Narrow Results

In this work, the existence of curves of homoclinic connections that bi-spiral around a T-point between two saddle-focus equilibria is detected in Chua's equation. That is, the homoclinic curve emerges spiraling from a T-point in a parameter bifurcation plane and ends, by a different spiral, at the same T-point. This new phenomenon is related to the existence of more than one intersection between the two-dimensional manifolds of the involved equilibria at the T-point.

We study perturbations of cubic planforms, proving there exists perturbations with homoclinic cycles between persistent steady states. Our results do not depend on the representation of the symmetry group of the lattice, and are thus quite general.

The problem is studied using group theory rather than direct methods. We use the abstract action of the symmetry group of the perturbation on the group orbit to determine the existence of zero- and one-dimensional flow-invariant subspaces. The residual symmetry of the perturbation constrains the flows on these subspaces and, in certain cases, homoclinic cycles are guaranteed to exist.

Cubic planforms are physically interesting due to their relevance to certain physical systems. Applications to reaction–diffusion systems, nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction are discussed.

A saddle-center transition of eigenvalues in the linearization about Hamiltonian equilibria, and the attendant planar homoclinic bifurcation, is one of the simplest and most well-known bifurcations in dynamical systems theory. It is therefore surprising that anything new can be said about this bifurcation. In this tutorial, the classical view of this bifurcation is reviewed and the lifting of the planar system to four dimensions gives a new view. The principal practical outcome is a new formula for the nonlinear coefficient in the normal form which generates the homoclinic orbit. The new formula is based on the intrinsic curvature of the lifted path of equilibria.

The Earth–Moon libration points are of interest for future missions and have been proposed for both storage of propellant and supplies for lunar missions and as locations to establish space-based facilities for human missions. Thus, further development of an available transport network in the vicinity of the Moon is valuable. In this investigation, a methodology to search for transfers between periodic lunar libration point orbits is developed, and a catalog of these transfers is established, assuming the dynamics associated with the Earth–Moon circular restricted three-body problem. Maneuver-free transfers, i.e. heteroclinic and homoclinic connections, are considered, as well as transfers that require relatively small levels of $\mathrm{\Delta }v$. Considering the evolution of Earth–Moon transfers as the mass parameter is reduced, a relationship emerges between the available transfers in the Earth–Moon system and maneuver-free transfers that exist within the Hill three-body problem. The correlation between transfers in these systems is examined and offers insight into the existence of solutions within the catalog. To demonstrate the persistence of the catalog transfers in a higher-fidelity model, several solutions are transitioned to a Sun–Earth–Moon ephemeris model with the inclusion of solar radiation pressure and lunar gravity harmonics. The defining characteristics are preserved in the high-fidelity model, validating both the techniques employed for this investigation and the solutions computed within the catalog.

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex.14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal.33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.