Global Bifurcation Techniques Research Articles

Existence of positive solutions of a fourth-order boundary value problem

We consider the fourth-order boundary value problem u ″ ″ = f ( t , u , u ″ ) , 0 < t < 1 u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 where f( t, u, p) = au − b p + ∘(∣( u, p)∣) near (0, 0), and f( t, u, p) = cu − d p + ∘(∣( u, p)∣) near ∞. We give conditions on the constants a, b, c, d that guarantee the existence of positive solutions. The proof of our main result is based upon global bifurcation techniques.

Size Structured Competition Model with Periodic Coefficients

A model of Diekmann et al. [14] for two size-structured populations competing for a single resource is extended in this paper to allow for periodic variation in parameter values (having a common period). The hyperbolic partial differential equation model is reduced to systems of ordinary differential equations describing the dynamic of two size-structured species competing for a single unstructured resource. The existence of nontrivial periodic solutions of those systems is considered. Under fairly general conditions, global bifurcation techniques as used by Gushing are used to show the existence of a continuum of solutions that bifurcate from a noncritical periodic solution of the reduced single species system. The positivity and the stability of the bifurcating branch solutions are studied. The stability is shown to depend on the stability of the solutions of the reduced system and the direction of bifurcation. The results and techniques can be generalized to the case of n, n ≥ 1, size-structured species competing for a single resource. It is also shown that the average size of individuals of each species is governed by a nonautonomous logistic equation whose solution under fairly general conditions approaches the unique periodic solution of the classical periodic logistic equation.

Ponies on a merry-go-round in large arrays of Josephson junctions

Numerical simulation of periodic solutions in large arrays of Josephson junctions indicates the existence of periodic solutions where each junction oscillates with the same waveform, but with equal phase lags. These solutions are called ponies on a merry-go-round or POMs for short. The authors prove the existence of POMs in the equations modelling large arrays of Josephson junctions by using global bifurcation techniques. The basic idea is to view the period of the solution and the phase lag as independent parameters and to prove, using a priori estimates, that the synchronous solution (with phase lag set to zero) can be continued to a solution with phase lag equal to (1/N)th of the period, a POM.

On the Global Bifurcation Characteristics of Adaptive Systems

This paper discusses the bifurcation behavior and nonlinear dynamics of some representative model-based adaptive control schemes. It focuses on situations where the desired set-point is locally but not globally attracting. In such situations, undesired attractors, attributable to plant/mood mismatch and/or unmodeled dynamics, are generated by the control scheme. When the boundaries of the basins of attraction of these attractors are computed, a picture of the global phase-space structure of the system emerges. This provides a quantitative measure of the robustness of the control scheme to both state and model perturbations. The dependence of the dynamics on the parameters of the system are studied via numerical (both local and global) bifurcation techniques to reveal which controller designs globally stabilize the set-point . This information can be used to suggest modifications to the control schemes that eliminate the multistability problem.

Periodic Two-Predator, One-Prey Interactions and the Time Sharing of a Resource Niche

A competition model involving two competing predator species and a single renewable resource prey species is studied under the assumption that the system parameters are periodic in time. It is shown by means of global bifurcation techniques that a continuum of positive periodic solutions exists as a function of a selected (averaged) parameter and that the stability of these solutions (at least locally near bifurcation) depends on the direction of bifurcation. In the special autonomous case of constant parameters the bifurcation is vertical and the spectrum of the continuum is a discrete point. This autonomous case supports the principle of competitive exclusion in that coexistence of the predators on the single resource prey is possible (in the sense that the system equilibrates) only on a parameter set of measure zero. In the more general case of periodic coefficients, however, it is shown that the spectrum can be an interval of positive length provided the predator parameter oscillations are out-of-phas...