Self-excited Attractors Research Articles

Multistable Systems with Hidden and Self-Excited Scroll Attractors Generated via Piecewise Linear Systems

In this work, we present an approach to design a multistable system with one-directional (1D), two-directional (2D), and three-directional (3D) hidden multiscroll attractor by defining a vector field on ℝ3 with an even number of equilibria. The design of multistable systems with hidden attractors remains a challenging task. Current design approaches are not as flexible as those that focus on self-excited attractors. To facilitate a design of hidden multiscroll attractors, we propose an approach that is based on the existence of self-excited double-scroll attractors and switching surfaces whose relationship with the local manifolds associated to the equilibria lead to the appearance of the hidden attractor. The multistable systems produced by the approach could be explored for potential applications in cryptography, since the number of attractors can be increased by design in multiple directions while preserving the hidden attractor allowing a bigger key space.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

Multistability and coexisting transient chaos in a simple memcapacitive system**Project supported by the National Natural Science Foundation of China (Grant No. 51377124) and the Science Fund for New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors, coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally, field programmable gate array (FPGA) is used to implement the proposed chaotic system.

A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractors

Abstract A novel amplitude control method (ACM) is proposed to construct multiple self-excited or hidden attractors by scaling partial or total variables without changing their dynamic and topological properties. Various attractors including nested attractor, axisymmetric attractor, and centrosymmetric attractor can be obtained by multiplying signals with different amplitudes. An universal pulse control module is designed to realize the amplitude scale. Different number of scrolls can be adjusted by regulating the pulse signals without redesigning the nonlinear circuit. The classical Lorenz system and Jerk system are employed as examples to generate nested hidden multi-butterfly and multiscroll attractors. Some novel properties of ACM, such as nested morphology, amplitude modulation, and constant Lyapunov exponential spectrum, are analyzed theoretically and simulated numerically. The circuit design and PSpice simulation results are implemented to verify the availability and feasibility of the proposed approach.

A new oscillator with mega-stability and its Hamilton energy: Infinite coexisting hidden and self-excited attractors.

In this paper, we introduce an interesting new megastable oscillator with infinite coexisting hidden and self-excited attractors (generated by stable fixed points and unstable ones), which are fixed points and limit cycles stable states. Additionally, by adding a temporally periodic forcing term, we design a new two-dimensional non-autonomous chaotic system with an infinite number of coexisting strange attractors, limit cycles, and torus. The computation of the Hamiltonian energy shows that it depends on all variables of the megastable system and, therefore, enough energy is critical to keep continuous oscillating behaviors. PSpice based simulations are conducted and henceforth validate the mathematical model.

Hidden hyperchaotic attractors in a new 4D fractional order system and its synchronization.

The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor-corrector method of the Adams-Bashforth-Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method.

Dynamical analysis, FPGA implementation and its application to chaos based random number generator of a fractal Josephson junction with unharmonic current-phase relation

The dynamical characteristics and its applications to random number generator of a fractal Josephson junction with unharmonic current-phase relation (FJJUCPR) described by a linear resistive-capacitive-inductance shunted junction (LRCLSJ) model are investigated in this paper. The dependence of the equilibrium points of the system to the external current source or the unharmonic current-phase relation (UCPR) parameter is revealed and their stability are analysed. The inclusion of unharmonic current-phase relation in an ideal or a fractal Josephson junction leads to transform the spiking, bursting and relaxations oscillations to an excitable mode. While the inclusion of fractal characteristics in insulating layer of Josephson junction leads to an increase of the amplitude of the spiking, bursting and relaxations oscillations. The numerical simulations results also indicate that FJJUCPR exhibits self-excited chaotic attractors and two different shapes of hidden chaotic attractors. The FJJUCPR is implemented in field programmable gate arrays (FPGA) in order to validate the numerical simulations results. In addition, random number generator design is performed using chaotic signals of the FJJUCPR. The random number generator design results are successful in the NIST SP 800-22 test.

Dynamics, control and symmetry breaking aspects of an infinite-equilibrium chaotic system

It is well known that the symmetry break deliberately induced in a nonlinear system may help to discover new nonlinear patterns. In this work, we investigate the impact of an explicit symmetry break on the dynamics of a recently introduced chaotic system with a curve of equilibriums (Pham et al. in Circuits Syst Signal Process 37(3):1028–1043, 2018). We demonstrate that the symmetry break engenders rich and striking nonlinear phenomena including the coexistence of multiple asymmetric stable states, the presence of parallel bifurcation branches, hysteresis, and critical transitions as well. A simple control strategy based on linear augmentation technique that enables to drive the system from the state of four coexisting self-excited attractors to a monostable state is successfully adapted. To the best of the authors’ knowledge, this work represents the first report on symmetry breaking for a chaotic system with infinite equilibriums and thus deserves dissemination.

