强迫耗散系统的有序结构和系统的发展(Ⅱ)，广义能量极小值原理和系统的发展

利用大气非线性动力学证明了广义能量极小值原理，进一步从理论上证明它是同大气非平衡 态热力学最小熵产生原理在物理上是一致的.前者表明，强迫耗散动力系统的终态广义能量 达极小值；而最小熵产生原理表明，远离热力学平衡态的开放系其终态时系统内部的不可逆 过程最弱.而且，系统广义能量达极小值和系统熵产生达极小值的终态一般是一个稳定的定 态，它对应着系统的某种有序结构.也就是说它是一个“低耗高效”的有序定态.大气系统作 为自然界一个典型的物理复杂系统，其最小熵产生原理和广义能量极小值原理隐示了自然界 复杂系统的一个一 关键词： 强迫耗散系统 能量极值原理 有序结构 动力系统     

外强迫作用下正压大气非线性特征数值模拟

在两种边界条件下用正压大气非线性位势涡度方程模拟了强迫、耗散和非线性共同作用下大气运动的若干特征.在南北两端为齐次边界条件时，单纯的热力强迫下只出现了波状定常解；在只有经向加热和波状地形共同强迫下出现了在两种基本流型之间振荡的准周期解，十分类似于旋转地球上大气运动的纬向和经向流型之间的准周期循环，前者对应于强的西风环流，后者对应于弱西风流型.在周期边界条件下，弱的下垫面加热导致定常解，较强的加热强迫可以得到周期解.尽管下垫面的波状地形也能强迫一个类似的准周期振荡，但弱西风流型是对称的，不像大气中的弱西风流型.因此，在周期边界条件下用低阶截谱模式求出的解析或数值解，不一定适合描述中高纬度大气的非线性动力特征. 关键词： 正压大气 热力强迫 地形起伏 周期解 准周期振荡     

有扩散不稳定性的四阶反应-扩散系统的空间周期结构

本文利用分叉理论研究一个含参数的有扩散不稳定性的四阶反应-扩散系统的空间周期结构问题,其中扩散项服从Cahn-Hillard广义扩散定律。首先通过稳定性和奇异性的分析得到从空间均匀定态产生空间周期定态的判别准则,然后利用奇异摄动方法获得这个空间周期定态的二阶近似解。本文的理论结果很好地符合文献[3]中的数值结果。 关键词：     

一类大气等离子体反应扩散模型的解法

研究了一类大气等离子体反应扩散模型. 利用奇摄动方法得到了问题的渐近解, 并用微分不等式理论证明了解的一致有效性.     

耗散对孤波与局地地形相互作用的影响

利用扰动法，由包括耗散和地形的准地转位涡度方程导出了强迫mKdV-Burgers方程，求得了小耗散情形下mKdV-Burgers方程的近似分析解，分析了mKdV孤波质量和能量的时间演变特性。对给定的局地地形，利用拟谱法对强迫mKdV-Burgers方程进行了数值求解。结果显示，小耗散的存在使弧波的振幅和移速随时间缓慢地减小，孤波宽度则随时间缓慢增大；在耗散和地形强迫的非线性系统中，在孤波与地形的相互作用中，耗散的存在使孤波在强迫区附近振荡传播，这有利于大振幅扰动的形成。     

具有一般非线性弹性力和广义阻尼力的相对转动非线性系统的周期解问题

讨论了一类相对转动非线性动力系统的周期解问题.首先建立了一类具有一般非线性弹性力、广义阻尼力和强迫周期力项的相对转动非线性动力系统;其次得到了对应自治系统的周期解不存在性结果,以及运用Mawhin重合度理论得到了该模型的周期解存在性结果,推广了已有的结果;最后举例证明本文结果的正确性.     

用连续法计算五维对流模型的定常解和周期解

利用连续算法(Continuation algorithm)对五维对流非线性动力系统的定常解和周期解进行了数值计算。在参数平面Ri-Re上计算出实分岔点曲线、极限点曲线、Hopf分岔点曲线,绘出了分岔图。在分岔图上的不同区域,存在性质不同的稳定解如定常吸引子、周期吸引子等。分析了定常解、周期解的分岔过程。计算结果很好地说明大气中由基本态到对流态再到波动态最后到湍流态的物理转换过程。 连续算法对研究非线性动力系统的分岔以及耗散结构是很有效的计算方法。     

一类相对转动非线性动力学系统周期解的唯一性与精确周期解

研究了一类具有线性刚度、非线性阻尼力和强迫周期力项的相对转动非线性动力学系统周期解的唯一性和精确周期解.讨论了一类自治系统极限环的唯一性与稳定性.应用定性分析方法,给出了一类相对转动非线性动力学系统具有唯一周期解的必要条件,并在一定条件下得到了系统的一类精确周期解.     

