Normal form analysis of stochastically forced dynamical systems

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Normal form analysis of stochastically forced dynamical systems

C. Nicolis

Znstitut d'Ae'ronomie Spatiale de Belgique, Avenue Circulaire 3, 1180 Bruxelles, Belgium

and G. Nicolis

Faculte' des Sciences de 1'Universite' Libre de Bruxelles, Campus Plaine CP. 226, Boulevard du Triomphe, 1050 Bruxelles, Belgium

Abstract

The possibility of reducing the dynamics of a system undergoing a Hopf bifurcation and forced by multiplicative noise to a universal normal form is examined on a simple model of geophysical interest. It is shown that the normal form equations and the stationary probability distribution depend on the way the noise is coupled to the original system. The universality of the normal forms is thus severely limited in the presence of noise.

1. Introduction

Stochastically forced dynamical systems are very common in nature and have been the subject of extensive investigations (Horsthemke and Lefever, 1984;

Gardiner, 1983). Of special interest is the phenomenon of 'noise induced transitions', which may arise when the forcing is coupled with the internal dynamics in a multiplicative fashion. For instance, an initially unforced system operating in the vicinity of a bifurcation may experience, under the effect of noise, a shift of the most probable value of the underlying probability distribution by an amount which depends on the variance of the noise as well as on its intrinsic parameters (Horsthemke and Lefever, 1984).

The possibility to reduce the description of a deterministic dynamical system undergoing a bifurcation to a universal normal form displaying a limited number of variables is well established, at least for simple transitions like pitchfork or Hopf bfurcation (Guckenheimer and Holmes, 1983). Whether a comparable universality prevails in the presence of a multiplicative stochastic forcing is a different matter, since the transformations leading to the normal form are likely to affect the noise process in a system-dependent manner. The purpose of the present paper is to illustrate this point on a simple model, and to stress in this way the need for a careful study of the passage between the original system and the normal form in the presence of stochastic forcing.

@ Oxford University Press 1986

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250

C. NICOLIS A N D G . NICOLIS

2. Normal form of a nonlinear oscillator in the presence of noise We consider the following dynamical system, arising in geophysics in connection with the modelling of long-term climatic variability (Saltzman et al., 1982):

i = - r ] + 8 , e = - ~ r ] + b 8 - r ] ~ 8 . (1) This system describes the interaction between the mean excess ocean temperature 8 and the excess sea-ice extent

r]

around certain reference values. Here a and b are positive parameters depending on the albedo, the atmospheric composition, the hydrological cycle and so forth. In view of the variability of these factors we regard b as a random process,

where q << 1 and F is a Gaussian white noise of unit variance. One can easily check that in the absence of noise and as long as a > b, b

= 1

is a Hopf bifurcation point such that for b > 1 a stable limit cycle solution of (1) is generated. Our main objective here is to study the response of this bifurcation to the stochastic forcing described by (2).

We first reduce equations (I), (2) to their normal form. Thanks to the absence of quadratic terms, a linear transformation T, constructed from the eigenvectors of the linearized stability operator and diagonalizing the noise-free linear part, is sufficient (Nicolis, 1984):

where

=

i(6 - I),

w, =

(a - 6)i.

Operating on both sides of (1) by T-' we find that, in the dominant order in 6 , the transformed variable z , defined through

obeys the following equation:

The nonlinear part of the right-hand side contains contributions coming from resonant terms (in z 1 ~ 1 ~ ) and non-resonant terms (in z3, z * ~ ,

Z *

1 ~ 1 ~ ) . According to the theory of dynamical systems (Guckenheimer and Holmes, 1983) non- resonant terms may be eliminated as long as the system remains in the vicinity of the bifurcation point 6

=

0. We have verified that in the presence of the forcing term F(t) these terms remain negligible to this order. The situation is different as far as the contribution of z and z* terms in the coefficient multiplying F(t) is concerned. At first sight one would tend to neglect the term in z* on the grounds

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that it contains a 'fast' angular dependence. This indeed turns out to be legitimate if F ( t ) represents a (deterministic) periodic forcing having a frequency close to

o,.

On the other hand, if F(t) is a white noise process both z and z* turn out to contribute terms of the same order. The reason for this is to be sought in the presence of an infinity of harmonics in F(t), which cancel the fast angular dependencies that may be induced by multiplicative terms. This point is also clearly recognized in a recent paper by Coullet et al. (1985).

Separating real and imaginary parts we finally arrive at the following set of stochastic differential equations:

where x, y are related to be the original variables q,

8

through

Equations (6) bring out quite clearly the rather stringent conditions that the effective random forces have to satisfy, as a result of the structure of the original system. For instance, suppose that the deterministic part of the dynamics is first expressed in terms of the radial and phase variables r, g, through

x =

r cos g,, y

=

r sin q, (Graham, 1982). The resulting equations would be

If the effect of multiplicative noise was modelled by adding phenomenologically a random part to the bifurcation parameter in equations (6) one would clearly miss some of the parameter dependencies and even some of the terms built in (6) and therefore, ultimately, the specificity of the system that one was supposed to study and the very possibility to make quantitative predictions. It is worth noting, in this respect, that the approximation of neglecting non-resonant terms in (5) is intimately related to the fact that the coupling between system and noise is linear.

