In this paper, the exponential stability problem is investigated for a class of Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. By using the analysis method, inequality technique and the properties of an M-matrix, several novel sufficient conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium point are derived. Moreover, the exponential convergence rate is estimated. The obtained results are less restrictive than those given in the earlier literature, and the boundedness and differentiability of the activation functions and differentiability of the time-varying delays are removed. Two examples with their simulations are given to show the effectiveness of the obtained results.
This paper is concerned with the existence of travelling wave solutions in a class of delayed reaction-diffusion systems without monotonicity, which concludes two-species diffusion-competition models with delays. Previous methods do not apply in solving these problems because the reaction terms do not satisfy either the so-called quasimonotonicity condition or non-quasimonotonicity condition. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of travelling wave solutions. More precisely, by using such a new cross-iteration, we reduce the existence of travelling wave solutions to the existence of an admissible pair of upper and lower solutions which are easy to construct in practice. To illustrate our main results, we study the existence of travelling wave solutions in two delayed two-species diffusion-competition systems.
We establish sufficient conditions for the regularity of solutions of the Navier-Stokes system based on conditions on one component of the velocity. The first result states that if del u(3) is an element of (LtLxr)-L-s, where 2/s + 3/r <= 11/6 and 54/23 <= r <= 18/5, then the solution is regular. The second result is that if u(3) is an element of (LtLxr)-L-s, where 2/s + 3/r <= 5/8 and 24/5 <= r <= infinity, then the solution is regular. These statements improve earlier results on one component regularity.
Inspired by the remarkable performance of the Leray-alpha (and the Navier-Stokes alpha (NS-alpha), also known as the viscous Camassa-Holm) subgrid scale model of turbulence as a closure model to Reynolds averaged equations (RANS) for flows in turbulent channels and pipes, we introduce in this paper another subgrid scale model of turbulence, the modified Leray-alpha (ML-alpha) subgrid scale model of turbulence. The application of the ML-alpha to infinite channels and pipes gives, due to symmetry, similar reduced equations as Leray-alpha and NS-alpha. As a result the reduced ML-alpha model in infinite channels and pipes is equally impressive as a closure model to RANS equations as NS-alpha and all the other alpha subgrid scale models of turbulence (Leray-alpha and Clark-alpha). Motivated by this, we present an analytical study of the ML-alpha model in this paper. Specifically, we will show the global well-posedness of the ML-alpha equation and establish an upper bound for the dimension of its global attractor. Similarly to the analytical study of the NS-alpha and Leray-a subgrid scale models of turbulence we show that the ML-alpha model will follow the usual k(-5/3) Kolmogorov power law for the energy spectrum for wavenumbers in the inertial range that are smaller than I/a and then have a steeper power law for wavenumbers greater than 1/alpha (where alpha > 0 is the length scale associated with the width of the filter). This result essentially shows that there is some sort of parametrization of the large wavenumbers (larger than 1/alpha) in terms of the smaller wavenumbers. Therefore, the ML-alpha model can provide us another computationally sound analytical subgrid large eddy simulation model of turbulence.
This paper is concerned with the semilinear strongly damped wave equation partial derivative(tt)u - Delta partial derivative(t)u - Delta u + partial derivative(u) = f. The existence of compact global attractors of optimal regularity is proved for nonlinearities partial derivative of critical and supercritical growth.
In this paper, a novel coupling scheme with different coupling delays is presented to achieve generalized synchronization (complete synchronization (CS), anticipating synchronization (AS) and lag synchronization (LS)). The Lyapunov-Krasovskii functional method is employed to investigate the global asymptotic stability of the error dynamical system. The theoretical analysis indicates that delayed chaotic neural networks coupled in this way can switch arbitrarily among CS, AS and LS under the newly proposed coupling configuration. Switching among different kinds of synchronization can be done by changing the transformation time of the coupling signal. Numerical simulations agree with the theoretical analysis.
We consider dynamical systems on domains that are not invariant under the dynamics-for example, a system with a hole in the phase space-and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue. Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.
The occurrence of chaos in basic Lotka - Volterra models of four competing species is studied. A brute-force numerical search conditioned on the largest Lyapunov exponent (LE) indicates that chaos occurs in a narrow region of parameter space but is robust to perturbations. The dynamics of the attractor for a maximally chaotic case are studied using symbolic dynamics, and the question of self-organized critical behaviour (scale-invariance) of the solution is considered.
