Advanced Synergetics

Haken, Hermann

doi:10.1007/978-3-642-45553-7

Cited by 855 publications

(562 citation statements)

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“…If one expands F (ᾱ(ǫ)t, k) w.r.t. k around k = 1 withᾱ(ǫ) fixed, one has 56) up to O(ǫ 2 ). Notice that the expansion of the elliptic functions at k = 1 is subtle; we have made the manipulation as follows;…”

Section: The First Order Equation Readsmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

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Section: The First Order Equation Readsmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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“…In the sense of Haken [1], we feel that all traditional disciplines in physics, which are concerned with the macroscopic behavior of multicomponent systems, require new ideas and concepts based on the synergetic approach, in order to cope with selforganizing systems.…”

Section: Discussionmentioning

confidence: 99%

“…Close to such instability points the dynamics of the system and its emerging structures are determined by a set of, in general, a few collective variables, often called order parameters. The underlying synergetic approach introduced by Haken [1] can explain the unexpected order and coherence arising on the macroscopic scale, regardless of the large number of competing physical forces interacting on the microscopic scale. Motivation for the intensive study of cooperative dynamics and pattern formation phenomena during the past few years derives from an increasing appreciation of the remarkable diversity of behavior encountered in nonlinear systems and of universal features shared by entire classes of similar nonlinear dynamic processes.…”

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confidence: 99%

Classification of current instabilities during low-temperature breakdown in germanium

et al. 1989

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Abstract. We present experimental investigations on the spatio-temporal nonlinear current flow in the post-breakdown regime of p-germanium at liquid-helium temperatures. The basic nonlinear effects are characterized in terms of the underlying semiconductor physics, taking into account the influence of different experimental parameters. 05.45. + b, 72.20.Ht, 72.70. + m It is well known that a large number of physical and nonphysical systems show spontaneous formation of spatial or temporal structures as a result of instability. Close to such instability points the dynamics of the system and its emerging structures are determined by a set of, in general, a few collective variables, often called order parameters. The underlying synergetic approach introduced by Haken [1] can explain the unexpected order and coherence arising on the macroscopic scale, regardless of the large number of competing physical forces interacting on the microscopic scale. Motivation for the intensive study of cooperative dynamics and pattern formation phenomena during the past few years derives from an increasing appreciation of the remarkable diversity of behavior encountered in nonlinear systems and of universal features shared by entire classes of similar nonlinear dynamic processes. PACS:So far, it appears that the subject of such complex nonlinear behavior is dominated by theoretical investigations and computer studies, whereas experimental measurements on real physical systems represent the minority. Among the various objects which can be studied experimentally, solid-state turbulence in semiconductors appears particularly interesting [2]. Nonlinear current transport behavior during lowtemperature avalanche breakdown of extrinsic germanium comprises the self-sustained development of spatio-temporal dissipative structures in the formerly homogeneous semiconductor [3]. This kind of nonequilibrium phase transition between different conducting states results from the autocatalytic nature of impurity impact ionization generating mobile charge carriers [4]. The simple and direct experimental accessibility via advanced measurement techniques favors semiconductors as a nearly ideal study object for complex nonlinear dynamics compared to other physical systems. Further representing a convenient model reaction-diffusion system that exhibits distinct universal features, the present semiconductor system may acquire general significance for many synergetic systems in nature. Finally, in view of the rapidly growing application of semiconductor technologies, the understanding, control, and possible exploitation of sources of instability in these systems have considerable practical importance. This paper gives a classification of our experimental investigations on the spatio-temporal nonlinear current flow in the post-breakdown regime of p-germanium at liquid-helium temperatures. Section 1 briefly outlines an example of a set of nonlinear current instabilities obtained from our semiconductor system. Section 2 reports the characterization of the basic...

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“…Various techniques have been derived by several researchers and mathematicians to study the complexity of nonlinear stochastic differential equations (Haken, 1983;Nisbet & Gurney, 1982). The efficacy of moment technique to linearize the above system is mentioned by many researchers (e.g.…”

Section: Statistical Linearization: Moment Equationsmentioning

confidence: 99%

Deterministic and stochastic analysis of a prey–predator model with herd behaviour in both

Maiti¹,

Sen

Samanta

2016

Systems Science & Control Engineering

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This paper aims to study the dynamical behaviours of a prey-predator system, where both the prey and predator show herd behaviours. Positivity, boundedness, stability of equilibrium points, et cetera, are discussed in deterministic environment. To incorporate the effect of fluctuating environment, we have perturbed the birth rate of prey species and death rate of predator species by Gaussian white noises. Then the resulting model is cultured by the method of statistical linearization to study the stability and non-equilibrium fluctuation of the populations in stochastic environment. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically. ARTICLE HISTORY

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Advanced Synergetics

Cited by 855 publications

References 0 publications

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Classification of current instabilities during low-temperature breakdown in germanium

Deterministic and stochastic analysis of a prey–predator model with herd behaviour in both

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