Bifurcations of optimal vector fields in the shallow lake system

The shallow lake problem is a nonconvex production-pollution dynamic optimisation problem whose solution structure depends nonlinearly on the system parameters. We perform a bifurcation analysis to investigate the consequences of varying the relative cost of pollution and the discount rate.

Bifurcations of optimal vector fields in the shallow lake model

The solution structure of the set of optimal solutions of the shallow lake problem, a problem of optimal pollution management, is studied as we vary the values of the system parameters: the natural resilience, the relative importance of the resource for social welfare and the future discount rate. We find parameter values at which qualitative changes occur. Using theoretical results on the bifurcations of the solution structure to infinite horizon optimization problems obtained earlier, we give a fairly complete bifurcation analysis of the shallow lake problem. In particular, we show how the increase of the discount rate affects the parameter regions where an oligotrophic steady state, corresponding to low pollution level, is globally stable or locally stable under optimal dynamics. Asymptotically, an increase of the discount rate can be offset with a proportional increase of the relative social weight of the resource.

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations

Bifurcations to quasi-periodic tori in a two parameter family of vector fields are studied. At criticality, the vector field has an equilibrium point with a zero eigenvalue and a pair of complex conjugate eigenvalues. This situation has been studied by Langford, Iooss, Holmes and Guckenheimer. Here we provide explicitly computed conditions under which the stability of the secondary branch of tori, and whether the flow on them is quasiperiodic, can be determined. The results are applied to "Brusselator" system of reaction diffusion equations

Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure