Bifurcation diagram

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

Traveling gravity water waves in two and three dimensions

Abstract: This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.

Bifurcation of switched nonlinear dynamical systems

Abstract: This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincare mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we analyze the behavior of a two-dimensional nonlinear autonomous system whose state space is constrained on two half planes concerned with state dependent switching characteristics. From investigation of bifurcation diagrams, we conclude that the tangent and global bifurcations play an important role for generating various periodic solutions and chaos. Some theoretical results are confirmed by laboratory experiments.

A numerical analysis of chaos in the double pendulum

Abstract: We analyse the double pendulum system numerically, using a modified mid-point integrator. Poincare sections and bifurcation diagrams are constructed for certain, characteristic values of energy. The largest Lyapunov characteristic exponents are also calculated. All three methods confirm the passing of the system from the regular low-energy limit into chaos as energy is increased.

Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling

Abstract: We consider a ring of identical elements with time delayed, nearest-neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The bifurcation and stability of nontrivial asynchronous oscillations from the trivial solution are analysed using equivariant bifurcation theory and centre manifold construction.