Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system

In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented. The chaotic behaviors are validated by the positive Lyapunov exponents. Furthermore, the fractional Hopf bifurcation is investigated. It is found that the system admits Hopf bifurcations with varying fractional order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the Hopf bifurcations are proposed. Numerical simulations are given to illustrate and verify the results.     

A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system

This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyper-chaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent double-wing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.     

Dynamical analysis in a 4D hyperchaotic system

Inspirited by Li and Jin (Nonlinear Dyn. 67:2857?C2864 2012), this paper investigates the Hopf bifurcation of a four-dimensional hyperchaotic system with only one equilibrium. A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, the standard normal form theory is applied to determine the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods. In addition, numerical simulations are used to justify theoretical results.     

Generalized synchronization of fractional-order hyperchaotic systems and its DSP implementation

In this paper, we investigate synchronization and its DSP implementation of fractional-order simplified Lorenz hyperchaotic systems by employing the Adomian decomposition method. The active controller and linear feedback controller are designed. Numerical simulation of the synchronized systems is carried out, and it is found that the synchronization phenomenon can be observed in both state variables and intermediate variables. Moreover, the synchronized systems are implemented in two TMS320F2-8335 DSP boards which are connected by a serial port and the output signals are exhibited by an oscilloscope. The experiment results show that the proposed implementation method works well on DSP.     

Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci

This paper reports a new fractional-order Lorenz-like system with one saddle and two stable node-foci. First, some sufficient conditions for local stability of equilibria are given. Also, this system has a double-scroll chaotic attractor with effective dimension being less than three. The minimum effective dimension for this system is estimated as 2.967. It should be emphasized that the linear differential equation in fractional-order Lorenz-like system seems to be less ??sensitive?? to the damping, introduced by a fractional derivative, than two other nonlinear equations. Furthermore, mixed synchronization of this system is analyzed with the help of nonlinear feedback control method. The first two pairs of state variables between the interactive systems are anti-phase synchronous, while the third pair of state variables is complete synchronous. Numerical simulations are performed to verify the theoretical results.     

Analysis and control of a hyperchaotic system with only one nonlinear term

In order to promote the development of chaos in nonlinear systems, and explore more convenient controllers for the engineering application, a four-dimensional nonlinear dynamic system with only one nonlinear term was constructed and its complex dynamic characteristics were analyzed, including the phase trajectory map, Lyapunov exponents, and so on. Furthermore, the recursive backstepping method was proposed to design a different controller; the hyperchaotic system was controlled to an equilibrium point and a periodic orbit. Theoretical analysis is in agreement with simulation results. The results show that the recursive backstepping control method can wipe off chaos, and make the hyperchaotic system achieve stable states. The control process is a smooth transition, and the transition time is short.     

Estimations for ultimate boundary of a new hyperchaotic system and its simulation

This paper has investigated the boundedness of a new hyperchaotic Rabinovich system. We have obtained the global exponential attractive set and the ultimate bound Ω _λ for this system. Furthermore, we can conclude that the rate of the trajectories of the system going from the exterior of the set Ω _λ,2 to the interior of the set Ω _λ,2 is an exponential rate. The estimate of the trajectories rate is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.     

The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system

Little seems to be known about the homoclinic orbits of hyperchaotic system. Through the deep researches of a 4D Lorenz-type hyperchaotic system, with the help of Fishing Principle, we obtain the existence conditions of homoclinic orbits of this hyperchaotic system. In order to justify the theoretical analysis, by using the numerical methods, a set of approximate bifurcation parameters and its corresponding homoclinic orbits are obtained.     

A new 5D hyperchaotic system based on modified generalized Lorenz system

This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincaré projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.     

Ultimate boundary estimation and topological horseshoe analysis on a parallel 4D hyperchaotic system with any number of attractors and its multi-scroll

Nonlinear Dynamics - This paper constructs a new four-dimensional autonomous hyperchaotic system with complex dynamic behaviors, and its boundary is estimated based on the proposed method and the...     

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.     

Dynamic analysis and control of a new hyperchaotic finance system

In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.     

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.     

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.     

Analysis and synchronization for a new fractional-order chaotic system with absolute value term

A new fractional-order chaotic system with absolute value term is introduced. Some dynamical behaviors are investigated and analyzed. Furthermore, synchronization of this system is achieved by utilizing the drive-response method and the feedback method. The suitable parameters for achieving synchronization are studied. Both the theoretical analysis and numerical simulations show the effectiveness of the two methods.     

Existence of attractor and control of a 3D differential system

This paper analyzes the orbit of a three-dimensional differential system based on the Shilnikov criterion. It also applies the Shilnikov method of constructing a heteroclinic connection between saddle focus equilibrium points of the system, proving that the system possesses “horseshoe” chaos. In addition, adaptive backstepping design is used to control this system with three unknown key parameters, and an algorithm of this controller is presented. Finally, we give some numerical simulation studies of the system in order to verify the analytic results.