Bifurcation diagram

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

Streamline topologies of two-dimensional peristaltic flow and their bifurcations

Abstract: Streamline patterns and their local and global bifurcations in a two-dimensional planar and axisymmetric peristaltic flow for an incompressible Newtonian fluid have been investigated. An analytical solution for the stream-function is found under a long-wavelength and low-Reynolds number approximation. The problem is solved in a moving coordinate system where a system of nonlinear autonomous differential equations can be established for the particle paths. Local bifurcations and their topological changes are inspected using methods of dynamical systems. Three different flow situations manifest themselves: backward flow, trapping or augmented flow. The transition between backward flow to trapping corresponds to a bifurcation of co-dimension one, in which a non-simple degenerate point changes its stability to form heteroclinic connections between saddle points that enclose recirculating eddies. The transition from trapping to augmented flow is a bifurcation of co-dimension two, in which heteroclinic saddle connections of adjacent waves coalesce below wave troughs. The coalescing of saddle nodes on the longitudinal axis produces a degenerate point with six heteroclinic connections (degenerate saddle). As the parameter is increased, the degenerate saddle bifurcates to saddles nodes which lift off the centerline. These bifurcations are summarized in a global bifurcation diagram. Theoretical results are compared with the experimental data.

Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification.

Abstract: A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Benard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton's method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk toward parallel rolls and the requirement imposed by the boundary conditions that primary bifurcations be toward states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; threefold-symmetric patterns called Mercedes, Mitsubishi, marigold, and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number.

Experimental Verification of Hopf Bifurcation in DC--DC Luo Converter

Abstract: DC-DC converters have been reported as exhibiting a wide range of bifurcations and chaos under certain conditions. This paper analyzes the bifurcations in current-controlled Luo topology operating in continuous conduction mode using continuous-time model. The stability of the system is analyzed by studying the locus of the complex eigenvalues, and the characteristic multipliers locate the onset of Hopf bifurcation. The 1-periodic orbit loses its stability via Hopf bifurcation, and the resulting attractor is a quasi-periodic orbit. This later bifurcates to chaos via border collision bifurcation. A computer simulation using MATLAB/SIMULINK confirms the predicted bifurcations. It has also been inferred from the experimental results that the margin of system stability decreases as the load decreases.

A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control

Abstract: In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincare maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.