Dynamic Hopf bifurcations generated by nonlinear terms

We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order α>1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function ψ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of ψ at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of ψ.     

Partial stability analysis of some classes of nonlinear systems

A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results.     

New convergence behavior of high-order hopfield neural networks with time-varying coefficients

In this paper the convergence behavior of the delayed high-order Hopfield neural networks (HHNNs) with time-varying coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge to zero point, which are new and complement of previously known results.     

Multiple solutions for strongly resonant periodic systems

We considered a semilinear, second order periodic system. We assumed that the differential operator x→−x^″−Ax$x\to -x^{″}-Ax$ has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F(t,x)$F(t,x)$. Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.     

Exponentially asymptotically stable dynamical systems

In this paper, we give some sufficient conditions which guarantee that the zero solution of the differential equation with two delayed terms x(t) = a(t)x(t-p(t)) - b(t)x(t-r(t))is uniformly stable     

One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species

The one-dimensional Poisson-Nernst-Planck (PNP) system is a basic model for ion flow through membrane channels. If the Debye length is much smaller than the characteristic radius of the channel, the PNP system can be treated as a singularly perturbed system. We provide a geometric framework for the study of the steady-state PNP system involving multiple types of ion species with multiple regions of piecewise constant permanent charge. Special structures of this particular problem are revealed, which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations. Near each singular orbit, an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of a solution of the singular boundary value problem. A new phenomenon on multiplicity and spatial behavior of steady-states involving three or more types of ion species is discovered in an example. (The phenomenon cannot occur when only two types of ion species are involved.)     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

Positive Periodic and Homoclinic Solutions for Nonlinear Differential Equations with Nonsmooth Potential

We study the existence of positive solutions and of positive homoclinic (to zero) solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and having a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory and our results extend the recent works of Korman–Lazer (Electronic JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).     

Stability of differential equations with piecewise constant arguments of generalized type

In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006.     

On camel-like traveling wave solutions in cellular neural networks

This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.     

Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: Frequency domain method

We use the frequency domain method to prove that the zero solution of certain third order nonlinear delayed differential equations is asymptotically stable, (when there is no forcing term). We also prove the existence of a bounded solution which is exponentially stable, (when there is a bounded forcing term). The situation for which the non-linear term is delayed is also proved.     

Normal form for linear systems with respect to its vector relative degree

For multi-input multi-output (MIMO) linear systems with existing vector relative degree a normal form is constructed. This normal form is not only structural simple but allows to characterize the system’s zero dynamics for the design of feedback controllers. A characterization of the zero dynamics in terms of the normal form is given.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Exponential convergence for high-order recurrent neural networks with a class of general activation functions

In this paper, we consider high-order recurrent neural networks with a class of general activation functions. By using some mathematical analysis techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.     

Exponential convergence behavior of shunting inhibitory cellular neural networks with time-varying coefficients

In this paper the convergence behavior of delayed shunting inhibitory cellular neural networks with time-varying coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge exponentially to the zero point, which are new and complement previously known results.     

Differential equations with state-dependent piecewise constant argument

A new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations—the existence and uniqueness of solutions, the existence of periodic solutions, and the stability of the zero solution—are obtained. Appropriate examples are constructed.     

The Solow model improved through the logistic manpower growth law

In this paper the neo-classical economic Solow-Swan model (1956) has been improved replacing its Malthusian manpower law with the Verhulst (logistic) one. The relevant ordinary differential equation for the ratio capital/work has been then integrated in closed form via the Hypergeometric function₂ F ₁. The logistic growth injection for the manpower is detected to induce a more slow dynamics onto the Solow-Swan system, which keeps its stability. Increasing developments are displayed as the technologic progress rises. Further sceneries are tested and the congruence of the new solution with the classical one is shown switching to zero the selflimitation coefficent in the logistic law. Research supported by MURST grant:Metodi matematici in economia     

A comparison theorem and an inequality of redhefferf†

For equations of the form w″+B(e^z)w = 0, where B(ζ) is a rational function which is analytic on 0<|ζ|∞, we determine the regions where the bulk of the zeros of a solution must be located. In the special case of the general Mathieu equation, these results complement earlier results of E. Hille (1924) who considered the special case of real Mathieu equations     

Differential equations on variable time scales

We introduce a class of differential equations on variable   time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B$B$-equivalence of the system [E. Akalín, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981–995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079–1086] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043–2060]. Appropriate examples are constructed to illustrate the theory.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.