A coupled partial differential equation model for interaction between mountain pine beetles (MPBs) and lodgepole pine is reviewed. An asymptotic cone-tree' solution is examined and the time-scale of the chemical response discussed. The equations are decoupled making an adiabatic assumption for MPB chemotaxis, and a 'local' projection is made using the leading eigenfunction for the MPB density equation. This projection yields a system of ordinary differential equations for the spatio-temporal response ui a single tree. These equations are analyzed, nnd their behavior compared with observations.

We show by explicit construction that any system described by a Markov process having a stationary density may be put in generalized detailed balance adding some odd parameters. A physical interpretation of the construction is given in terms of the different time-scales involved

We estimate the threshold value of the perturbation strength needed to cause magnetic field-line stochasticity in the peripheral region of a Tokamak zvith an ergodic magnetic limiter, accoriding to a Hamiltonian description for field lines. We model the limiter action as a periodic sequence of impulsive excitations. Two global stochasticity prescriptions are used: Chirikov's overlapping criterion and the renormalization scheme of Escande, Doveil and Benkadda.

We examine the qualitative behaviour of an NPZ (nutrient-phyto- plankton-zooplankton) model for parameter ranges consistent with values used in the literature. The wide range of values partly reflects variations of conditions in different environtments for the plankton, but in many cases is a measure of the difficulties in making observations and consequent uncertainties. We pay particular attention to the bifurcational behaviour of the system, and to the regions of parameter space for which oscillatory behaviour is possible; such oscillatory behaviour has recently been found in both observational data and in more complex ecosystem models. In some regions of parameter space, we also find that multiple attractors occur. Finally, we examine in more detail the behaviour for a range of values of nutrient input.

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurcations, period doubling, period adding, and sub- and superharmonic coexistence of solutions. The solutions are shown both in the phase plane and as Poincare map fixed points under parameter continuation using the package AUTO. Such results illustrate the possibility of the coexistence of 'dangerous', large-amplitude responses at the same point of parameter space as 'safe' solutions. The feasibility of experimental verification of the results is discussed.

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations – both, without temperature feedback and in the presence of temperature feedback – have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded

A new method to investigate asymptotic properties of linear differential equations with strong threshold and switching effects is presented. The method is applied to systems of equations of the form dx/dt = F(x) - yx, where y = constant and the dependence of F on x is mediated by sigmoid functions. Using a special sigmoid function called a logoid, which rises monotonically from zero to one in a narrow interval surrounding the threshold value, exact analytical expressions for the limiting value of all steady points can be found in the limit when the logoid approaches a step function. The limiting values are independent of the shape of the logoid for a large class of logoids. Relations between steady points and limit cycles of the equations with logoids, their step function limit and the corresponding piecewise linear equations are derived. It is found that the approximation of sigmoids by the step function idealization is not always warranted. The results strongly suggest the use of logoids instead of othe...

The parametrically excited pendulum exhibits a wide variety of non-linear behaviour. We consider numerical studies of periodic motion; the simple periodic solutions are also studied analytically using the harmonic balance method. For more complicated motions the problem is tackled by solving algebraic non-linear equations either using the definition of fixed points for Poincare map, or, where convenient, using a Galerkin technique; in both techniques the stability of solutions can also be checked. The cell mapping technique is used to study the attracting basins and to systematically assess the various types of periodic motions.

The stability of small oscillations of non -holonomic systems is discussed. In contrast to Neimark and Fufaev, we have shown here that a non-holonomic system does indeed possess an isolated equilibrium state under certain conditions. We provide conditions under which there exists asymptotic stability of small oscillations of a non-holonomic system about its equilibrium position. The connection between a holonomic and non-holonomic system is also discussed. The results obtained here can be useful for the control of a non-holonomic system