Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.     

自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

Effective approaches to explore rich dynamics of delay-differential equations

Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed.     

Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective

We conjecture that the well-known oscillations (3- to 5-yr and 10-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude—a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.     

Rich dynamic of a stage-structured prey–predator model with cannibalism and periodic attacking rate

The dynamic behavior of a stage-structure prey–predator model with cannibalism for prey and periodic attacking rate for predator is investigated. Firstly, the permanence, locally and globally asymptotic stability analyses of the model with constant attacking rate are explored. After that, sufficient conditions for the permanence of the corresponding nonautonomous system with periodic attacking rate are obtained. Furthermore, numerical simulations are presented to illustrate the effects of periodic attacking rate. Simulation results show that the system with periodic attacking rate shows a rich behaviors, including period-doubling and period-having bifurcations, chaos and windows of periodicity.     

Resonances for multistratified acoustic waveguides

The equation of acoustic oscillations in multistratified waveguides is considered. It is assumed that the properties of the medium do not depend on the longitudinal coordinate in a neighbourhood of infinity and may be different at different ends of the waveguides. It is proved that the truncated resolvent of the corresponding operator admits an analytical continuation through the continuous spectrum. The singularities (poles, branching points) of the truncated resolvent on the continuous spectrum are investigated. The large time asymptotic behavior of the compulsory oscillations due to periodic forces is obtained.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

A homogenization approach for the motion of motor proteins

We consider the asymptotic behavior of an evolving weakly coupled Fokker–Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are, respectively, proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile.     

Impulsive perturbations of a three-trophic prey-dependent food chain system

The dynamics of an impulsively controlled three-trophic food chain system with general nonlinear functional responses for the intermediate consumer and the top predator are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive controls act in a periodic fashion, the constant impulse (the biological control) and the proportional impulses (the chemical controls) acting with the same period, but not simultaneously. Sufficient conditions for the global stability of resource and intermediate consumer-free periodic solution and of the intermediate consumer-free periodic solution are established, the latter corresponding to the success of the integrated pest management strategy from which our food chain system arises. In this regard, it is seen that, theoretically speaking, the control strategy can be always made to succeed globally if proper pesticides are employed, while as far as the biological control is concerned, its global effectiveness can also be reached provided that the top predator is voracious enough or the (constant) number of top predators released each time is large enough or the release period is small enough. Some situations which lead to chaotic behavior of the system are also investigated by means of numerical simulations.     

Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features

By adapting a pre-existing model to include the effects of Vascularization within a tumor or multicell spheroid, a predator-prey system describing the cell populations of a solid tumor and reactive lymphocytes is formulated. The paper serves as a review of the minimal deterministic approach to tumor-host immune system interactions while examining, in a qualitative manner, the modifications to the dynamics induced by a simple representation of the vascularized tumor. In addition, the possibility of limit-cycle behavior is studied by regarding each of six parameters present in the model as a bifurcation parameter. Thus, in principle, well-defined and periodic oscillations in both lymphocyte and tumor cell populations may occur under appropriate circumstances; whether or not such oscillations are sustainable by the host, and their stability, amplitude and period depend on aquisition of more quantitative information concerning the relevant parameter ranges.     

Regular low frequency cardiac output oscillations observed in critically ill surgical patients

A serendipitous finding during development of an automated “electronic flow chart” system to gather data on ICU patients [1] was the observation of low frequency oscillations in blood pressure that were not explained by systematic variability in the environment. [2] This finding has since been confirmed by others. [3,4] In the present report, hemodynamic data for critically ill surgical patients was continuously collected and visualized on a computer workstation to search for patterns not noted by standard monitoring. With this system, we observed low-frequency periodic oscillations in the cardiac output of ten patients, with regular periodicities of 4 to 280 minutes (average = 34 minutes). The mortality rate in these patients was 40%, while the mortality was only 10.8% in 83 similarly monitored intensive care unit (ICU) patients who did not develop regular oscillations in cardiac output. Interestingly, these oscillatory patterns appear to be associated with inadequate resuscitation of increased metabolic rates. The mathematical definition of “chaos” refers to irregular behavior that appears to be random but is actually deterministic. [5] A surprising finding concerning transitions between states of apparent randomness and order in nonlinear systems is that many systems become more organized after being disturbed. Chaotic behavior in biological systems may represent a normal physiologic state, while the loss of chaotic behavior may herald a pathophysiologic state. [6] The mechanism of the regular low frequency oscillations in cardiac output remains to be determined, but the high mortality rate suggests that it is a pathophysiologic marker, perhaps due to inadequate oxygen delivery in under-resuscitated shock. © 1997 John Wiley & Sons, Inc.     

