An experiment of dynamical behaviours in an erbium-doped fibre-ring laser with loss modulation

This paper reports a systematic experimental investigation on the dynamics in the low-frequency region in an erbium-doped fibre-ring laser with loss modulation. A rich variety of bifurcation is analyzed through the bifurcation diagram and structured with the concept of the winding numbers. The coexistence of multiple attractors and the crisis that appear in the saddle-node bifurcations, and an interesting structure of bifurcation which is similar to the bifurcations in high-frequency range, have been observed.     

A fuzzy crisis in a Duffing-van der Pol system

A crisis in a Duffing--van del Pol system with fuzzy uncertainties is studied by means of the fuzzy generalised cell mapping (FGCM) method. A crisis happens when two fuzzy attractors collide simultaneously with a fuzzy saddle on the basin boundary as the intensity of fuzzy noise reaches a critical point. The two fuzzy attractors merge discontinuously to form one large fuzzy attractor after a crisis. A fuzzy attractor is characterized by its global topology and membership function. A fuzzy saddle with a complicated pattern of several disjoint segments is observed in phase space. It leads to a discontinuous merging crisis of fuzzy attractors. We illustrate this crisis event by considering a fixed point under additive and multiplicative fuzzy noise. Such a crisis is fuzzy noise-induced effects which cannot be seen in deterministic systems.     

Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincar\'{e} maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.     

Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion

Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.     

A memristor oscillator based on a twin-T network

A novel inductance-free nonlinear oscillator circuit with a single bifurcation parameter is presented in this paper. This circuit is composed of a twin-T oscillator, a passive RC network, and a flux-controlled memristor. With an increase in the control parameter, the circuit exhibits complicated chaotic behaviors from double periodicity. The dynamic properties of the circuit are demonstrated by means of equilibrium stability, Lyapunov exponent spectra, and bifurcation diagrams. In order to confirm the occurrence of chaotic behavior in the circuit, an analog realization of the piecewise-linear flux-controlled memristor is proposed, and Pspice simulation is conducted on the resulting circuit.     

Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincare′ map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.     

Chaotic Behavior of a Brownian Particle in a Periodic Potential

The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically.     

Phase Propagations in a Coupled Oscillator-Excitor System of FitzHugh-Nagumo Models

A one-dimensional array of 2N ＋ 1 automata with FitzHugh-Nagumo dynamics, in which one is set to be oscillatory and the others are excitable, is investigated with hi-directional interactions. We find that 1 ： 1 rhythm propagation in the array depends on the appropriate couple strength and the excitability of the system. On the two sides of the 1 ： 1 rhythm area in parameter space, two different kinds of dynamical behaviour of the pacemaker, i.e. phase-locking phenomena and canard-like phenomena, are shown. The latter is found in company with chaotic pattern and period doubling bifurcation. When the coupling strength is larger than a critical value, the whole system ends to a steady state.     

Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.     

Generation of a novel spherical chaotic attractor from a new three-dimensional system

A new three-dimensional （3D） system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.     

Slip Magnetohydrodynamic Viscous Flow over a Permeable Shrinking Sheet

The magnetohydrodynamic （MHD） flow under slip conditions over a shrinking sheet is solved analytically. The solution is given in a closed form equation and is an exact solution of the full governing Navier-Stokes equations. Interesting solution behavior ＆ observed with multiple solution branches for certain parameter domain. The effects of the mass transfer, slip, and magnetic parameters are discussed.     

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincar'e map of the system is constructed. Using the Poincar'e map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.     

One new fractional-order chaos system and its circuit simulation by electronic workbench

This paper proposes a new chaotic system and its fractional-order chaotic system. The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained. It finds that this new fractional-order system is chaotic for q 〉 0.783 if the system parameter m=6. The chaotic attractors for q=0.8, and q=0.9 are obtained. A circuit is designed to realize its fractional-order chaos system for q=0.9 by electronic workbench.     

