proposed fractal-fractional derivative operators, are investigated. Three different types of fractal-fractional derivative operators called the Caputo, Caputo-Fabrizio-Caputo and the Atangana-Baleanu-Caputo operators are employed to comprehend the dynamics of the attractors with the help of varying fractional-order $ \Omega$ and fractal-dimension $ \Delta$ parameters, where $ \Omega,\Delta\in ]0, 1]$. The new strange behaviour of the attractors obtained in this study was not possible earlier in the literature for either classical or fractional cases. Various new three-dimensional graphs are presented for the numerical simulations, which have been carried out with the application of well-convergent methods devised to find approximate solutions for the fractal-fractional order systems.

Strange nonchaotic and chaotic attractors in a self-excited thermoacoustic oscillator subjected to external periodic forcing

We experimentally investigate the synchronization dynamics of a self-excited thermoacoustic system forced beyond its phase-locked state. The system consists of a laminar premixed flame in a tube combustor subjected to periodic acoustic forcing. On increasing the forcing amplitude above that required for phase locking, we find that the system can transition out of phase locking and into chaos, which is consistent with the Afraimovich-Shilnikov theorem for the breakdown of a phase-locked torus. However, we also find some unexpected behavior, most notably the emergence of a strange nonchaotic attractor (SNA) before the onset of chaos. We verify the existence of the SNA and chaotic attractor by examining the correlation dimension, the autocorrelation function, the power-law scaling in the Fourier amplitude spectrum, the permutation entropy in a pseudoperiodic surrogate test, and the permutation spectrum. In summary, this study explores the SNA and chaotic dynamics of a thermoacoustic system forced beyond its phase-locked state, opening up new pathways for the development of alternative strategies to control self-excited thermoacoustic oscillations in combustion devices such as gas turbines and rocket engines.

Chaotic and non-chaotic strange attractors of a class of non-autonomous systems.

In this paper, the dynamics of a class of non-autonomous systems, generated from a unified chaotic autonomous system, is studied. It is found, via parameter modulation, that they have chaotic and non-chaotic strange attractors (NCSA). Several representative systems are constructed to illustrate the complex strange dynamics. The first example exhibits Lorenz-like behavior and Chen-like behavior at different time intervals. The second illustrates the existence of NCSA, which is constructed by "joining" the chaotic Chen system and a system with regular dynamics. The third is constructed based on the topological structure of the original autonomous systems, which has complex transient dynamics at the beginning, with a periodic orbit as the omega-limit set. The last one has quasi-periodic coefficients, yielding strange dynamics. These examples demonstrate that non-autonomous systems can have extremely rich and interesting dynamics under certain conditions.

Coupled Lorenz oscillators near the Hopf boundary: Multistability, intermingled basins, and quasiriddling.

We investigate the dynamics of coupled identical chaotic Lorenz oscillators just above the subcritical Hopf bifurcation. In the absence of coupling, the motion is on a strange chaotic attractor and the fixed points of the system are all unstable. With the coupling, the unstable fixed points are converted into chaotic attractors, and the system can exhibit a multiplicity of coexisting attractors. Depending on the strength of the coupling, the motion of the individual oscillators can be synchronized (both in and out of phase) or desynchronized and in addition there can be mixed phases. We find that the basins have a complex structure: the state that is asymptotically reached shows extreme sensitivity to initial conditions. The basins of attraction of these different states are characterized using a variety of measures and depending on the strength of the coupling, they are intermingled or quasiriddled.

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.

Extreme multistability analysis of memristor-based chaotic system and its application in image decryption

An active charge-controlled memristive Chua’s circuit is implemented, and its basic properties are analyzed. Firstly, with the system trajectory starting from an equilibrium point, the dynamic behavior of multiple coexisting attractors depending on the memristor initial value and the system parameter is studied, which shows the coexisting behaviors of point, period, chaos, and quasic-period. Secondly, with the system motion starting from a non-equilibrium point, the dynamics of extreme multistability in a wide initial value domain are easily conformed by new analytical methods. Furthermore, the simulation results indicate that some strange chaotic attractors like multi-wing type and multi-scroll type are observed when the observed signals are extended from voltage and current to power and energy, respectively. Specially, when different initial conditions are taken, the coexisting strange chaotic attractors between the power and energy signals are exhibited. Finally, the chaotic sequences of the new system are used for encrypting color image to protect image information security. The encryption performance is analyzed by statistic histogram, correlation, key spaces and key sensitivity. Simulation results show that the new memristive chaotic system has high security in color image encryption.