An image encryption algorithm based on a hidden attractor chaos system and the Knuth–Durstenfeld algorithm

Chaotic systems have been widely applied in digital image encryption due to their complex properties such as ergodicity, pseudo randomness and extreme sensitivity to their initial values and parameters. An image encryption algorithm based on a hidden attractor chaos system and Knuth–Durstenfeld algorithm is proposed. First, a hidden attractor chaos system is used to encrypt digital image. Compared to a self-excited attractor, the hidden attractor's attracting basin does not intersect with any small neighbourhoods of the equilibria. It is difficult for attackers to reconstruct the attractor by finding equilibrium points. Therefore, the hidden attractor chaotic system is difficult to decrypt. Meanwhile, the hidden attractor chaos system is very sensitive to initial values and parameters. Second, the Knuth–Durstenfeld algorithm has good randomness. In addition, the Knuth–Durstenfeld algorithm can reduce the time complexity and the space complexity of the permutation while achieving good permutation effects. Thus, Knuth–Durstenfeld algorithm is used to permutate the digital image. Finally, DNA sequence operations are used to diffuse image pixels values. Some experimental analyses have been applied to measure the new scheme, and the experimental results illustrate the scheme possesses better encryption performances. This method can be applied in secure image communication fields.

Simple Chaotic Jerk Flows With Families of Self-Excited and Hidden Attractors: Free Control of Amplitude, Frequency, and Polarity

Although chaotic systems with self-excited and hidden attractors have been discovered recently, there are few investigations about relationships among them. In this paper, using a systematic exhaustive computer search, three elementary three-dimensional (3D) dissipative chaotic jerk flows are proposed with the unique feature of exhibiting different families of self-excited and hidden attractors. These systems have a variable equilibrium for different values of the single control parameter. In the family of self-excited attractors, these systems can have a single all-zero-eigenvalue non-hyperbolic equilibrium or two symmetrical hyperbolic equilibria. Besides, for a particular value of the parameter, these systems have no equilibria, and therefore all the attractors are readily hidden. The proposed systems represent a rare class of chaotic systems in which a single system exhibits three different types of equilibria for different values of the single control parameter. In particular, for a single all-zero-eigenvalue non-hyperbolic equilibrium, the coefficient of the two linear terms provides a simple means to rescale the amplitude and frequency, while the introduction of a new constant in the variable x provides a polarity control. Therefore, a free-controlled chaotic signal can be obtained with the desired amplitude, frequency, and polarity. When implemented as an electronic circuit, the corresponding chaotic signal can be controlled by two independent potentiometers and an adjustable DC voltage source, which is convenient for constructing a chaos-based application system.

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

In this work, we report a new 4D chaotic hyperjerk system and present a detailed dynamic analysis of the new system with Lyapunov exponents, bifurcation plots, etc. We find that the new hyperjerk system exhibits multistability and coexisting chaotic attractors. The hyperjerk system has a unique saddle-focus rest point at the origin, which is unstable. This shows that the new chaos hyperjerk system has a self-excited chaotic attractor. As an application of backstepping control, we obtain new results for the global chaos complete synchronisation of pair of chaotic hyperjerk systems. A circuit model using MultiSim of the new chaotic hyperjerk system is designed for applications in practice. Finally, an FPGA-based implementation of the new chaotic hyperjerk dynamical system is performed by applying two numerical methods and their corresponding hardware resources are given.

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

Self-Reproducing Hidden Attractors in Fractional-Order Chaotic Systems Using Affine Transformations

This article proposes a unified approach for hidden attractors control in fractional-order chaotic systems. Hidden attractors have small basins of attractions and are very sensitive to initial conditions and parameters. That is, they can be easily drifted from chaotic behavior into another type of dynamics, which is not suitable for encryption applications that require quite wide initial conditions and parameters ranges for encryption key design. Hence, a systematic coordinate affine transformation framework is utilized to construct transformed systems with self-reproducing attractors. Simulation results of two three-dimensional fractional-order chaotic systems with hidden attractors validate that the proposed framework supports attractors geometric structure design and multi-wing generation. Hidden attractor size, polarity, phase, shape and position control while preserving the chaotic dynamics is indicated by strange attractors, spectral entropy, maximum Lyapunov exponent and bifurcation diagrams. Simulations demonstrate the capability of multi-wing generation from fractional-order hidden attractors with no equilibria using non-autonomous parameters as opposed to the classical equilibria extension techniques suitable only for self-excited attractors. The self-reproduced multiple wings can share the same center point or be distributed along an arbitrary line, curve or surface thanks to the non-autonomous translation parameters. Multi-wing attractors widen the basin of attraction and enlarge the state space volume. For practical applications, the proposed technique makes fractional-order systems with hidden attractors suitable for circuit implementations that require specific signal level and polarity conditions. In addition, for digital encryption applications, the relatively wide range of the extra parameters enhances the key space and hence the robustness against brute force attacks.

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.

CAMO: Self-Excited and Hidden Chaotic Flows

In this paper, we announce a novel 4D chaotic system which belongs to the self-excited attractor and hidden attractor family depending on the parameter values. Lyapunov exponents, bifurcation diagram and bicoherence plot of the CAMO (Camouflage) chaotic system are investigated. Also, fractional-order model of the proposed CAMO system (FOCAMO) is derived and analyzed. FOCAMO chaotic system is then implemented in Field Programmable Gate Array (FPGA) using Adomian decomposition method. Also, power efficiency analysis for various fractional-orders is investigated. The paper helps build a better understanding of chaotic systems with self-excited or hidden attractors.

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.

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.

Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form

There are many works on self-excited and hidden attractors. However the relationship between them is less investigated. In this study we present a system which can have both hidden self-excited attractors. Dynamical properties of the chaotic system are studied using the equilibrium points and Eigenvalues analysis, Lyapunov exponents and bifurcation plots. Since fractional order models are more interesting in engineering applications, the fractional order version of the proposed system is derived using Adomian decomposition method. Bifurcation and stability analysis of the fractional order model shows the existence of chaotic oscillations. To demonstrate the engineering importance of the fractional order model, we have designed a digital communication system with SCSK (Symmetric Chaos Shift Keying) modulation method separately for both self-excited attractor and hidden attractor, the bit error rate performance was compared.

Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria

Multistability, i.e., coexisting attractors, is one of the most exciting phenomena in dynamical systems. This paper presents a new category of coexisting hidden attractor: five-dimensional (5D) systems with a curve of equilibria. Based on the segmented disc dynamo, a new 5D hyperchaotic system is proposed. The paper studies not only coexisting self-excited attractors but also coexisting hidden attractors in the new system with four types of equilibria: a curve of equilibria, a line equilibrium, a stable equilibrium, and no equilibria. Furthermore, the paper proves that the degenerate Hopf and pitchfork bifurcations occur in the system. Numerical simulations demonstrate the emergence of the two bifurcations.

Chaotic system with bondorbital attractors

Two kinds of chaotic attractors with nontrivial topologies are found in a 4D autonomous continuous dynamical system. Since the equilibria of the system are located on both sides of the basic structures of these attractors, and the basic structures of these attractors are bond orbits, we call them bondorbital attractors. They have fractional dimensions and their Kaplan–Yorke dimensions are greater than 3. The generation mechanisms of the two types of attractors are explored and analyzed based on the Shilnikov’s theorems. The type I attractors generated by system with parameter $$P_{1}$$ possess coexistence features, and the type II attractors generated by system with parameter $$P_{2}$$ have the ability to realize consecutive bond orbits. Furthermore, the type I attractors have continuous attracting basins with diagonal distribution and can be caught by means of a method of shorting capacitors in hardware experiments, whereas the type II attractors possess discrete basins of attraction and are difficult to be captured in hardware experiments. The difference between the two types of bondorbital attractors and traditional self-excited attractors in generation method is analyzed, and we also verify that they are not hidden attractors. Based on the step function sequence f(x, M, N), the type II attractors with at most ( $$N+M+1$$ )-fold structures can be generated in the system with parameter $$P_{2}$$ . Two sets of symmetric specific initial conditions are used to verify that the system with parameter $$P_{2}$$ can generate bondorbital attractors with fourfold and fivefold basic structures based on f(x, 2, 2). Some characteristics of the two classes of bondorbital attractors are listed in tabular form.