In this paper we study the nonlinear evolution of patterns at near-critical conditions on domains with spatially periodic boundaries; the amplitude of the periodic effect is assumed to be small. By considering a simple one-dimensional model problem we are able to focus on the essence of the periodic effects. We find that the wave number p of the imposed periodicity has a significant influence on the behaviour of the solutions. We have to distinguish between non-resonant p, resonant p and small p. We derive a number of modulation equations which differ structurally from the classical Ginzburg-Landau equation and study the behaviour of the solutions to these equations.

We provide a multiple time scales analysis for the Swift–Hohenberg equation with delayed feedback via Pyragas control, with and without additive noise. An analysis of the pattern formation near onset indicates both the possibility of either standing waves (rolls) or traveling waves via Turing or Turing–Hopf bifurcations, respectively, depending on the product of the strength of the feedback and the length of the delay. The remainder of the paper is focused on Turing bifurcations, where the delay can drive the appearance of an additional time scale, intermediate to the usual slow and fast time scales observed in the modulation of rolls without delay. In the deterministic case, a Ginzburg–Landau-type modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein–Uhlenbeck-type equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy roll-type patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient.

We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model we are able to explain results using analytical expressions and compare these with numerical investigations.

A multiscale approach is used to derive stochastic amplitude and phase dynamics for a canonical noise-sensitive model exhibiting coherence resonance. Explicit expressions for the dependence on noise levels and model type are compared with computational coherence measures.

We develop a multi-scale analysis for stochastic differential equations. Such models are particularly sensitive to noise when the system is near a critical point, such as a Hopf bifurcation, which marks a transition to oscillatory behavior. In particular, we are interested in the case when the combined effects of the noise and the bifurcation amplify oscillations which would decay in the deterministic system. The derivation of reduced equations for the envelope of the oscillations provides an efficient analysis of the dynamics by separating the influence of the noise from the intrinsic oscillations over long time scales.

Manifold Data Mining has developed innovative demographic and household spending pattern databases for six-digit postal codes in Canada. Their collection of information consists of both demographic and expenditure variables which are expressed through thousands of individually tracked factors. This large collection of information about consumer behaviour is typically referred to as a mine. Although very large in practice, for the purposes of this report, the data mine consisted of $m$ individuals and $n$ factors where $m \sim 2000$ and $n \sim 50$ . Ideally, the first algorithm would identify a few factors in the data mine which would differentiate customers in terms of a particular product preference. Then the second algorithm would build on this information by looking for patterns in the data mine which would identify related areas of consumer spending. To test the algorithms two case studies were undertaken. The first study involved differentiating BMW and Honda car owners. The algorithms developed were reasonably successful at both finding questions that differentiate these two populations and identifying common characteristics amongst the groups of respondents. For the second case study it was hoped that the same algorithms could differentiate between consumers of two brands of beer. In this case the first algorithm was not as successful as differentiating between all groups; it showed some distinctions between beer drinkers and non-beer drinkers, but not as clearly defined as in the first case study. The second algorithm was then used successfully to further identify spending patterns once this distinction was made. In this second case study a deeper factor analysis could be used to identify a combination of factors which could be used in the first algorithm.

Explicit expressions valid near expiry are derived for the values and the optimal exercise boundaries of American put and call options on assets with dividends. The results depend sensitively on the ratio of the dividend yield rate D to the interest rate r. For D > r the put boundary near expiry tends parabolically to the value rK/D where K is the strike price, while for D less than or equal to r the boundary tends to K in the parabolic-logarithmic form found for the case D = 0 by Barles et al. (1995) and by Kuske and Keller (1998). For the call, these two behaviors are interchanged: parabolic and tending to rK/D for D < r, as was shown by Wilmott, Dewynne, and Howison (1993), and parabolic-logarithmic and tending to K for D greater than or equal to r. The results are derived twice: once by solving an integral equation, and again by constructing matched asymptotic expansions.

We study the deformation of ail elastic strut oil a nonlinear Winkler foundation subjected to ail axial compressive load P. Using multi-scale analysis and numerical methods we describe the localized., cellular, post-buckled state of the system when P is removed from the critical load P := 2. The solutions, and their modulation frequencies, differ significantly from those predicted by weakly nonlinear analysis very close to P = 2. In particular, when P approaches the Maxwell load P-M, the localized solutions approach a large-amplitude heteroclinic connection between ail unbuckled solution and a periodic solution. An asymptotic description of P-M in terms of the system parameters is given. The agreement between the numerical calculations and the asymptotic approximations is striking.

