Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique

In this paper, active backstepping design technique is applied to achieve reduced-order hybrid combination synchronization and reduced-order projective hybrid combination synchronization of three chaotic systems consisting of: (i) two third-order chaotic Josephson junctions as drives and one second-order chaotic Josephson junction as response system; (ii) one third-order chaotic Josephson junction as the drive and two second-order chaotic Josephson junctions as the slaves. Numerical simulations are performed to verify the feasibility and effectiveness of the analytical results. Reduced-order combination synchronization has more valuable practical applications to information processing in physical, biological, and social systems than the normal one master system and one slave system synchronization scheme.     

Nonlinear optimal control of population systems: applications in ecosystems

This article investigates the chaotic Lotka–Volterra system as an optimal nonlinear design problem for biological pest control strategies. In the biological control strategy, natural enemies are introduced such that the pest density is stabilized below the economic injury level, and the population of natural enemies remains sufficiently high to control the pests. Applying dynamic programming, this problem was reduced to the Hamilton–Jacobi–Bellman equation. The functions satisfying the reduced equation were obtained among the correspondent Lyapunov functions of the considered Lotka–Volterra system. A closed-form optimal feedback control law was derived. The effectiveness of the method is verified by numerical simulations of biological pest control. The biological implications of these results are also discussed.     

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

Hyperchaos,chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points

Based on three-dimensional (3D) Lü chaotic system, we introduce a four-dimensional (4D) nonlinear system with infinitely many equilibrium points. The Lyapunov-exponent spectrum is obtained for the 4D chaotic system. A hyperchaotic attractor and a chaotic attractor are emerged in this 4D nonlinear system. Furthermore, to verify the existence of hyperchaos, the chaotic dynamics of this 4D nonlinear system is also studied by means of topological horseshoe theory and numerical computation.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

A hyperchaotic system without equilibrium

This article introduces a new chaotic system of 4-D autonomous ordinary differential equations, which has no equilibrium. This system shows a hyper-chaotic attractor. There is no sink in this system as there is no equilibrium. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincaré maps. There is little difference between this chaotic system and other chaotic systems with one or several equilibria shown by phase portraits, Lyapunov exponents and time series methods, but the Poincaré maps show this system is a chaotic system with more complicated dynamics. Moreover, the circuit realization is also presented.     

A new chaotic attractor from hybrid optical bistable system

In this work, a new three-dimensional autonomous chaotic system has been introduced by modifying a hybrid optical system. The single quadratic nonlinearity is replaced by a single cubic nonlinearity; the new system can display two 1-scroll chaotic attractors simultaneously or one 2-scroll chaotic attractor. The bifurcation diagram is obtained and Lyapunov spectrum is calculated for the proposed system. The results show that the new system exhibits rich complexity features such as stable, periodic, and chaotic dynamics.     

Chaos control and modified projective synchronization of unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control

This paper proposes the chaos control and the modified projective synchronization methods for unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a regular or periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise-like, synchronized chaotic systems can be used as cipher generators for secure communication. Obviously, the importance of obtaining these objectives is specified when the dynamics of gyroscope system are unknown. In this paper, using the neural backstepping control technique, control laws are established which guarantees the chaos control and the modified projective synchronization of unknown chaotic gyroscope system. In the neural backstepping control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimators are derived in the sense of Lyapunov function. Thus, the unknown chaotic gyroscope system can be guaranteed to be asymptotically stable. Also, the control objectives have been achieved.     

Chaotic behavior in piecewise-linear sampled-data control systems

This paper discusses the sufficient conditions for chaotic behavior in piecewise-linear sampled-data control systems by applying Shiraiwa-Kurata's theorem. First, it is shown that a discrete-time system with a piecewise-linear element is chaotic if the lower-dimensional system induced from the original system has a snapback repeller. Next, this result is applied to a piecewise-linear sampled-data control system. Finally, the chaotic region for a two-dimensional sampled-data control system with a dead zone element is studied, and two types of transition from a fixed point to a chaotic attractor are studied by numerical simulation.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications

A novel charge-controlled memcapacitor 3D chaotic oscillator with two unstable equilibriums is proposed. Various dynamic properties of the proposed system are derived and investigated to show the existence of chaotic oscillations. Fractional-order analysis of the chaotic oscillator shows that the maximum value for the largest positive Lyapunov exponent is exhibited in fractional order. Adomian decomposition method is used to discretize the fractional-order system. Field-programmable gate arrays are used to realize the proposed oscillator. In addition, random number generator is designed by employing this novel chaotic system in its fractional-order form.     

Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors

Nonlinear Dynamics - Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its...     

Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation

A three-dimensional autonomous chaotic system is presented and physically implemented. Some basic dynamical properties and behaviors of this system are described in terms of symmetry, dissipative system, equilibria, eigenvalue structures, bifurcations, and phase portraits. By tuning the parameters, the system displays chaotic attractors of different shapes. For specific parameters, the system exhibits periodic and chaotic bursting oscillations which resemble the conventional heart sound signals. The existence of Shilnikov type of heteroclinic orbit in the three-dimensional system is proven using the undetermined coefficients method. As a result, Shilnikov criterion guarantees that the three-dimensional system has the horseshoe chaos. The corresponding electronic circuit is designed and implemented, exhibiting experimental chaotic attractors in accord with numerical simulations.     

Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography

In this paper, the stability conditions and chaotic behaviors of new different fractional orders of reverse butterfly-shaped dynamical system are analytically and numerically investigated. Designing an appropriate feedback controller, the fractional order chaotic system is synchronized. Applying the synchronized fractional order systems in digital cryptography, a well secured key system is obtained. The numerical simulations are given to validate the correctness of the proposed synchronized fractional order chaotic systems and proposed key system.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.