Inspired by the discovery of chaotic behaviour determined by simple differential equations, applied mathematicians have adopted in the last 20 years a strategy familiar to artists: instead of trying to include every detail, settle for a depiction that captures the essentials as simply as possible -- a cartoon rather than a photographic reproduction. Bifurcation theory (the classification of these cartoons) has proved to be immensely powerful tool for gaining a qualitative understanding of complicated dynamics.
Several issues need to be addressed before low-dimensional differential equations can be deemed to have predictive power. One is the influence of noise -- an element of randomness added to a deterministic rule. Any differential equation modelling a physical system ignores some influences. For example, low order models in fluid mechanics are often derived by ignoring higher Fourier modes. Nevertheless, if a model is well chosen, one can hope that the ignored influences will be small, rapidly varying, and have mean zero. In these circumstances, it makes sense to call them noise and to model them using a stochastic process.
The effect of noise on dynamics in continuous time is too seldom considered; the appropriate tool for doing so is stochastic calculus , the calculus of continuous but irregular paths driven by random forces. Since the development of Ito 's integral, this calculus has developed as part of probability theory. Paths, or trajectories, of stochastic processes can be decomposed into deterministic and random increments. No matter how short a time interval is chosen, the random increment in the interval cannot be predicted from knowledge of previous increments. Therefore an SDE describes not just one path but an ensemble of paths. (The mean value of a quantity y, denoted < y >, refers to to an average over the ensemble of paths.)
It is not possible to base stochastic calculus on the concept of a derivative because stochastic paths are typically not differentiable. Instead, paths are characterised by their incremental mean and mean square. Having to abandon the derivative and consider squares of infinitesimals seems counter-intuitive, but only because our intuition is improperly trained. As a rule to be implemented on a computer, there is no insurmountable difficulty. At each timestep, a random and a deterministic increment are added. The deterministic one is the same as it would be were there no noise; the random one is a Gaussian random variable with variance proportional to the timestep. In this way, an approximation is generated to the value taken by one path of a stochastic process at a set of times. Means can be evaluated by repeating the procedure numerous times.
Why use an SDE instead of the more traditional Fokker-Planck equation? I mention three reasons. The first, elegance, is subjective. The second is that the numerical methods used for SDEs are natural extensions of their deterministic counterparts, whereas a Fokker-Planck equation is already a partial differential equation (and a functional Fokker-Planck equation is harder still). Numerics for stochastic partial differential equations are also little more difficult than for their deterministic counterparts. In this case a computer program generates a succession of configurations that approximate the values taken by one realisation of a stochastic process (formally taking values in an infinite-dimensional space) at a discrete set of times. The third reason is that SDEs, dealing with one trajectory at a time, fit more naturally into the new culture of applied mathematics: describe a complicated system by a considering trajectories satisfying a simplified differential equation that includes explicitly only the most important variables. Such a differential equation is a more realistic model if noise in included. One of the reasons this is not done more often is that SDEs are not widely understood.
Traditionally in statistical physics, the role of noise is to rattle a system around in a potential derived from a deterministic equation. In the standard static theory of the influence of noise on bifurcations and phase transitions, for example, the main effect of noise with (r.m.s.) magnitude is to smear out the bifurcation in an O () region around the critical point. In this framework, multiplicative noise is typically reckoned to have a more interesting effect than additive noise, but noise is needed to produce effects. A more dramatic effect is seen in dynamic bifurcations , when a parameter is slowly increased through a critical point. In this case the effect of noise is a function of the parameter , where is the rate of change of the parameter and the amplitude of the noise. (In most of the phenomena of interest in this thesis, additive noise is more important than multiplicative noise, and the effect coloured noise differs little from that of white noise.)
The term noise, referring to the random part of a differential equation, originates from the study of electrical circuits; the connotation of an unwanted, negative influence lingers. There are situations, however, where noise has a positive, organising effect. There are, for example, are noise-controlled transitions, with a critical noise level above which the shape of a static probability distribution changes, and stochastic resonance, in which random and periodic inputs can produce a signal-to-noise ratio at the output that is maximised at a non-zero noise level. Slow-fast dynamics is a more dramatic example of organisation by noise. Here all the properties used to describe trajectories (the size and period of orbits, the position of bifurcations, the existence or not of chaotic trajectories, whether trajectories go to to infinity, etc) are controlled by tiny amounts of additive noise. The sensitivity to noise is due to the existence of an invariant manifold -- a subset of the phase space that a trajectory cannot leave. For long periods, the trajectory is close to an invariant manifold and a distinction can be made between variables measuring distance in the direction parallel to the manifold (driving variables), and those measuring distance from the manifold (slave variables). These slow phases are occasionally interrupted by short-lived `revolts' of the slave variables (fast phases).
Analysis of slow-fast dynamics exploits the separation into two phases, and is conveniently captured by a map. Those familiar with the analysis of homoclinic chaos or of relaxation oscillations will recognise some parallels. The most important effect on the map of adding small-amplitude noise to the differential equation is deterministic and ; it is not equivalent to adding noise to the deterministic map. In the analysis of the slow phase, when trajectories are close to an invariant manifold, I make use of SDEs. This leads to an accurate analytical expression for the map.
In a few years, it might be possible to write a summary of stochastic partial differential equations (SPDEs) in the same spirit. For now, however, the theory of SPDEs is not sufficiently complete. In Chapter 3, I introduce some ideas from the current theory of SPDEs and move quickly to worked examples of numerical methods for adding space-time white noise to parabolic partial differential equations in one and two space dimensions.
In Chapter 4 I analyse the effect of noise on the delay phenomenon in dynamic bifurcations: the replacement of the smooth transition between states, expected from a na\"\i ve interpretation of bifurcation theory, by a delay followed by a sudden jump. If , the rate of change of the parameter, is sufficiently small, the characteristic delay is , where is the size of additive perturbations.
Chapter 5 contains the first analysis of the phenomenon of noise and dynamic bifurcations in systems described by partial differential equations. The correlation length that emerges from the noise during the slow sweep through the critical point is frozen in by the nonlinearity to form domains with characteristic size . I solve SPDEs for the Ginzburg-Landau and Swift-Hohenberg models in one and two space dimensions. In Chapter 6 I consider noise-controlled dynamics in a set of ordinary differential equations proposed as a model of a shear instability in convection. Analysing separately the (noise-controlled) slow phase and (more strongly nonlinear) fast phase, I derive a map which concisely captures the dynamics and explicitly includes the effect of noise. The most important parameter is , where is the noise level and ) is the inverse of the (slow) timescale for the dynamics near the invariant manifolds. I also consider a model set of nonlinear partial differential equations in one space dimension. Noise-controlled slow-fast dynamics, similar to those seen in the ordinary differential equations, are found when the spatial domain is small.
In Chapter 7 I consider noise-controlled dynamics in a model for the resonant interaction of wave modes. This was one of the first examples reported of the dramatic simplifying effect of noise: in some parameter ranges, for example, the trajectories are chaotic without noise and almost periodic with noise (the size of successive loops varies according to a distribution with standard deviation proportional to ).
In the final Chapter I discuss a normal form for an unfolding near a codimension-2 bifurcation, appropriate as a description of the nonresonant interaction of modes in systems with O(2) symmetry. Well behaved orbits are produced by adding noise in parameter ranges where the noiseless equations have no stable solution. Thus noise can be viewed as an alternative to the commonly-added higher order terms.
Throughout this thesis, I compare my analysis with results obtained from numerical simulation of stochastic trajectories.
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