Effects of Particle Size on Dilute Particle Dispersion in a Karman Vortex Street Flow

It is shown that in a Karman vortex street flow,particle size influences the dilute particle dispersion.Together with an increase of the particle size,there is an emergence of a period-doubling bifurcation to a chaotic orbit,as well as a decrease of the corresponding basins of attraction.A crisis leads the attractor to escape from the central region of flow.In the motion of dilute particles,a drag term and gravity term dominate and result in a bifurcation phenomenon.     

Mode shift and stability control of a current mode controlled buck-boost converter operating in discontinuous conduction mode with ramp compensation

By establishing the discrete iterative mapping model of a current mode controlled buck-boost converter, this paper studies the mechanism of mode shift and stability control of the buck-boost converter operating in discontinuous conduction mode with a ramp compensation current. With the bifurcation diagram, Lyapunov exponent spectrum, time-domain waveform and parameter space map, the performance of the buck-boost converter circuit utilizing a compensating ramp current has been analysed. The obtained results indicate that the system trajectory is weakly chaotic and strongly intermittent under discontinuous conduction mode. By using ramp compensation, the buck-boost converter can shift from discontinuous conduction mode to continuous conduction mode, and effectively operates in the stable period-one region.     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.     

Identification of Chaotic Systems with Application to Chaotic Communication

We propose and develop a novel method to identify a chaotic system with time-varying bifurcation parameters via an observation signal which has been contaminated by additive white Gaussian noise.This method is based on an adaptive algorithm,which takes advantage of the good approximation capability of the radial basis function neural network and the ability of the extended Kalman filter for tracking a time-varying dynamical system.It is demonstrated that,provided the bifurcation parameter varies slowly in a time window,a chaotic dynamical system can be tracked and identified continuously,and the time-varying bifurcation parameter can also be retrieved in a sub-window of time via a simple least-square-fit method.     

Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory

In this paper, controlling chaos when chaotic ferroresonant oscillations occur in a voltage transformer with nonlinear core loss model is performed. The effect of a parallel metal oxide surge arrester on the ferroresonance oscillations of voltage transformers is studied. The metal oxide arrester(MOA) is found to be effective in reducing ferroresonance chaotic oscillations. Also the multiple scales method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes sub-harmonic, quasi-periodic, and also chaotic oscillations. In this paper, the chaotic behavior and various ferroresonant oscillation modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as period doubling bifurcation(PDB), saddle node bifurcation(SNB), Hopf bifurcation(HB), and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via the multiple scales method to obtain Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system.     

Design and FPGA implementation of multi-wing chaotic switched systems based on a quadratic transformation

Based on a quadratic transformation and a switching function,a novel multi-wing chaotic switched system is proposed.First,a 4-wing chaotic system is constructed from a 2-wing chaotic system on the basis of a quadratic transformation.Then,a switching function is designed and by adjusting the switching function,the number and the distribution of the saddle-focus equilibrium points of the switched system can be regulated.Thus,a set of chaotic switched systems,which can produce 6-to-8-12-16-wing attractors,are generated.The Lyapunov exponent spectra,bifurcation diagrams,and Poincarémaps are given to verify the existence of the chaotic attractors.Besides,the digital circuit of the multi-wing chaotic switched system is designed by using the Verilog HDL fixed-point algorithm and the state machine control.Finally,the multi-wing chaotic attractors are demonstrated via FPGA platform.The experimental results show that the number of the wings of the chaotic attractors can be expanded more effectively with the combination of the quadratic transformation and the switching function methods.     

Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile

Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation.The complex flow patterns beyond the symmetry-breaking at Re■320 are presented by a bifurcation diagram for Re 2500.The results of stable symmetric and asymmetric solutions are consistent with those reported in literature,and a new unstable asymmetric branch is obtained besides the stable branches.A novel hysteresis phenomenon is observed in the range of2208 Re 2262,where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist.The vortices near the sidewall appear when the Reynolds number increases,which correspond to the bifurcation of topology structure,but not the bifurcation of Navier-Stokes equations.The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.