Bifurcations and traveling waves in a delayed partial differential equation

Here cell population dynamics in which there is simultaneous proliferation and maturation is considered. The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters. The solution behavior depends on the initial function varphi(x) and a three component parameter vector P=(delta,lambda,r). For strictly positive initial functions, varphi(0) greater, similar 0, there are three homogeneous solutions of biological (i.e., non-negative) importance: a trivial solution u(t) identical with 0, a positive stationary solution u(st), and a time periodic solution u(p)(t). For varphi(0)=0 there are a number of different solution types depending on P: the trivial solution u(t), a spatially inhomogeneous stationary solution u(nh)(x), a spatially homogeneous singular solution u(s), a traveling wave solution u(tw)(t,x), slow traveling waves u(stw)(t,x), and slow traveling chaotic waves u(scw)(t,x). The regions of parameter space in which these solutions exist and are locally stable are delineated and studied.     

Excitations of nonlinear local waves described by the sinh-Gordon equation with a variable coefficient

We explore novel excitations in the form of nonlinear local waves, which are described by the sinh-Gordon (SHG) equation with a variable coefficient. With the aid of the self-similarity transformation, we establish the relationship between solutions of the SHG equation with a variable coefficient and those of the standard SHG equation. Then, using the Hirota bilinear method, we obtain a more general bilinear form for the standard SHG equation and find new one- and two-soliton waves whose forms involve two arbitrary self-similarity functions. By an appropriate choice of the smooth self-similarity functions, we determine and display novel localized waves, and discuss their properties. The method used here can be extended to the three- and higher order soliton solutions.     

Dynamic correlations in a thermalized system described by the Burgers equation

For the field u(x, t) governed by the Burgers equation with a thermal noise, short-time asymptotics of multipoint correlators are obtained. Their exponential parts are independent of the correlator number. This means that they are determined by a single rare fluctuation and exhibit an intermittency phenomenon.     

Self-preservation of large-scale structures in a nonlinear viscous medium described by the Burgers equation

We use the asymptotic solution of the one-dimensional Burgers equation to study the self-preservation of large-scale random structures. We show that in the process of their evolution, large-scale structures remain stable against small-scale perturbations for the case of a continuous initial spectrum with a spectral index smaller than unity. We study both analytically and numerically the correlation coefficient of a large-scale structure and of the same structure with a high-frequency perturbation and show that with the passage of time the coefficient tends to unity. Using the asymptotic formulas of the theory of random excursion of stochastic processes, we study the statistical properties of the perturbing field and find that the effect of high-frequency perturbations is equivalent to the introduction of effective viscosity. Zh. éksp. Teor. Fiz. 115, 564–583 (February 1999)     

Stabilization of chaos systems described by nonlinear fractional-order polytopic differential inclusion

In this paper, sliding mode control is utilized for stabilization of a particular class of nonlinear polytopic differential inclusion systems with fractional-order-0?相似文献   

Periodic solutions of a singularly perturbed delay differential equation

A singularly perturbed differential delay equation of the form

(1)     

Waves described by higher approximations of the nonlinear schrödinger equation

We analyze nonlinear waves within the framework of the basic equation of the third approximation of the theory of dispersion of nonlinear waves. This equation, which includes terms of nonlinear and linear dispersion of the third order, adquately describes short wave packets with a width of up to several wavelengths. Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 2, pp. 222–242, February, 1998.     

Uncertainty features of microscopic particles described by nonlinear Schrüdinger equation

The uncertainty relationship between position and momentum of the microscopic particles is calculated by nonlinear quantum theory in which the states of the particles are described by a nonlinear Schrüdinger equation. The results show that the uncertainty relation differs from that in the quantum mechanics and has a minimum value in this case. This means that the position and momentum of the particles could be determined simultaneously to a certain degree, which could be caused by the wave–corpuscle duality of the microscopic particles described by the nonlinear Schrüdinger equation.     

Interaction of solitary waves in the system described by the Kadomtsev-Petviashvili equation with a self-consistent source

Solitary waves moving with nonconstant velocity are found in the nonlinear integrable system described by the Kadomtsev-Petviashvili equation with a self-consistent source. Explicit expressions are derived for the solutions describing the interaction of an arbitrary number of these waves. It is shown that in contrast with the decay and fusion of solitons, the decay and fusion of the above solitary waves are not of the resonance nature and proceed in the general case. The obtained results are relevant to some problems of hydrodynamics, solid state physics, plasma physics, etc.     

Accelerated expansion of the Friedmann Universe filled with the perfect liquid described by a nonlinear equation of state

A spatially flat Friedmann model of the Universe filled with the perfect liquid with a nonlinear homogeneous time-dependent equation of state is discussed. A gravitational equation of motion is solved. It is shown that in this case, there can result a periodic Universe rerunning cycles of space acceleration of the phantom (non-phantom) type with occurrence of cosmological singularities.     

Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).     

Dynamics of two-dimensional radiating vortices described by the nonlinear Schrödinger equation

The paper considers the dynamics of dark charged solitons (vortices) described by the two-dimensional (2D) nonlinear Schrödinger equation (NSE) with a repulsive potential. The dynamics of these point-like vortices in the NSE is quite different in comparison with the vortices in an incompressible liquid because of the possibility of wave-like emission of energy, momentum, and angular momentum in the first case. Another important feature is the characteristic scale of the problem, namely the screening parameter. Related problems of the collapse of a vortex dipole and the decay of a multicharged vortex in a region bounded by an absolutely reflecting shell are investigated both analytically and numerically. The conditions and scaling of a vortex dipole collapse and the limitations on the decay of a multicharge dipole in a bounded region are obtained.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Stationary waves described by a generalized KdV-burgers equation

We have obtained solutions in the form of stationary waves within the framework of the generalized KdV-Burgers equation, which contains nonconservative terms of linear pump, linear HF dissipation, and nonlinear dissipation. Both periodic and solitary waves are analyzed. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 10, pp. 1241–1248, October, 1997.     

Solution multistability in first-order nonlinear differential delay equations

The dependence of solution behavior to perturbations of the initial function (IF) in a class of nonlinear differential delay equations (DDEs) is investigated. The structure of basins of attraction of multistable limit cycles is investigated. These basins can possess complex structure at all scales measurable numerically although this is not necessarily the case. Sensitive dependence of the asymptotic solution to perturbations in the initial function is also observed experimentally using a task specific electronic analog computer designed to investigate the dynamics of an integrable first-order DDE.     

Ensemble and trajectory statistics in a nonlinear partial differential equation

We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value() problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean and variance are well-characterized functions of the noise statistics, and and depend on . When noise is continuously present along the trajectory, and are independent of the noise statistics and . However, in their evolution toward and , both _u (t, x) and _u ² (t, x) depend on the noise and.     

Dynamics of a nonlinear parametrically excited partial differential equation

We investigate a parametrically excited nonlinear Mathieu equation with damping and limited spatial dependence, using both perturbation theory and numerical integration. The perturbation results predict that, for parameters which lie near the 2:1 resonance tongue of instability corresponding to a single mode of shape cos nx, the resonant mode achieves a stable periodic motion, while all the other modes are predicted to decay to zero. By numerically integrating the p.d.e. as well as a 3-mode o.d.e. truncation, the predictions of perturbation theory are shown to represent an oversimplified picture of the dynamics. In particular it is shown that steady states exist which involve many modes. The dependence of steady state behavior on parameter values and initial conditions is investigated numerically. (c) 1999 American Institute of Physics.     

Development of stochastic oscillations in a one-dimensional dynamical system described by the Korteweg-de Vries equation

The behavior of the solution of the Korteweg-de Vries equation for large-scale oscillating periodic initial conditions prescribed on the entire x axis is considered. It is shown that the structure of small-scale oscillations arising in a Korteweg-de Vries system as t→∞ loses its dynamical properties as a consequence of phase mixing. This process can be called the generation of soliton turbulence. The infinite system of interacting solitons with random phases developing under these conditions leads to oscillations having a stochastic character. Such a system can be described using the terms applied to a continuous random process, the probability density and correlation function. It is shown that for this it suffices to determine from the prescribed initial conditions amplitude distribution function of the solitons and their mean spatial density. The limiting stochastic characteristics of the mixed state for problems with initial data in the form of an infinite sequence of isolated small-scale pulses are found. Also, the problem of stochastic mixing under arbitrary initial conditions in the dispersionless limit (the Hopf equation) is completely solved. Zh. éksp. Teor. Fiz. 115, 333–360 (January 1999)