Hidden Chaotic Attractors Research Articles

Bistability and hidden attractors in the paradigmatic Rössler'76 system.

In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.

Zero–Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in [Formula: see text] with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in [Formula: see text] depending on a real parameter [Formula: see text], which, for [Formula: see text], coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value [Formula: see text]. We prove that, for [Formula: see text], all the 23 considered systems have a nonisolated zero–Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero–Hopf bifurcation takes place at this point for [Formula: see text], which leads to the creation of three periodic orbits bifurcating from it for [Formula: see text] small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when [Formula: see text] are obtained by period-doubling route to chaos.

Hidden attractors and multistability in a modified Chua’s circuit

The first hidden chaotic attractor was discovered in a dimensionless piecewise-linear Chua’s system with a special Chua’s diode. But designing such physical Chua’s circuit is a challenging task due to the distinct slopes of Chua’s diode. In this paper, a modified Chua’s circuit is implemented using a 5-segment piecewise-linear Chua’s diode. In particular, the coexisting phenomena of hidden attractors and three point attractors are noticed in the entire period-doubling bifurcation route. Attraction basins of different coexisting attractors are explored. It is demonstrated that the hidden attractors have very small basins of attraction not being connected with any fixed point. The PSIM circuit simulations and DSP-assisted experiments are presented to illustrate the existence of hidden attractors and coexisting attractors.

Multistability and Hidden Attractors in a Three-Dimensional Chaotic System

This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory.

A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller

In the present work, a new nonequilibrium four-dimensional chaotic jerk system is presented. The proposed system includes only one constant term and has coexisting and hidden attractors. Firstly, the dynamical behavior of the system is investigated using bifurcation diagrams and Lyapunov exponents. It is illustrated that this system either possesses symmetric equilibrium points or does not possess an equilibrium. Rich dynamics are found by varying system parameters. It is shown that the system enters chaos through experiencing a cascade of period doublings, and the existence of chaos is verified. Then, coexisting and hidden chaotic attractors are observed, and basin attraction is plotted. Moreover, using the multiscale C0 algorithm, the complexity of the system is investigated, and a broad area of high complexity is displayed in the parameter planes. In addition, the chaotic behavior of the system is studied by field-programmable gate array implementation. A novel methodology to discretize, simulate, and implement the proposed system is presented, and the successful implementation of the proposed system on FPGA is verified through the simulation outcome. Finally, a robust sliding mode controller is designed to suppress the chaotic behavior of the system. To deal with unexpected disturbances and uncertainties, a disturbance observer is developed along with the designed controller. To show the successful performance of the designed control scheme, numerical simulations are also presented.

A fractional map with hidden attractors: chaos and control

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings.

A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method

Four-dimensional chaotic systems are a very interesting topic for researchers, given their special features. This paper presents a novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors, which includes only one constant term. The proposed system presents the phenomenon of multi-stability, which means that two or more different dynamics are generated from different initial conditions. It is one of few published works in the last five years belonging to the aforementioned category. Using Lyapunov exponents, the chaotic behavior of the dynamical system is characterized, and the sensitivity of the system to initial conditions is determined. Also, systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits and bifurcation transition diagrams. Moreover, a design technique of a new fuzzy adaptive sliding mode control (FASMC) for synchronization of the fractional-order systems has been offered. This control technique combines an adaptive regulation scheme and a fuzzy logic controller with conventional sliding mode control for the synchronization of fractional-order systems. Applying Lyapunov stability theorem, the proposed control technique ensures that the master and slave chaotic systems are synchronized in the presence of dynamic uncertainties and external disturbances. The proposed control technique not only provides high performance in the presence of the dynamic uncertainties and external disturbances, but also avoids the phenomenon of chattering. Simulation results have been presented to illustrate the effectiveness of the presented control scheme.

Analysis and FPGA implementation of an autonomous Josephson junction snap oscillator

An autonomous Josephson junction (JJ) snap oscillator is designed and investigated in this paper. Depending on DC bias current, the proposed snap oscillator has two or no equilibrium points. The stability analysis of the two equilibrium points shows that one of the equilibrium point is unstable and the existence of Hopf bifurcation is established for the other equilibrium point. During the numerical analysis, some interesting dynamical behaviors such as chaotic self-excited attractors, chaotic hidden attractors, antimonotonicity, chaotic bubble hidden attractors, bistable period-1-bubble and coexistence between periodic and chaotic hidden attractors are found. Finally, the Field Programmable Gate Array (FPGA) of proposed snap oscillator is implemented. The results obtained from the FPGA implementation of proposed snap oscillator are qualitatively the same to the one obtained during the numerical simulations.

Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor

Memristor and fractional-order derivatives are feasible options for constructing new systems with complex dynamics. This paper presents a new fractional-order chaotic system based on a fractional-order memristor (fracmemristor). It is worth noting that this chaotic system based on a fracmemristor does not have any equilibrium points but generates infinitely many hidden chaotic attractors and other dynamical behaviors. Systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits, bifurcation diagrams, Lyapunov exponents, and riddled basins of attraction.

