Bifurcation diagram

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

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

Abstract: The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Karman equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincare maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.

Bifurcation analysis of reaction-diffusion Schnakenberg model

Abstract: Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling Schnakenberg chemical reaction The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns

Primary and Secondary Bifurcations in the Couette-Taylor Problem

Abstract: We study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry. The difficulty is that the linearized stability analysis leads to multiple eigenvalues for the most simple flows. Only a consideration of the symmetry-group action on the critical eigenvectors allows us to derive and to solve the bifurcation equations. We recover through this analysis the different patterns which are observed in experiments as the Reynolds number is increased: Steady Taylor vortices and bifurcation of either wavy, or twisted vortices from the Taylor vortex flow in the case of co-rotating cylinders; spiral vortices in the case of (strongly) counterrotating cylinders, and ribbon-cells, which have not yet been observed in experiments. Then we show that, under natural assumptions on the loss of stability of these oscillatory flows, the next bifurcation leads to quasi-periodic flows without frequency locking, whose different patterns are studied.

The dynamic complexity of a three species food chain model

Abstract: In this paper, a three-species food chain model is analytically investigated on theories of ecology and using numerical simulation. Bifurcation diagrams are obtained for biologically feasible parameters. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics.