Bifurcation diagram

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

Anti-control of Hopf bifurcations

Abstract: Bifurcation control generally means to design a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some desirable dynamical behaviors. A typical objective is to delay and/or stabilize an existing bifurcation. In this paper, we consider the problem of anti-controlling bifurcations, that is, a certain bifurcation is created at a desired location with preferred properties by appropriate control. Washout-filter-aided dynamic feedback control laws are developed for the creation of Hopf bifurcations. As Hopf bifurcations give rise to limit cycles, anti-control of Hopf bifurcations suggests a new approach for designing limit cycles with specified oscillatory behaviors into a system via feedback control when such dynamical behaviors are desirable,.

Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback.

Abstract: The bifurcation diagram of a single-mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods. For this, we show how to cope with the special symmetry properties of the equations. As the feedback strength is increased, branches of modes and antimodes appear, and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions (bifurcation bridges). Such connections seem generically present as new pairs of modes and antimodes appear. We subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase. Our results extend and confirm existing results and hypotheses reported in the literature. For large values of the linewidth enhancement factor (alpha=5-6), bridges break through homoclinic orbits. Changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes, allowing different types of connections between modes.

Finite Element Modal Formulation for Hypersonic Panel Flutter Analysis with Thermal Effects

Abstract: A finite element time-domain modal formulation for analyzing nonlinear flutter of panels subjected to hypersonic airflow has been developed. Von Karman large deflection plate theory is used for description of the structural nonlinearity, and third-order piston theory is employed to consider the aerodynamic nonlinearity. The thermal loadings of uniformly distributed surface temperatures and temperature gradients through the panel thickness are considered. By the application of the modal truncation technique, the number of governing equations of motion is reduced dramatically so that the computational costs are reduced significantly. All possible types of panel behavior, including flat and stable, buckled but dynamically stable, limit cycle oscillation (LCO), periodic motion, and chaotic motion were observed. Examples of the applications of the proposed methodology were flutter responses of isotropic and composite panels. Special emphasis was placed on the boundary between LCO and chaos and on the route to chaos. Time history, phase plane plot, Poincare map, bifurcation diagram, and Lyapunov exponent are employed in the chaos study. It is found that at low or moderately high dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos, whereas, at high dynamic pressures, the route generally involves bursts of chaos and rejuvenations of periodic motions.

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

Abstract: This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.