A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorous...

This Rapid Communication presents an analytical study of the bouncing of a completely inelastic ball on a vertically vibrated plate. The interplay of saddle-node and period-doubling bifurcations leads to an intricate structure of the bifurcation diagram with uncommon properties, such as an infinity of bifurcation cascades in a finite range of the control parameter Gamma. A pseudochaotic behavio...

The problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type is addressed. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a pe...

We present experimental evidence of coexisting periodic attractors in a semiconductor laser subject to external optical injection. The coexisting attractors appear after the semiconductor laser has undergone a Hopf bifurcation from the locked steady state. We consider the single-mode rate equations and derive a third-order differential equation for the phase of the laser field. We then analyse ...

This paper is concerned with chaos of a family of logistic maps. It is first proved that a regular and nondegenerate snap-back repeller implies chaos in the sense of both Devaney and Li-Yorke for a map in a metric space. Based on this result, it is shown that the logistic system is chaotic in the sense of both Devaney and Li-Yorke, and has uniformly positive Lyapunov exponents in an invariant s...

This paper illustrates transcritical and Hopf bifurcations in a realistic ac/dc power system model. These bifurcations are thoroughly analyzed based on bifurcation theory to understand and characterize their e ect on the dynamic behavior of the system, particularly on voltage stability. A new technique to trace transcritical bifurcations diagrams is proposed, based on the generic characteristic...

The parametric representation method has been shown to be a useful geometric tool to study tangent and Hopf bifurcations in chemical systems. As an example, we investigate the behaviour of a two-dimensional system of differential equations. This general model includes some known catalytic oxidation models as special cases. The general system can have one or three stationary solutions depending ...

In mechanical applications it is essential that unexpected dynamics are avoided. The industry wants to build reliable machines that are not sensitive to initial conditions. Therefore, a simple method has been developed to extract all sets of stable bifurcation diagrams. The method gives a designer a good overview of possible dynamics and thereby the possibility to select a safe operating region...

An elastic membrane subject to an external pressure and electrostatic force is considered. The resulting elliptic problem is analyzed for the strip and disk geometry. The structure of the solution set is explored with the aid of phase plane techniques, perturbation methods, and bifurcation analysis. In each case, the stability of the solution is determined using linear stability theory with the...

We investigate globally coupled stochastic three-state oscillators, which we consider as general models of stochastic excitable systems. We compare two situations:in the first case the transitions between the three states of each unit 1 → 2 → 3 → 1 are determined by Poissonian waiting time distributions. In the second case only transition 1 → 2 is Poissonian whereas the others are deterministic...