Collective behavior of interacting locally synchronized oscillations in neuronal networks

Local circuits in the cortex and hippocampus are endowed with resonant, oscillatory firing properties which underlie oscillations in various frequency ranges (e.g. gamma range) frequently observed in the local field potentials, and in electroencephalography. Synchronized oscillations are thought to play important roles in information binding in the brain. This paper addresses the collective behavior of interacting locally synchronized oscillations in realistic neural networks. A network of five neurons is proposed in order to produce locally synchronized oscillations. The neuron models are Hindmarsh–Rose type with electrical and/or chemical couplings. We construct large-scale models using networks of such units which capture the essential features of the dynamics of cells and their connectivity patterns. The profile of the spike synchronization is then investigated considering different model parameters such as strength and ratio of excitatory/inhibitory connections. We also show that transmission time-delay might enhance the spike synchrony. The influence of spike-timing-dependence-plasticity is also studies on the spike synchronization.     

A Holling II functional response food chain model with impulsive perturbations

In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.     

Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix

We consider a linear periodic control system such that the ranks of the matrix multiplying the control and the extended matrix consisting of the averaged coefficient matrix and the matrix multiplying the control are the same. We assume that the control has the form of feedback linear in the state variables and is periodic with the same period as the system itself. We pose the problem of control of the frequency spectrum of strongly irregular periodic oscillations with an objective set, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies. We obtain necessary and sufficient conditions for the solvability of this problem.     

DISCONTINUOUS FORCING OF PERIODIC SOLUTIONS IN C1 VECTOR FIELDS WITH APPLICATIONS TO POPULATION MODELS

Averaging methods are used to compare solutions of two-dimensional systems of ordinary differential equations with constant or periodic forcing. The asymptotic separation of solutions of the periodically forced equations from the solutions of the constantly forced equations is proportional to the L¹ norm of the periodic forcing terms. This result is applied to population models of Kolmogorov-type with climax fitness functions where forcing represents stocking or harvesting of a population. The asymptotic behavior of such systems may be controlled, to some extent, by varying the period and/or amplitude of the forcing functions.     

Optimizing the dynamics of a two-cell DC–DC buck converter by time delayed feedback control

A study of the dynamical behavior of a two-cell DC–DC buck converter under a digital time delayed feedback control (TDFC) is presented. Various numerical simulations and dynamical aspects of this system are illustrated in the time domain and in the parameter space. Without TDFC, the system may present many undesirable behaviors such as sub-harmonics and chaotic oscillations. TDFC is able to widen the stability range of the system. Optimum values of parameters giving rise to fast response while maintaining stable periodic behavior are given in closed form. However, it is detected that in a certain region of the parameter space, the stabilized periodic orbit may coexist with a chaotic attractor. Boundary between basins of attraction are obtained by means of numerical simulations.     

Analysis of competitive chemostat models with the Beddington–DeAngelis functional response and impulsive effect

In this paper, we introduce and study a competitive system with Beddington–DeAngelis type functional response in periodic pulsed chemostat conditions. We investigate the subsystem with substrate and one of the microorganisms and study the stability of the periodic solutions, which are the boundary periodic solution of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate and one of the microorganism. Further, we prove that the system is permanent if the impulsive period less than some critical value. Therefore, our results are valuable for the manufacture of products by genetically altered organisms.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.