Branches Of Periodic Orbits Research Articles

Nonlinear modes of large-amplitude ship motion with roll-pitch coupling

This research pertains to the dynamics of a ship taking into account the nonlinear coupling of roll and pitch degrees of freedom. Such nonlinearity has critical implications for large-amplitude ship motion, and can thus be useful in understanding conditions leading to capsize. The nonlinear normal modes of the conservative system are determined numerically and are displayed in a frequency-energy plot, which clarifies the bifurcations that connect the various branches of periodic orbits. Numerical simulations show that, although the solutions on most branches result in capsize if their energy is beyond a common critical value, a few branches contain stable solutions at higher energies that do not lead to capsize, suggesting possible methods of capsize mitigation. We study a class of solutions analytically using a complexification-averaging technique. Finally, we run a simulation in the case of light damping in roll with high-energy initial conditions, illustrating transitions between branches that occur in the direction of decreasing energy on the frequency-energy plot.

Nonlinear modes of large-amplitude ship motion with roll-pitch coupling

This research pertains to the dynamics of a ship taking into account the nonlinear coupling of roll and pitch degrees of freedom. Such nonlinearity has critical implications for large-amplitude ship motion, and can thus be useful in understanding conditions leading to capsize. The nonlinear normal modes of the conservative system are determined numerically and are displayed in a frequency-energy plot, which clarifies the bifurcations that connect the various branches of periodic orbits. Numerical simulations show that, although the solutions on most branches result in capsize if their energy is beyond a common critical value, a few branches contain stable solutions at higher energies that do not lead to capsize, suggesting possible methods of capsize mitigation. We study a class of solutions analytically using a complexification-averaging technique. Finally, we run a simulation in the case of light damping in roll with high-energy initial conditions, illustrating transitions between branches that occur in the direction of decreasing energy on the frequency-energy plot.

Numerical Computation of Grazing Bifurcation for Periodic Orbit

In this paper, a numerical algorithm for computing grazing bifurcation of periodic orbit in autonomous planar systems is derived and analyzed. We propose an analytical condition which to ensure the orbit quadratically intersects the switching manifold. Based upon this, we take into a new phase condition which ensure there is a branch of periodic orbits. The nondegenerate condition with respect to its bifurcation parameter is presented to ensure the defining equation well posed. A numerical simulation is carried out for the bifurcation diagram of the van der pol system, which agrees with analytical results very well.

From steady solutions to chaotic flows in a Rayleigh–Bénard problem at moderate Rayleigh numbers

The dynamics of a Rayleigh–Bénard convection problem in a cubical cavity at moderate values of the Rayleigh number ( Ra ≤ 10 5 ) and a Prandtl number of Pr = 0.71 (with extensions to Pr = 0.75 and 0.80) was investigated. The cubical cavity was heated from below and had perfectly conducting sidewalls and uniform temperature distributions on the two horizontal walls. A system of ordinary differential equations with a dimension of typically N ≈ 11 000 was obtained when the conservation equations were discretized by means of a Galerkin method. Previous knowledge of the bifurcation diagram of steady solutions, reported in the literature, was used to identify the origin of several branches of periodic orbits that were continued with Ra. Half a dozen of such periodic orbits were found to be stable within narrow ranges of Ra (at most, some 5000 units wide). An attracting two-torus, restricted to a very narrow region of Ra, was also identified. It was found that the instabilization of periodic orbits quite often resulted into the development of complex dynamics such as the creation of homoclinic and heteroclinic orbits. Instances of both types of global bifurcations were analyzed in some detail. One particular instance of chaotic dynamics (a strange attractor) was also identified. Chaotic dynamics has been found at Pr = 0.71 in a flow invariant subspace, which can be interpreted as a fixed-point subspace in terms of equivariant theory; this subspace is not attracting. However, some regions of attracting chaotic dynamics for moderate Rayleigh numbers ( 9 × 10 4 ≤ R a ≤ 10 5 ) were found at values of Pr slightly above 0.71. The role of a particular homoclinic solution found at Pr = 0.71 in the generation of these chaotic regions was analyzed.

Numerical Continuation in a Physical Experiment: Investigation of a Nonlinear Energy Harvester

In this paper, we demonstrate the use of control-based continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. By employing the methodology of Sieber et al. (2008, “Experimental Continuation of Periodic Orbits Through a Fold,” Phys. Rev. Lett., 100(24), p. 244101), a branch of periodic orbits is continued through a saddle-node bifurcation and along the associated branch of unstable periodic orbits using a modified time-delay controller. At each step in the continuation, the pseudo-arclength equation is appended to a set of equations that ensure that the controller is noninvasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.