激光脉冲放大器传输波的计算

研究了一个非线性激光脉冲放大器传输系统，利用广义泛函-变分迭代理论和方法，得到了相应系统的近似解.从而为设计激光脉冲放大器提供了便利. 关键词： 激光 放大器 非线性     

具有一般广义阻尼力和强迫周期力的相对转动非线性动力学模型的周期解

研究了一类具有一般广义阻尼力和强迫周期力项的相对转动非线性动力学模型的周期解问题.讨论了对应自治系统的周期解问题.运用Mawhin重合度理论,得到了该模型的周期解存在惟一性结果,推广了已有的结果,并且列举了具体的例子来说明结果的新颖性.     

Thermodynamics of nonlinear,interacting irreversible processes. II

The scope of the thermodynamic theory of nonlinear irreversible processes is widened to include the nonlinear stability analysis of system motion. The emphasis is shifted from the analysis of instantaneous energy flows to that of the average work performed by periodic nonlinear processes. The principle of virtual work separates dissipative and conservative forces. The vanishing of the work of conservative forces determines the natural period of oscillation. Stability is then determined by the variations of the dissipative forces with amplitude of oscillation. If the work is a minimum, under certain conditions, the motion is stable. Reduction to linear analysis shows the coincidence with the impedance analysis of electrical circuit theory. The theory is applied to the analysis of temporal interactions of nonlinear irreversible processes in the particular cases of synchronization and hysteresis. Characteristic nonequilibrium phenomena of directional energy transfers, self-excitation, system passivity, wave modulation, and beat phenomena are observed. Possible relationships with biological processes are discussed.     

Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons

The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV–Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.     

Exploding dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one and two spatial dimensions

We review the work on exploding dissipative solitons in one and two spatial dimensions. Features covered include: the transition from modulated to exploding dissipative solitons, the analogue of the Ruelle-Takens scenario for dissipative solitons, inducing exploding dissipative solitons by noise, two classes of exploding dissipative solitons in two spatial dimensions, diffusing asymmetric exploding dissipative solitons as a model for a two-dimensional extended chaotic system. As a perspective we outline the interaction of exploding dissipative solitons with quasi one-dimensional dissipative solitons, breathing quasi one-dimensional solutions and their possible connection with experimental results on convection, and the occurence of exploding dissipative solitons in reaction-diffusion systems. It is a great pleasure to dedicate this work to our long-time friend Hans (Prof. Dr. Hans Jürgen Herrmann) on the occasion of his 60th birthday.     

Quantum theory of a dissipative oscillator

The nonlinear equation of dissipative quantum mechanics is considered in the relaxation-time approximation. It is shown that the steady current-free state does not change when dissipation is taken into account; in particular, there is no ground-state damping, and the zero energy is conserved. A solution is obtained to the problem of the excitation of a harmonic oscillator, serving as a model of single-mode radiation in an open resonator; the solution obtained describes the evolution of the oscillator from an arbitrary steady state under the action of a constraining force. Transition probabilities between the oscillator steady states are calculated. The results are found to be in agreement with the classical theory of damping oscillations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 99–105, September, 1978.     

A comment on the stochastic quantization of friction

It is shown that Hashimoto's recent construction of a nonlinear wave equation on the basis of stochastic mechanics for a dissipative system submitted to a general velocity-dependent frictional force is not valid, except for the trivial one-dimensional case.     

Nonlinear analysis of a reaction-diffusion system: Amplitude equations

A reaction-diffusion system with a nonlinear diffusion term is considered. Based on nonlinear analysis, the amplitude equations are obtained in the cases of the Hopf and Turing instabilities in the system. Turing pattern-forming regions in the parameter space are determined for supercritical and subcritical instabilities in a two-component reaction-diffusion system.     

Manipulation of self-aggregation patterns and waves in a reaction-diffusion system by optimal boundary control strategies

Reaction-diffusion systems are of considerable importance in many areas of physical sciences. For many reasons, an external manipulation of the dynamics of these processes is desirable. Here we show numerically how spatiotemporal behavior like pattern formation and wave propagation in a two component nonlinear reaction-diffusion model of bacterial chemotaxis can be externally controlled. We formulate the control goal as an objective functional and apply numerical optimization for the solution of the resulting control problem.