White and coloured external noise and transition phenomena in nonlinear systems

Arnold, Ludwig; Horsthemke, Werner; Lefever, René

doi:10.1007/bf01324036

Cited by 107 publications

(38 citation statements)

References 19 publications

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“…Hence, in the limit T?-" 2 --0, which is of interest here, the evolution of 9 is completely correlated to that of z : The spectral density S(v) of z T converges to a 8 peak located at the frequency i^ = 0, i.e., lim 2^0 S(v) = (cF 2 /2)8(v). This is exactly what one expects to happen when the evolution of 9 is so fast that at every instant the value of this variable is able to "equilibrate" to the fluctuations of the noise (for more details see Arnold, Horsthemke, and Lefever, 18 and Ref. 1, p. 226).…”

Section: P*( U ) = F R F 0 V Dzd0p(u9z)mentioning

confidence: 52%

Sensitivity of a Hopf Bifurcation to Multiplicative Colored Noise

Lefever

Turner

1986

Phys. Rev. Lett.

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The location of a Hopf bifurcation in parameter space may be postponed or advanced by multiplicative colored noise depending on the interplay between the time scale of the noise and the rotation period of the phase variable.PACS numbers: 05.40. +jThe effect of multiplicative noise on nonequilibrium systems is currently the object of considerable research, particularly into the capacity of such noise to give rise to so-called "noise-induced transitions." 1 The latter can be viewed, in most experimental examples known so far, 2 " 6 as a shift of a preexisting deterministic instability: The noise displaces, by an amount proportional to its intensity, the location in parameter space of an instability already present under noiseless conditions.In the analysis of this behavior it is usual (i) to take as a starting point the Landau equation, or more generally the normal-form equation, giving the long-time deterministic dynamics of some slow mode(s) near the instability 7 ; (ii) to take the noise into account by letting some parameter fluctuate in this reduced evolution equation; and (iii) to consider that the noise is white. This is in line with (i): The noise indeed is not affected by the instability; hence, the closer one is to the instability point, the more separated the time scales of the slow mode(s) and of the noise become and the more appropriate the white-noise idealization appears.The reliability of these simplifications, however, does not depend solely on the occurrence of a critical slowing down; it also requires that the coupling of the noise with the "fast" variables, which do not slow down at the instability, be appropriately modeled. This condition has been scarcely studied. Usually the role of the fast variables is overlooked 8 and the adequacy of (i)-(iii) is simply postulated; the justification comes a posteriori insofar as the outcome of the analysis agrees with the experimental results. This is, however, not always the case; e.g., the noise-induced shifts of electrohydrodynamic instabilities in liquid-crystal systems cannot be explained by the addition of white noise to the bifurcation parameter of the Landau equation for the unstable mode. 6 * 9,10Our motivation here is to investigate the coupling between noise and a fast variable in the case of a wellknown chemical model (Brusselator) ] l presenting an instability frequently met in nonequilibrium systems: the Hopf bifurcation. We find that even in the limit of vanishingly small noise intensity, it is the interplay between the noise and the "fast" phase variable which determines whether the bifurcation is postponed or advanced.The effect of multiplicative white noise has been considered recently in diverse physicochemical oscillators. 12 The approaches based on normal forms 13 " 15 predict that the noise tends to stabilize the trivial state. 16 Furthermore if, as recommended in Ref. 1, the bifurcation of the most probable value of the probability density is taken as index of bifurcation, one finds that the noise postpones the instability by an amo...

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Section: P*( U ) = F R F 0 V Dzd0p(u9z)mentioning

confidence: 52%

Sensitivity of a Hopf Bifurcation to Multiplicative Colored Noise

Lefever

Turner

1986

Phys. Rev. Lett.

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“…The properties established that way, are retained at least qualitatively under more realistic noise conditions. This point is discussed in detail elsewhere (Arnold et al, 1978).…”

Section: Introductionmentioning

confidence: 89%

“…For more details on this point see Arnold (1973). To analyze the stationary solutions of (9) we pass to the corresponding Fokker-Planck equation:…”

Section: Efl= Fl E(fl~-fl)(flc-fl)=az6(t-t')mentioning

confidence: 99%

“…Remarkably, it comprises systems whose average environmental conditions are such that if fluctuations were neglected the possibility of instabilities would physically be completely excluded. It also comprises systems which in deterministic treatments can never become unstable whatever the values of their parameters or their physical conditions (Arnold et al, 1978).…”

Section: Introductionmentioning

confidence: 99%

“…Throughout the paper, we shall make use of Itocalculus (Arnold, 1973). Its distinct advantage is that it is mathematically rigourous in contrast to the Langevin formalism widely used to describe the influence of noise in physical and chemical systems.…”

Section: Introductionmentioning

confidence: 99%

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