The rate of convergence to a stable law is determined for the probability density of the normalized sum of n independent identically distributed random variables, as $n\rightarrow \infty$. Methods are given for using these results to fit data to such a law.

The delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise. A new asymptotic approximation for the time-dependent probability density function gives a complete description of the process into the transition region, and is easily interpreted in terms of the noisy dynamics. It is also used to calculate mean transition times. The method is applied to two nonlinear systems with noise: a one-dimensional canonical model for a steady bifurcation and the noisy FitzHugh–Nagumo model.

New modulation equations for hexagonal patterns in reaction–diffusion systems are derived for parameter régimes corresponding to the onset of patterns. These systems include additional nonlinearities which are not present in Rayleigh–Bénard convection or Swift–Hohenberg type models. The dynamics of hexagonal and roll patterns are studied using a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations. The investigation demonstrates instabilities and new phenomena not found in other systems, and is applied to patterns of flame fronts in a certain model of burner stabilized flames.

The optimal exercise boundary near the expiration time is determined for an American put option. It is obtained by using Green's theorem to convert the boundary value problem for the price of the option into an integral equation for the optimal exercise boundary. This integral equation is solved asymptotically for small values of the time to expiration. The leading term in the asymptotic solution is the result of Barles et al. An asymptotic solution for the option price is obtained also.

The probability density for the solution y(n) of a stochastic difference equation is considered. Following Knessl et al, [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium, A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a traveling wave solution as a basic state. We determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability. Specifically, we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that are coupled to an equation for a mean field, generated by the translation of the basic state in the direction of its propagation. The modulation equations are then employed to determine stability boundaries to long wave perturbations for both unidirectional and counterpropagating traveling waves. The stability analysis is delicate because the results depend on the order in which transverse and longitudinal perturbation wavenumbers are taken to zero. For the unidirectional wave we demonstrate that it is sufficient to consider the cases of (i) purely transverse perturbations, (ii) purely longitudinal perturbations, and (iii) longitudinal perturbations with a small transverse component. These yield Eckhaus type, zigzag type, and skew type instabilities, respectively. The latter arise as a specific result of interaction with the mean field. We also consider the degenerate case of very small group velocity, as well as other degenerate cases, which yield several additional instability boundaries. The stability analysis is then extended to the case of counterpropagating traveling waves.

We consider the behaviour of a premixed flame anchored on a flat burner. For Lewis numbers L < L * < 1, stationary spatially periodic solutions corresponding to stationary cellular flames bifurcate from the basic solution which corresponds to a steady planar flame. We study the existence and stability of two-dimensional patterns which correspond to certain imposed symmetries by considering the evolution of N pairs of wave vectors, each of which is separated from the next by angle π/ N . In the neighbourhood of the critical Lewis number L *, we derive evolution equations for the amplitudes corresponding to N = 2, which corresponds to square patterns, and N = 3, which corresponds to triangular or hexagonal patterns. We determine existence and stability results in terms of m ∈(0, 1), the flow rate of the fuel, and K > 2/ e , the scaled heat loss to the burner. Square patterns exist for L < L * and are stable for values of m and K above a stability boundary in the m − K plane, which has a maximum at K = K * ∼ 4.77, so that for K > K * square patterns are stable for all m . The stability of the square patterns does not vary with L . Hexagonal patterns exist for L < L H , where 1 > L H > L *. The size of the stability region increases with decreasing L < L *. For a range of values of L there is bistability, that is, for given parameter values rolls and hexagons are simultaneously stable, each with its own domain of attraction.

We consider the behavior of a premixed flame anchored on a flat burner. For Lewis numbers L > L ∗ > 1 L > {L^*} > 1 , one-dimensional stationary spatially periodic solutions corresponding to stationary one-dimensional cellular flames (rolls) bifurcate from the basic solution which corresponds to a steady planar flame. We derive and analyze an equation for the evolution of the amplitude of the roll solution just beyond the critical Lewis number L ∗ {L^*} . That is, we consider the case of supercritical bifurcation ( L > L ∗ ) \left ( {L > {L^*}} \right ) and determine the ranges of wave numbers of perturbations corresponding to both the Eckhaus instability (to longitudinal perturbations) and the zigzag instability (to transverse perturbations) of the bifurcating solution. We determine these ranges in terms of the flow rate m ∈ ( 0 , 1 ) m \in \left ( 0, 1 \right ) and the scaled heat loss to the burner K > 2 / e K > 2/e . For wave numbers k > 0.25 k > 0.25 we find that the zigzag instability occurs for all allowed values of K K and for m m bounded away from 1 and 0. As k k increases, the range of values of m m and K K for which this instability occurs decreases. For k ≥ 0.4 k \ge 0.4 the zigzag instability no longer occurs for any allowed value of m m and K K . For each value of L L there is a minimum value m = m ∗ ( L ) m = {m_*}\left ( L \right ) above which the Eckhaus instability does not occur. As L L approaches L ∗ , m ∗ ( L ) {L^*}, {m_*}\left ( L \right ) increases.

The stationary Schrödinger equation on a one-dimensional lattice endowed with a random potential is considered. Specifically, the equation studied is $u_{n + 1} + u_{n - 1} = ( E - \varepsilon V_n )u_n $, where$\varepsilon V_n $ is the random potential at site n . When $\varepsilon = 0$, the band of allowed energies is given by$E = 2\cos \pi r,\, | r | < 1$, and only this band is considered A singular perturbation expansion of the stationary probability density $p(x, E, \varepsilon)$ of the random process $X_n = u_n /u_{n - 1} $ is constructed in the limit of weak disorder $( \varepsilon \ll 1)$. The coefficients in the expansion are analytic functions of r for $\varepsilon > 0$. They contain internal layers at rational values of r, which were previously termed "anomalies." The expansion approximates $p( x,E,\varepsilon )$ uniformly for all r inside the band, away from band-center $( r = \tfrac{1}{2} )$ and band-edge $(r = 0)$. It is used to calculate the first term in the expansion of the Lyapunov exponent $\gamma( E,\varepsilon )$, which determines the localization length of the wave function, thus confirming the Thouless formula for $\gamma ( E,\varepsilon )$ inside the band and the Kappus–Wegner formula in band-center. Band-center and band-edge expansions are constructed, which match the in-band limits, allowing a uniform approximation for the Lyapunov exponent in all regions of the energy band.

We develop a two-dimensional adaptive pseudo-spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions. The method is based on introducing two-dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated. The method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel. Spatio-temporal patterns are computed as a parameter μ, related to the activation energy, is increased above a critical value μc. The spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder, together with a standing wave pattern in the direction of the azimuthal angle ψ For small values of μ − μc the solutions exhibit a nearly sinusoidal dependence in both time and ψ As μ is increased further relaxation oscillations in both time and ψ occur. Beyond a critical value of μ stable time-periodic solutions are no longer found and the solution exhibits a quasi-periodic time dependence.

Two coupled lasers exhibiting oscillatory intensities are known to synchronize in phase or out-of-phase and with equal intensities. But a different form of synchronization - called localization - has been discussed recently in the literature of coupled oscillators. Localization means that the two lasers may exhibit different intensities. We show that this phenomenon is possible in a system of two coupled solid state lasers differing only by their detunings. We determine the bifurcation diagram of the localized states and obtain analytical conditions for stable localization.

We apply multi-scale analysis to stochastic delay-dieren tial equations, deriving approx- imate stochastic equations for the amplitudes of oscillatory solutions near critical delays of deterministic systems. Such models are particularly sensitive to noise when the system is near a critical point, which marks a transition to sustained oscillatory behavior in the deter- ministic system. In particular, we are interested in the case when the combined eects of the noise and the proximity to criticality amplify oscillations which would otherwise decay in the deterministic system. The derivation of reduced equations for the envelope of the oscillations provides an ecien t analysis of the dynamics by separating the inuence of the noise from the intrinsic oscillations over long time scales. We focus on two well-known problems: the linear SDDE, and the logistic equation with delay. In addition to the envelope equations, the analysis identies scaling relationships between small noise and the proximity of the bifurcation due to the delay which enhances the resonance of the noise with the intrinsic oscillations of the systems.