Generation of Extremely Multistable Systems Based on Lurie Systems

Chaotic signals and systems are widely used in image encryption, secure communications, weak signal detection, and radar systems. Many researchers have focused in recent years on the development of systems with an infinite number of coexisting chaotic attractors. We propose some approaches in this work to the generation of self-reproducing systems with an infinite number of coexisting self-excited or hidden chaotic attractors with the same Lyapunov exponents based on mathematical models of Lurie systems. The proposed approach makes it possible to generate extremely multistable systems using numerous known examples of the existence of chaotic attractors in Lurie systems without resorting to an exhaustive computer search. We illustrate the proposed methods by constructing extremely multistable systems with 1-D and 2-D grids of hidden chaotic attractors using the generalized Chua system, in which hidden attractors were first discovered by G.A. Leonov and N.V. Kuznetsov.

A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis.

When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors.

Complex Dynamical Behaviors in a 3D Simple Chaotic Flow with 3D Stable or 3D Unstable Manifolds of a Single Equilibrium

This paper shows some examples of chaotic systems for the six types of only one hyperbolic equilibrium in changed chameleon-like chaotic system. Two of the six cases have hidden attractors. By adjusting the parameters in the system and controlling the stability of only one equilibrium, we can further obtain chaos with four kinds of conditions: (1) index-0 node; (2) index-3 node; (3) index-0 node foci; (4) index-3 node foci. Based on the method of focus quantities, we study three limit cycles (the outmost and inner cycles are stable, and the intermediate cycle is unstable) bifurcating from an isolated Hopf equilibrium. In addition, one periodic solution can be obtained from a nonisolated zero-Hopf equilibrium. The system may help us in better understanding, revealing an intrinsic relationship of the global dynamical behaviors with the stability of equilibrium point, especially hidden chaotic attractors.

Hidden Chaotic Attractors and Synchronization for a New Fractional-Order Chaotic System

Although a large number of hidden chaotic attractors have been studied in recent years, most studies only refer to integer-order chaotic systems and neglect the relationships among chaotic attractors. In this paper, we first extend LE1 of sprott from integer-order chaotic systems to fractional-order chaotic systems, and we add two constant controllers which could produce a novel fractional-order chaotic system with hidden chaotic attractors. Second, we discuss its complicated dynamic characteristics with the help of projection pictures and bifurcation diagrams. The new fractional-order chaotic system can exhibit self-excited attractor and three different types of hidden attractors. Moreover, based on fractional-order finite time stability theory, we design finite time synchronization scheme of this new system. And combination synchronization of three fractional-order chaotic systems with hidden chaotic attractors is also derived. Finally, numerical simulations demonstrate the effectiveness of the proposed synchronization methods.

Self-excited and hidden attractors in a novel chaotic system with complicated multistability

In this paper, a new 3D chaotic system with trigonometric function term as a nonlinear controller is proposed. Depending on the nonlinear controller and the value of the parameters, the proposed system exhibits self-excited attractor with an unstable equilibrium, and hidden attractor with no equilibrium or a stable equilibrium. In addition, the unusual and striking dynamic behavior of the coexistence of self-excited chaotic with periodic attractors, two self-excited chaotic attractors with periodic attractor, three periodic attractors, hidden chaotic with point attractors, two hidden chaotic attractors, and four hidden chaotic attractors are explored by selecting appropriate initial values. Consequently, the proposed system has high sensitivity to its initial values and parameters, hence it can be applied in chaos-based cryptographic applications. Thus, the non-periodicity of coexisting attractors of the system is investigated through Lyapunov exponents and Sample Entropy. To demonstrate the performance of the system in real applications, we construct a pseudo-random number generator (PRNG) based on the hidden attractor case. The randomness test results show that the generated PRNG pass all the statistical tests.

A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors.

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a ‘hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation

In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations.

Comment on “A hidden chaotic attractor in the classical Lorenz system”

In this short communication, we comment on the recent report of a hidden attractor in the classical Lorenz system. We contend that the reported system gives instead a chaotic transient whose duration approaches infinity at a critical value of the parameters. We caution others who are searching for hidden attractors to consider carefully the possibility that the attractor is instead a transient chaotic set.

A new hidden chaotic attractor with extreme multi-stability

The topic of hidden attractors is a very hot topic in nonlinear dynamics. Multi-stability is another important topic in that area. One recent important category of multi-stable systems is “systems with extreme multi-stability”. In this paper we introduce a new five-dimensional chaotic system with both hidden attractors and extreme multi-stability. Such systems are very rare in literature. Also, we show its feasibility by the help of electronic circuit implementation.

Hidden attractors in a new complex generalised Lorenz hyperchaotic system, its synchronisation using adaptive contraction theory, circuit validation and application

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters $$b,\ c$$ and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.

Shadowing in hidden attractors

Hidden attractors found in physical systems are different from self-exited attractors and may have a small basin of attraction. The issue of shadowing in these attractors using dynamical noise is discussed. We have particularly considered two classes of dynamical systems which have hidden attractors in their state space. In one of the systems, there is no fixed point but only a hidden attractor in the state space, while in the other, the system has one unstable fixed point along with a hidden attractor in the state space. The effect of dynamical noise on these dynamical systems is studied by using the Hausdorff distance between the noisy and deterministic attractors. It appears that, up to some threshold value of noise, the noisy trajectory completely shadows the noiseless trajectory in these attractors which is quite different from the results of self-exited attractors. We compare the results of hidden chaotic attractors with the self-exited chaotic attractors.