The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

Solution and dynamics analysis of a fractional‐order hyperchaotic system

Numerical solution and chaotic behaviors of the fractional‐order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional‐order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional‐order chaotic system is designed. Dynamics of the fractional‐order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C₀ complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order q is small. It lays a foundation for the practical application of the fractional‐order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.     

Stability and Hopf bifurcation analysis of novel hyperchaotic system with delayed feedback control

In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016     

Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

Hyperchaotic attractors from a linearly controlled Lorenz system

In this paper, a four-dimensional (4D) continuous-time autonomous hyperchaotic system with only one equilibrium is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the second equation of the 3D Lorenz system. Some complex dynamical behaviors of the hyperchaotic system are investigated, revealing many interesting properties: (i) existence of periodic orbit with two zero Lyapunov exponents; (ii) existence of chaotic orbit with two zero Lyapunov exponents; (iii) chaos depending on initial value $w_0$; (iv) chaos with only one equilibrium; and (v) hyperchaos with only one equilibrium. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are derived and studied.     

A new hyperchaotic system and its circuit implementation

In this paper, a new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The Multisim results of the hyperchaotic circuit were well agreed with the simulation results.     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems

Based on the Lyapunov stability and adaptive synchronization theory, optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail. The linear errors of corresponding variables and parameters are used to construct different appropriate positive Lyapunov functions V and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dV/dt?0. Particularly, an optional gain coefficient is selected in the parameter observers and positive Lyapunov function, which decides the transient period to identify the unknown parameters and reach synchronization. The scheme is used to study the synchronization of two non-identical hyperchaotic Rössler systems. The theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system. To confirm the model independence of this scheme, an alternative hyperchaotic system is investigated, whereby the results confirm that the five unknown parameters are identified rapidly and exactly, and that the two hyperchaotic systems reach complete synchronization as well.     

Maps serving the combined coupling for use in environmental models and their behaviour in the presence of dynamical noise

Many physical, biological as well as the environmental problems, can be described by the dynamics of driven coupled oscillators. In order to study their behaviour as a function of coupling strength and nonlinearity, we considered dynamics of two maps serving the combined coupling (diffusive and linear) in the above fields. Firstly, we have considered a logistic difference equation on extended domain that is a part of the maps, that is discussed using its bifurcation diagram, Lyapunov exponent, sample as well as the permutation entropy. Secondly we have performed the dynamical analysis of the coupled maps using Lyapunov exponent and cross sample entropy in dependence on two coupling parameters. Further, we investigated how dynamical noise can affects the structure of their bifurcation diagrams. It was done (i) by the noise entering in two specific ways, that disturbs either the logistic parameter on extended domain or (ii) by an additive “shock” to the state variables. Finally, we demonstrated the effect of forcing by parametric noise, introduced in all maps’ parameter, on Lyapunov exponent of coupled maps.     

A hyperchaotic system from a chaotic system with one saddle and two stable node-foci

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D new chaotic system with one saddle and two stable node-foci. Some complex dynamical behaviors such as ultimate boundedness, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding bounded hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorous derived and studied.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

A new hyperchaotic system from the Lü system and its control

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D Lü system. Some complex dynamical behaviors such as Hopf bifurcation, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Furthermore, the design is illustrated with both simulations and experiments. Finally, the control problem of a new hyperchaotic system is investigated using negative feedback control. Ordinary feedback control, dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

Dynamics of coupled non-identical systems with period-doubling cascade

We discuss the structure of bifurcation diagram in the plane of parameters controlling period-doublings for the system of coupled logistic maps. The analysis is carried out by computing the charts of dynamical regimes and charts of Lyapunov exponents giving showy and effective illustrations. The critical point of codimension two at the border of chaos is found. It is a terminal point for the Feigenbaum critical line. The bifurcation analysis in the vicinity of this point is presented.      

Adaptive synchronization of a hyperchaotic system with uncertain parameter

This paper addresses the synchronization problem of two Lü hyperchaotic dynamical systems in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is derived to make the states of two identical Lü hyperchaotic systems with unknown system parameters asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization schemes.     

Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay

In this paper, we identify the critical point for a Hopf-pitchfork bifurcation in a nonlinear financial system with delay, and derive the normal form up to third order with their unfolding in original system parameters near the bifurcation point by normal form method and center manifold theory. Furthermore, we analyze its local dynamical behaviors, and show the coexistence of a pair of stable periodic solutions. We also show that there coexist a pair of stable small-amplitude periodic solutions and a pair of stable large-amplitude periodic solutions for different initial values. Finally, we give the bifurcation diagram with numerical illustration, showing that the pair of stable small-amplitude periodic solutions can also exist in a large region of unfolding parameters, and the financial system with delay can exhibit chaos via period-doubling bifurcations as the unfolding parameter values are far away from the critical point of the Hopf-pitchfork bifurcation.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.