Had the coefficient of the cubic term of (1) been a fluctuating quantity, additional terms should have been incorporated in the deterministic part of the normal form (equivalent, presumably, to

a

nonlinearity of fifth order) in order to match the smallness of the stochastic part.

3. Stationary probability density and critical behaviour

In order to determine analytically the stationary probability distribution gen- erated by equations (6) we switch to a Fokker-Planck description, within which we can carry out nonlinear transformations of coordinates using the rules of ordinary calculus. We first adopt the Ito interpretation (no spurious drift in the (x, y)-representation). Applying the method given by Baras et al. (1982) we arrive, after a lengthy asymptotic calculation (B << 1, q << 1, Blq finite) at the

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C. NICOLIS A N D G . NICOLIS

following form of the leading part of the stationary probability density in the polar coordinates

x =

r cos q, y

=

r sin q :

1

Notice the dependence of Ps on the frequency

coo

of the unforced system. This important feature is due to the coupling between system and noise as described by (5). It would never arise were noise introduced phenomenologically through the bifurcation parameter B.

The extrema of distribution (9b) are easily found to be located at r

=

0 and at the non-trivial value

For

q =

0 we recover the radius of the deterministic limit cycle in the vicinity of the Hopf bifurcation point,

=

0. For q # O we see that the appearance of a real-valued non-trivial extremum is shifted in parameter space in a stabilizing direction. We have verified that if the term in

z *

in the coefficient of

F ( t )

in (5) is discarded one obtains a completely different result from (lo), in which the frequency

wo

appears explicitly.

Suppose now that the Stratonovich interpretation is adopted. Starting again from (6) and following the same procedure as above we find the following stationary probability distribution:

which, once again, displays a dependence on the frequency

wO.

The non-trivial extremum of this distribution is now located at

As in (10) the transition to a limit cycle is shifted in a stabilizing direction, although the amount of shift is different. Still, in both (10) and (12) the parameters appear in an identical fashion.

Since the most probable value is not always accessible easily to experiment one may wonder whether there exist other quantities whose behaviour reflects the effect of noise in a more transparent way. As is well known, for multiplicative stochastic processes critical behaviour in the form of a divergent variance or of a critical slowing down does not occur in general. However, there exist other criteria which can be attached to critical behaviour. For instance, if the distributions (9) or (11) become unnormalizable, the system will no longer admit a time-independent distribution. More to the point, even if the distribution remains normalizable, the averages of such quantities as r-" will become singular for some n. The lowest value for which this will happen is n

=

1. One then finds that

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where

for Ito interpretation, and

for Stratonovich interpretation. Contrary to (10) and (12), the threshold value PC

around which

( r - ' )

shows critical behaviour depends explicitly on the linearized frequency

w,

around the unstable focus of the unforced system. We again emphasize that this result could never be obtained from a phenomenological theory in which noise is simply added in the bifurcation parameter of the radial part of normal form.

4. Discussion

When perturbed by a multiplicative stochastic forcing the Hopf bifurcation appears to be sensitive to the specific structure of the system. It would be interesting to conduct analog-circuit experiments on (1) or (6) and check whether the frequency of the unforced motion does indeed interfere with the critical behaviour described by (13).

Acknowledgements

This work was supported by the EEC under contract number ST 25-0079-1- B(EBD) and by the US Department of Energy under contract number DE-ASOS- 81ER 10947.

References

Baras, F., Malek Mansour, M. and Van den Broeck, C. (1982) Asymptotic properties of coupled nonlinear Langevin equations in the limit of weak noise. 11: transition to a limit cycle. Journal of Statktical Physics. 28, 577-587.

Coullet, P., Elphick, C. and Tirapegui, E. (1985) Normal form of a Hopf bifurcation with noise. Physics Letters A l l l , 277-282.

Gardiner, C. (1983) Handbook of Stochastic Methods (Springer, Berlin).

Graham, R. (1982) Hopf bifurcation with fluctuating control parameter. Physical Review A25, 3234-3258.

Guckenheimer, J. and Holmes, Ph. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin).

Horsthemke, W. and Lefever, R. (1984) Noise-Induced Transitions (Springer, Berlin).

Nicolis, C. (1984) Self-oscillations and predictability in climate dynamics. Tellus A%, 1-10.

Saltzman, B., Sutera, A. and Hansen, A. R. (1982) A possible marine mechanism for internally generated long-period climate cycles. Journal of the Atmospheric Sciences 39, 2634-2637.

Suzuki, M., Kaneko, K. and Sasagawa, F. (1981) Phase transition and slowing down in nonequilibrium stochastic processes. Progress of Theoretical Physics 65, 828-849.

(Received August 1986)

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