Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections

We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available.Comment: 18 pages, 10 figure

Finding NHIM in 2 and 3 degrees-of-freedom with H\'enon-Heiles type potential

We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally hyperbolic invariant manifolds and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.Comment: 15 pages, 6 figures. This manuscript is better served as dessert to the main course: arXiv:1903.1026

Fast computation and characterization of forced response surfaces via spectral submanifolds and parameter continuation

For mechanical systems subject to periodic excitation, forced response curves (FRCs) depict the relationship between the amplitude of the periodic response and the forcing frequency. For nonlinear systems, this functional relationship is different for different forcing amplitudes. Forced response surfaces (FRSs), which relate the response amplitude to both forcing frequency and forcing amplitude, are then required in such settings. Yet, FRSs have been rarely computed in the literature due to the higher numerical effort they require. Here, we use spectral submanifolds (SSMs) to construct reduced-order models (ROMs) for high-dimensional mechanical systems and then use multidimensional manifold continuation of fixed points of the SSM-based ROMs to efficiently extract the FRSs. Ridges and trenches in an FRS characterize the main features of the forced response. We show how to extract these ridges and trenches directly without computing the FRS via reduced optimization problems on the ROMs. We demonstrate the effectiveness and efficiency of the proposed approach by calculating the FRSs and their ridges and trenches for a plate with a 1:1 internal resonance and for a shallow shell with a 1:2 internal resonance

Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behaviour, including period bubbling and period adding or Farey sequences. The complex bifurcation sequences, characterized by Mixed Mode Oscillations, exhibit partial features of Shilnikov and Gavrilov-Shilnikov scenario. Utilizing the fact that the model has disparate time scales of dynamics, we explain the origin of the relaxation oscillations using the geometrical structure of the bent-slow manifold. Based on a local analysis, we calculate the maximum number of small amplitude oscillations, $s$, in the periodic orbit of $L^s$ type, for a given value of the control parameter. This further leads to a scaling relation for the small amplitude oscillations. The incomplete approach to homoclinicity is shown to be a result of the finite rate of `softening' of the eigen values of the saddle focus fixed point. The latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back-to-back Hopf bifurcation.Comment: 14 Figures(Postscript); To Appear in Physica D : Nonlinear Phenomen

Nonlinear oscillatory mixing in the generalized Landau scenario

We present a set of phase-space portraits illustrating the extraordinary oscillatory possibilities of the dynamical systems through the so-called generalized Landau scenario. In its simplest form the scenario develops in N dimensions around a saddle-node pair of fixed points experiencing successive Hopf bifurcations up to exhausting their stable manifolds and generating N-1 different limit cycles. The oscillation modes associated with these cycles extend over a wide phase-space region by mixing ones within the others and by affecting both the transient trajectories and the periodic orbits themselves. A mathematical theory covering the mode-mixing mechanisms is lacking, and our aim is to provide an overview of their main qualitative features in order to stimulate research on it.Peer ReviewedPostprint (author's final draft