On the Darboux Integrability of the Hindmarsh–Rose Burster

We study the Hindmarsh–Rose burster which can be described by the differential system = y-x~3+ bx~2+ I-z,  = 1-5 x2~-y, z = μ(s(x-x_0)-z),where b, I, μ, s, x_0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.     

Control of chaotic solutions of the Hindmarsh–Rose equations

Irregular bursting and spiking solutions of the Hindmarsh–Rose model for the electrical activity of neuron cell bodies have been converted by a chaos control algorithm to periodic spike trains. A proportional feedback method is used to control both chaotic spike trains and chaotic bursting by applying controlling perturbations to membrane parameters.     

Global dynamics of nonautonomous Hindmarsh–Rose equations

Global dynamics of nonautonomous diffusive Hindmarsh–Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.     

Synchronization of boundary coupled Hindmarsh–Rose neuron network

In this work, we present a new mathematical model of a boundary coupled neuron network described by the partly diffusive Hindmarsh–Rose equations. We prove the global absorbing property of the solution semiflow and then the main result on the asymptotic synchronization of this neuron network at a uniform exponential rate provided that the boundary coupling strength and the stimulating signal exceed a quantified threshold in terms of the parameters.     

Exponential synchronization of memristive Hindmarsh–Rose neural networks

A new model of neural networks in terms of the memristive Hindmarsh–Rose equations is proposed. Globally dissipative dynamics is shown with absorbing sets in the state spaces. Through sharp and uniform grouping estimates and by leverage of integral and interpolation inequalities tackling the linear network coupling against the memristive nonlinearity, it is proved that exponential synchronization at a uniform convergence rate occurs when the coupling strengths satisfy the threshold conditions which are quantitatively expressed by the parameters.     

Local Bifurcations in the Cahn–Hilliard and Kuramoto–Sivashinsky Equations and in Their Generalizations

Computational Mathematics and Mathematical Physics - A periodic boundary value problem for a nonlinear evolution equation that takes the form of such well-known equations of mathematical physics as...     

Global dynamics of diffusive Hindmarsh–Rose equations with memristors

Global dynamics of the diffusive Hindmarsh–Rose equations with memristors as a new proposed model for neuron dynamics are investigated in this paper. We prove the existence and regularity of a global attractor for the solution semiflow through uniform analytic estimates showing the higher-order dissipative property and the asymptotically compact characteristics of the solutions by the approach of Kolmogorov–Riesz theorem. The quantitative bounds of the regions containing this global attractor respectively in the state space and in the regular space are explicitly expressed by the model parameters.     

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.     

Bifurcations in Kuramoto–Sivashinsky equations

We consider the local dynamics of the classical Kuramoto–Sivashinsky equation and its generalizations and study the problem of the existence and asymptotic behavior of periodic solutions and tori. The most interesting results are obtained in the so-called infinite-dimensional critical cases. Considering these cases, we construct special nonlinear partial differential equations that play the role of normal forms and whose nonlocal dynamics thus determine the behavior of solutions of the original boundary value problem.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Unidirectional synchronization for Hindmarsh–Rose neurons via robust adaptive sliding mode control

This paper presents an adaptive neural network (NN) based sliding mode control for unidirectional synchronization of Hindmarsh–Rose (HR) neurons in a master–slave configuration. We first give the dynamics of single HR neuron which may exhibit spike-burst chaotic behaviors. Then we formulate the problem of unidirectional synchronization control of two HR neurons and propose a NN based sliding mode controller. The controller consists of two simple radial basis function (RBF) NNs which are used to approximate the desired sliding mode controller and the uncertain nonlinear part of the error dynamical system, respectively. The control scheme is robust to the uncertainties such as approximate errors, ionic channel noise and external disturbances. The simulation results demonstrate the validity of the proposed control method.     

Adaptive synchronization of two coupled chaotic Hindmarsh–Rose neurons by controlling the membrane potential of a slave neuron

This paper addresses the problem of an adaptive synchronization of two chaotic Hindmarsh–Rose (HR) neurons coupled with a gap junction via a single control input. Two adaptive control approaches (i.e., a nonlinear control when the entire state variables are available and a linear feedback control when only the membrane potential is available) are proposed to guarantee the asymptotic synchronization of the state trajectories of two coupled HR neurons having unknown parameters. Numerical simulations demonstrating and verifying the effectiveness of the proposed control methods are provided.     

Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction–Diffusion Systems

The distribution of the roots of a second order transcendental polynomial is analyzed, and it is used for solving the purely imaginary eigenvalue of a transcendental characteristic equation with two transcendental terms. The results are applied to the stability and associated Hopf bifurcation of a constant equilibrium of a general reaction–diffusion system or a system of ordinary differential equations with delay effects. Examples from biochemical reaction and predator–prey models are analyzed using the new techniques.     

Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback

We investigate the principal parametric resonance of a Rayleigh–Duffing oscillator with time-delayed feedback position and linear velocity terms. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase of the oscillator. We study the effects of the frequency detuning, the deterministic amplitude, and the time-delay on the dynamical behaviors, such as stability and bifurcation associated with the principal parametric resonance. Moreover, the appropriate choice of the feedback gain and the time-delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time-delay can broaden the stable region of the non-trivial steady-state solutions and enhance the control performance. Theoretical stability analysis is verified through a numerical simulation.     

On the dynamics of the Rayleigh–Duffing oscillator

We give a complete algebraic characterization of the first integrals of the Rayleigh–Duffing oscillator. We prove the non existence of centers of such system and we study the form of the singular first integrals at the origin.     

Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation

In this paper, the Gerdjikov–Ivanov equation is investigated by using the bifurcation theory and the method of phase portraits analysis. The existence of every kind of travelling waves is proved, in some conditions, exact parametric representations of above travelling waves in explicit form are obtained.     

On the confusion and diffusion properties of Maiorana–McFarland's and extended Maiorana–McFarland's functions

A practical problem in symmetric cryptography is finding constructions of Boolean functions leading to reasonably large sets of functions satisfying some desired cryptographic criteria. The main known construction, called Maiorana–McFarland, has been recently extended. Some other constructions exist, but lead to smaller classes of functions. Here, we study more in detail the nonlinearities and the resiliencies of the functions produced by all these constructions. Further we see how to obtain functions satisfying the propagation criterion (among which bent functions) with these methods, and we give a new construction of bent functions based on the extended Maiorana–McFarland's construction.     

Bifurcations in a Diffusive Predator–Prey Model with Beddington–DeAngelis Functional Response and Nonselective Harvesting

In this paper, we discuss the dynamics of a predator–prey model with Beddington–DeAngelis functional response and nonselective harvesting. By using the Lyapunov–Schmidt reduction, we obtain the existence of spatially nonhomogeneous steady-state solution. The stability and existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the change of a specific parameter are investigated by analyzing the distribution of the eigenvalues. We also get an algorithm for determining the bifurcation direction of the Hopf bifurcating periodic solutions near the nonhomogeneous steady-state solution. Finally, we show some numerical simulations to verify our analytical results.

     

Bifurcations in a predator–prey model in patchy environment with diffusion

In this paper we formulate a predator–prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other one. Numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, i.e. the stable constant steady state loses its stability and spatially non-constant stationary solutions, a pattern emerge.