It is shown that the periodic discrete nonlinear Schrodinger equation, with cubic nonlinearity, possesses gap solutions, i.e. standing waves, with the frequency in a spectral gap, that are exponentially localized in the spatial variable. The proof is based on the linking theorem in combination with periodic approximations.
In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.
For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.
We consider the Navier-Stokes equations with Navier friction boundary conditions and prove two results. First, in the case of a bounded domain we prove that weak Leray solutions converge (locally in time in dimension >= 3 and globally in time in dimension 2) as the viscosity goes to 0 to a strong solution of the Euler equations, provided that the initial data converge in L 2 to a sufficiently smooth limit. Second, we consider the case of a half-space and anisotropic viscosities: we fix the horizontal viscosity, send the vertical viscosity to 0 and prove convergence to the expected limit system under a weaker hypothesis on the initial data.
In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multi-parametric bifurcations. It is demonstrated that an infinite number of twoparametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.
We develop a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional Henon-like maps are diffeomorphisms that limit on non-invertible one-dimensional maps. Wang and Young formulated hypotheses that suffice to prove the existence of chaotic attractors in these families. Three-dimensional singularly perturbed vector fields have return maps that are also two-dimensional diffeomorphisms limiting on one-dimensional maps. We describe a generic mechanism that produces folds in these return maps and demonstrate that the Wang-Young hypotheses are satisfied. Our analysis requires a careful study of the convergence of the return maps to their singular limits in the C-k topology for k >= 3. The theoretical results are illustrated with a numerical study of a variant of the forced van der Pol oscillator.
The formation of a polygonal configuration of proto-blood-vessels from initially dispersed cells is the first step in the development of the circulatory system in vertebrates. This initial vascular network later expands to form new blood vessels, primarily via a sprouting mechanism. We review a range of recent results obtained with a Monte Carlo model of chemotactically migrating cells which can explain both de novo blood vessel growth and aspects of blood vessel sprouting. We propose that the initial network forms via a percolation-like instability depending on cell shape, or through an alternative contact-inhibition of motility mechanism which also reproduces aspects of sprouting blood vessel growth.
The Keller-Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing-up solutions for large enough initial conditions in dimensions d >= 2, but all the solutions are regular in one dimension, a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller-Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyse this fractional Keller-Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller-Segel model.
Using an unusual yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at an asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable entropy. The measure mu we consider gives the mu-expansiveness property to the automaton. It is constructed with respect to a factor dynamical system based on simple 'counter dynamics'. As a counterpart, we prove that in the case of positively expansive automata, the perturbations move at positive linear speed over all the configurations.
In this paper, a selected analysis of the dynamics in an example impact microactuator is performed through a combination of numerical simulations and local analysis. Here, emphasis is placed on investigating the system response in the vicinity of the so-called grazing trajectories, i.e. motions that include zero-relative-velocity contact of the actuator parts, using the concept of discontinuity mappings that account for the effects of low-relative-velocity impacts and brief episodes of stick-slip motion. The analysis highlights the existence of isolated co-dimension-two grazing bifurcation points and the way in which these organize the behaviour of the impacting dynamics. In particular, it is shown how higher-order truncations of local maps of the near-grazing dynamics predict and enable the computation of global bifurcation curves emanating from such degenerate bifurcation points, thereby unfolding the near-grazing dynamics. Although the numerical results presented here are specific for the chosen model of an electrically driven and previously experimentally realized impact microactuator, the methodology generalizes naturally to arbitrary systems with impacts. Moreover, the qualitative nature of the near-grazing dynamics is expected to generalize to systems with similar nonlinearities.
We study a stochastic nonlocal partial differential equation, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially localized 'activity bumps'. This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we study the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar 'coarse' variable, i.e. reducing the dimensionality of the system. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately initialized bursts of direct simulation, and the effects of changing parameters on this potential can easily be studied. We demonstrate this approach in terms of (a) an experience-based 'intelligent' choice of the coarse variable and (b) a variable obtained through data-mining direct simulation results, using a diffusion map approach.