Narrow Results

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

Normally, conservative systems do not have attractors. However, in a system with escapes, the infinity acts as an attractor. Furthermore, attractors may appear as singularities at a finite distance. We consider the basins of escape in a particular Hamiltonian system with escapes and the rates of escape for various values of the parameters. Then we consider the basins of attraction of a system of two fixed black holes, with particular emphasis on the asymptotic curves of its unstable periodic orbits.

The discretization of ordinary differential equations is investigated for the case of the conservative Hénon–Heiles system. Starting from a discrete Hamiltonian function, which is invariant under time reversal, discrete equations of motion are analytically obtained using three different discretization schemes recently proposed and investigated in the literature. In the case where the discretization scheme successfully provide discrete systems in which the trace of the Jacobian matrix corresponding to the property required by a conservative system is preserved, it is shown that they are not necessarily invariant to time reversal. Such models are however quite robust when the time step is increased. For the schemes where the trace of Jacobian matrix does not match the condition required by conservative systems, it is shown that energy conservation is not achieved and the original dynamics is lost. Steps toward the solution to this problem are given.

In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2-element circuits, but in addition, it displays by construction, two n-element circuits which are both positive versus one positive and one negative, depending on the parity (even or odd) of the dimension n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever be the dimension, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (that we call for short "A1_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2_nD". A2_3D behaves essentially as A1_3D, in agreement with the fact that the signs of the circuits remain identical. A2_4D, as well as other arabesque systems with even dimension, has two positive n-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.

Lateral-torsional buckling of elastic structures under combined loading is considered in this paper. Closed-form solutions are available for a specific range of loading. More generally, approximation of the buckling curve (limit of the stable domain in the loading parameters space) is investigated from the stationary property of the Rayleigh's quotient. The approximation is then compared to a numerical approach, namely the iterative method of Vianello and Stodola. Closed-form solutions give upper bounds with relative error less than 0.2%. It is shown that the stable domain in the loading parameters space is convex. The Papkovitch–Schaefer theorem is extended from previous theoretical works of Plaut,¹² despite the nonlinear dependence of the equilibrium equations on the loading parameters for the one-dimensional system. The boundary of the stable domain is clearly nonlinear, but this nonlinearity is weak. It is shown that Dunkerley's lower bound is relevant for the two structural cases considered, and the maximum relative error induced by such a lower bound is lower than 2%.

In this note we show how very general continuation results can be used to describe (1) families of periodic orbits in conservative systems, and (2) families of symmetric and doubly-symmetric orbits in reversible conservative systems. We describe a general approach which repairs the lack of submersivity for such problems due to the presence of first integrals. Our results can in particular be applied in the Hamiltonian context; we very briefly describe how they can be used for the continuation of choreographies in the N-body problem.