BIFURCATION DIAGRAMS OF QUADRATIC DIFFERENTIAL SYSTEMS HAVING ONE FOCUS AND ONE SADDLE

This paper gives bifurcation diagrams of system (2) below in the (α,λ) para meter plane.

Bifurcation behavior and coexisting motions in a time-delayed power system

With the increase of system scale, time delays have become unavoidable in nonlinear power systems, which add the complexity of system dynamics and induce chaotic oscillation and even voltage collapse events. In this paper, coexisting phenomenon in a fourth-order time-delayed power system is investigated for the first time with different initial conditions.With the mechanical power, generator damping factor, exciter gain, and time delay varying, the specific characteristic of the time-delayed system, including a discontinuous ＂jump＂ bifurcation behavior is analyzed by bifurcation diagrams, phase portraits, Poincar′e maps, and power spectrums. Moreover, the coexistence of two different periodic orbits and chaotic attractors with periodic orbits are observed in the power system, respectively. The production condition and existent domain of the coexistence phenomenon are helpful to avoid undesirable behavior in time-delayed power systems.

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit （oscillation） in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.

A novel hyperchaos evolved from three dimensional modified Lorenz chaotic system

This paper reports a new four-dimensional continuous autonomous hyperchaos generated from the Lorenz chaotic system by introducing a nonlinear state feedback controller. Some basic properties of the system are investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. By numerical simulating, this paper verifies that the four-dimensional system can evolve into periodic, quasi-periodic, chaotic and hyperchaotic behaviours. And the new dynamical system is hyperchaotic in a large region. In comparison with other known hyperchaos, the two positive Lyapunov exponents of the new system are relatively more larger. Thus it has more complex degree.

A new piecewise linear Chen system of fractional-order:Numerical approximation of stable attractors

In this paper we present a new version of Chen＇s system： a piecewise linear （PWL） Chert system of fractional-order. Via a sigmoid-like function, the discontinuous system is transformed into a continuous system. By numerical simulations, we reveal chaotic behaviors and also multistability, i.e., the existence of small pararheter windows where, for some fixed bifurcation parameter and depending on initial conditions, coexistence of stable attractors and chaotic attractors is possible. Moreover, we show that by using an algorithm to switch the bifurcation parameter, the stable attractors can be numerically approximated.

Analysis of transition between chaos and hyper-chaos of an improved hyper-chaotic system

An improved hyper-chaotic system based on the hyper-chaos generated from Chen＇s system is presented, and some basic dynamical properties of the system are investigated by means of Lyapunov exponent spectrum, bifurcation diagrams and characteristic equation roots. Simulations show that the new improved system evolves into hyper-chaotic, chaotic, various quasi-periodic or periodic orbits when one parameter of the system is fixed to be a certain value while the other one is variable. Some computer simulations and bifurcation analyses are given to testify the findings.

Novel four-dimensional autonomous chaotic system generating one-,two-,three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.

Control and synchronization of a hyperchaotic system based on passive control

In this paper, a new hyperchaotic system is proposed, and the basic properties of this system are analyzed by means of the equilibrium point, a Poincar~ map, the bifurcation diagram, and the Lyapunov exponents. Based on the passivity theory, the controllers are designed to achieve the new hyperchaotic system globally, asymptotically stabilized at the equilibrium point, and also realize the synchronization between the two hyperchaotic systems under different initial values respectively. Finally, the numerical simulation results show that the proposed control and synchronization schemes are effective.

Phase synchronization and its transition in two coupled bursting neurons:theoretical and numerical analysis

It is crucially important to study different synchronous regimes in coupled neurons because different regimes may correspond to different cognitive and pathological states. In this paper, phase synchronization and its transitions are discussed by means of theoretical and numerical analyses. In two coupled modified Morris-Lecar neurons with a gap junction, we show that the occurrence of phase synchronization can be investigated from the dynamics of phase equation, and the analytical synchronization condition is derived. By defining the phase of spike and burst, the transitions from burst synchronization to spike synchronization and then toward nearly complete synchronization can be identified by bifurcation diagrams, the mean frequency difference and time series of neurons. The simulation results suggest that the synchronization of bursting activity is a multi-time-scale phenomenon and the phase synchronization deduced by the phase equation is actually spike synchronization.

Dynamical Complexity Analysis in a Model of Electroencephalogram

This paper studies a dynamical model of electroencephalogram(EEG).By linearizing the EEG model conditions for the stability of equilibrium point are obtained. Choosing excitatory inputs as bifurcation parameters, numerical simulations show that the EEG model exhibits a series of complex dynamics, including limit cycle,periodic doubling bifurcation, periodic halving bifurcation, chaotic bands with periodic windows.