Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.

Global Bifurcation Diagrams of Steady States of Systems of PDEs via Rigorous Numerics: a 3-Component Reaction-Diffusion System

Abstract: In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol. 53(4):617---641, 2006] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comput. 79(271):1565---1584, 2010], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.

Long-Term Dynamics

Abstract: There are many applications where one is concerned with the long-term behaviour of nonlinear ODEs. It is therefore of great interest to know whether this behaviour is accurately captured when they are solved by numerical methods.

Local Negative Shear and the Formation of Transport Barriers.

Abstract: We present a set of 3D nonlinear equations describing drift-resistive ballooning modes in a torus including the self-consistent modification of the local magnetic shear due to the finite $\ensuremath{\beta}$ shift of the flux surfaces. Simulations using these equations reveal that a bifurcation of the transport occurs when the local magnetic shear on the outside midplane reverses sign. A fully self-consistent bifurcation diagram is calculated which reveals significant hysteresis, i.e., the transport barrier is maintained at lower values of $\ensuremath{\beta}$ than is required for the formation of the barrier, as expected from the experimental observations.

On Tracing an Implicitly Defined Curve by Quasi-Newton Steps and Calculating Bifurcation by Local Perturbations

Abstract: An algorithm is presented which traces an implicitly defined curve ($H(x) = 0 $ for $H:\mathbb{R}^{N + 1} \to \mathbb{R}^N$). The Davidenko-IVP is integrated by a self-correcting predictor-corrector method which does not evaluate the Jacobians of H explicitly. Instead, a quasi-Newton method of C. Broyden [Math. Comp., 19 (1965), pp. 577–593] is used in the corrector phase. It is then shown how the algorithm may be used to locate bifurcation points and trace the bifurcating branches by introducing local perturbations of H in the sense of H. Jurgens, H. O. Peitgen, and D. Saupe [in Analysis and Computation of Fixed Points, S. M. Robinson, ed., Academic Press, New York]. Numerical results are reported on a difficult test problem and on the bifurcation of periodic solutions of a differential delay equation.

Codimension-two bursting analysis in the delayed neural system with external stimulations

Abstract: In the neural system, action potentials play a crucial role in many mechanisms of information communication. The quiescent state, spiking and bursting activities are important biological behaviors with the different neurocomputational properties. In this paper, based on the bifurcation mechanisms involved in the generation of action potentials, an interesting mathematical study of bursting behavior is obtained. The transition between the bursting and quiescence state is investigated,which shows that the time delay must be large enough for bursting behavior to occur in a delayed system. Two types of the codimension-two bifurcation, i.e., Bogdanov–Takens (BT) bifurcation and saddle-node homoclinic (SNH) bifurcation are investigated also. The bifurcation curves of the parameters and the phase portraits for the different regions are shown. The local existence of the homoclinic curve is achieved by using the center manifold reduction and normal form method. For occurrence of a periodic stimulation in the neighborhood of the SNH bifurcation, the system can switch over from an equilibrium state to an oscillatory state either through saddle-node on an invariant circle bifurcation (called circle bifurcation for simplicity) or saddle-node (SN) bifurcation, and back from the oscillatory state to the equilibrium state through the circle or homoclinic bifurcation. Complex bursting phenomena are displayed for the different values of delay couplings and stimulation intensities. Some types of bursting behaviors, such as Circle/Circle (Type II or parabolic bursting), Circle/Homoclinic, SN/Circle (triangular bursting), SN/Homoclinic (Type I or square-wave bursting), and Fold/Hopf bursting are obtained in the firing area. The results show that the different burstings are related to the delay coupling and external inputs.