Anti-Control of Hopf Bifurcation in the Chaotic Liu System with Symbolic Computation

The anti-control of bifurcation refers to the task of creating a certain bifurcation with particular desired properties and location by appropriate controls. We consider, via feedback control and symbolic computation, the problem of anti-control of Hopf bifurcation in the chaotic Liu system. We propose an anti-control scheme and show that compared with the uncontrolled system, the anti-controlled Liu system can exhibit Hopf bifurcation in a much larger parameter region. The anti-control strategy used keeps the equilibrium structure of the Liu system and can be applied to generate Hopf bifurcation at the desired location with preferred stability. We illustrate the etticiency of the anti-control approach under different operating conditions.     

Waves and patterns in ring lattices with delays

We investigate the existence and the stability of waves and phase locked states in rings of coupled oscillators with delayed interactions. Using center manifold reduction and the normal form method, we reduce the equation governing the dynamics of the whole network to an amplitude-phase model (i.e. a set of coupled ordinary differential equations describing the evolution of both the amplitudes and the phases of the oscillators). Then we prove the existence of traveling waves, in-phase and anti-phase locked oscillations, in both one-dimensional and two-dimensional lattices. The influence of the interaction strength and the number of oscillators is investigated, and the possible coexistence of waves and phase locked oscillations is shown.     

Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation

This Letter is concerned with bifurcation and chaos control in scalar delayed differential equations with delay parameter τ. By linear stability analysis, the conditions under which a sequence of Hopf bifurcation occurs at the equilibrium points are obtained. The delayed feedback controller is used to stabilize unstable periodic orbits. To find the controller delay, it is chosen such that the Hopf bifurcation remains unchanged. Also, the controller feedback gain is determined such that the corresponding unstable periodic orbit becomes stable. Numerical simulations are used to verify the analytical results.     

Induction of Hopf bifurcation and oscillation death by delays in coupled networks

This work explores a system of two coupled networks that each has four nodes. Delayed effects of short-cuts in each network and the coupling between the two groups are considered. When the short-cut delay is fixed, the arising and death of oscillations are caused by the variational coupling delay.     

Dynamical Analysis of Protein Regulatory Network in Budding Yeast Nucleus

Recent progresses in the protein regulatory network of budding yeast Saccharomyces cerevisiae have provided a global picture of its protein network for further dynamical research. We simplify and modularize the protein regulatory networks in yeast nucleus, and study the dynamical properties of the core 37-node network by a Boolean network model, especially the evolution steps and final fixed points. Our simulation results show that the number of fixed points N（k） for a given size of the attraction basin k obeys a power-law distribution N（k） χ k^-2.024. The yeast network is more similar to a scale-free network than a random network in the above dynamical properties.     

Bursting neurons with coupling delays

A pair of Hindmarsh–Rose neurons with delayed coupling is studied. Bifurcations due to time-lag and coupling and stability of the stationary state that corresponds to the quiescence behavior are analyzed. Bursting is created by coupling and its properties strongly depend on the time-lag. In particular, there is a domain of values of time-lags which renders the bursting of the two neurons exactly synchronous.     

Two Different Bifurcation Scenarios in Neural Firing Rhythms Discovered in Biological Experiments by Adjusting Two Parameters

Two different bifurcation scenarios, one is novel and the other is relatively simpler, in the transition procedures of neural firing patterns are studied in biological experiments on a neural pacemaker by adjusting two parameters. The experimental observations are simulated with a relevant theoretical model neuron. The deterministic non-periodic firing pattern lying within the novel bifurcation scenario is suggested to be a new case of chaos, which has not been observed in previous neurodynamical experiments.     

Chaos control in a discrete time system through asymmetric coupling

We study a pair of asymmetrically coupled identical chaotic quadratic maps. We investigate, via numerical simulations, chaos suppression associated with the variation of both parameters, the coupling parameter and the parameter which measures the asymmetry. This is a new technique recently introduced for chaos suppression in continuous systems and, as far we know, not yet tested for discrete systems. Parameter-space regions where the chaotic dynamics is driven towards regular dynamics are shown. Lyapunov exponents and phase-space plots are also used to characterize the phenomenon observed as the parameters are changed.     

The effect of time-delay on anomalous phase synchronization

Anomalous phase synchronization in nonidentical interacting oscillators is manifest as the increase of frequency disorder prior to synchronization. We show that this effect can be enhanced when a time-delay is included in the coupling. In systems of limit-cycle and chaotic oscillators we find that the regions of phase disorder and phase synchronization can be interwoven in the parameter space such that as a function of coupling or time-delay the system shows transitions from phase ordering to disorder and back.     

Global view on a nonlinear oscillator subject to time-delayed feedback control

We apply time-delayed feedback control to stabilise unstable periodic orbits of an amplitude-phase oscillator. The control acts on both, the amplitude and the frequency of the oscillator, and we show how the phase of the control signal influences the dynamics of the oscillator. A comprehensive bifurcation analysis in terms of the control phase and the control strength reveals large stability regions of the target periodic orbit, as well as an increasing number of unstable periodic orbits caused by the time delay of the feedback loop. Our results provide insight into the global features of time-delayed control schemes.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

Simplified models for vibrational energy transfer in proteins

We consider the transfer of vibrational energy in proteins and derive a simplified model for the resonant energy exchange between different vibrational modes. We use the parameters of an earlier study [K. Moritsugu, et al., Phys. Rev. Lett. 85 (2000) 3970 ] to compare our predictions with the results of the molecular dynamics simulations, and reveal an excellent agreement.     

Grazing-induced crises in hybrid dynamical systems

In hybrid dynamical systems including both continuous and discrete components, an interplay between a continuous trajectory and a discontinuity boundary can trigger a sudden qualitative change in the system dynamics. Grazing phenomena, which occur when a continuous trajectory hits a boundary tangentially, are well known as a representative of such phenomena. We demonstrate that a grazing phenomenon of a chaotic attractor can result in its sudden disappearance and initiate chaotic transients. The mechanism of this grazing-induced crisis is revealed in an illustrative example. Furthermore, we derive a formula to obtain the critical exponent of the power law on the mean duration of chaotic transients.     

Partial differential equations possessing Frobenius integrable decompositions

Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.     

Global analysis of the immune response

The immune system may be seen as a complex system, characterized using tools developed in the study of such systems, for example, surface roughness and its associated Hurst exponent. We analyze densitometric (Panama blot) profiles of immune reactivity, to classify individuals into groups with similar roughness statistics. We focus on a population of individuals living in a region in which malaria endemic, as well as a control group from a disease-free region. Our analysis groups individuals according to the presence, or absence, of malaria symptoms and number of malaria manifestations. Applied to the Panama blot data, our method proves more effective at discriminating between groups than principal-components analysis or super-paramagnetic clustering. Our findings provide evidence that some phenomena observed in the immune system can be only understood from a global point of view. We observe similar tendencies between experimental immune profiles and those of artificial profiles, obtained from an immune network model. The statistical entropy of the experimental profiles is found to exhibit variations similar to those observed in the Hurst exponent.     

Non-Smooth Bifurcation and Chaos in a DC-DC Buck Converter

A direct-current-direct-current （DC-DC） buck converter with integrated load current feedback is studied with three kinds of Poincaré maps. The external corner-collision bifurcation occurs when the crossing number per period varies, and the internal corner-collision bifurcations occur along with period-doubling and period-tripling bifurcations in this model. The multi-band chaos roots in external corner-collision bifurcation and often grows into 1-band chaos. A new kind of chaotic sliding orbits, which is more complex for non-smooth systems, is also found in this model.     

Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable non-linear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of plasma physics. Instead of pursuing the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single-soliton or multiple-soliton solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second order. The charge density displays a triple-layer structure. The analysis is carried out through the third order.     

From single- to multiple-soliton solutions of the perturbed KdV equation

The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiple-soliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton-anti-soliton scattering, soliton-anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works.     

Synchronization and Pattern Dynamics of Coupled Chaotic Maps on Complex Networks

The synchronization and pattern dynamics of coupled logistic maps on a certain type of complex network, constructed by adding random shortcuts to a regular ring, is investigated. For parameters where an isolated map is fully chaotic, the defect turbulence, which is dominant in the regular network, can be tamed into ordered periodic patterns or synchronized chaotic states when random shortcuts are added, and the patterns formed on the complex network can be grouped into two or three branches depending on the coupling strength.