Exponentially small splitting of invariant manifolds of parabolic points

We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system. We suppose that the origin is a parabolic xed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connexion associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincaré-Melnikov function

Normal Forms and Sternberg Conjugation Theorems for Infinite Dimensional Coupled Map Lattices

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same spatial decay. In the presence of resonances the conjugations are to a polynomial normal form that also has decaying properties

Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation

We consider a one parameter family of 2-DOF Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation. We briefly review the well-established normal form theory in this case. Then we focus on the invariant manifolds when there are homoclinic orbits to the complex-saddle equilibrium point, and we study the behavior of the splitting of the 2D invariant manifolds. The symmetries of the normal form are used to reduce the dynamics around the invariant manifolds to the dynamics of a family of area-preserving near-identity Poincaré maps that can be extended analytically to a suitable neighborhood of the separatrices. This allows, in particular, to use well-known results for area-preserving maps and derive an explicit upper bound of the splitting of separatrices for the Poincaré map. We illustrate the results in a concrete example. Different Poincaré sections are used to visualize the dynamics near the 2D invariant manifolds. Last section deals with the derivation of a separatrix map to study the chaotic dynamics near the 2D invariant manifolds

Whiskered Parabolic Tori in the Planar (n+1)-Body Problem

Abstract: The planar $(n+1)$-body problem models the motion of $n+1$ bodies in the plane under their mutual Newtonian gravitational attraction forces. When $n \geq 3$, the question about final motions, that is, what are the possible limit motions in the planar $(n+1)$-body problem when $t \rightarrow \infty$, ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar $(n+1)$-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocity while the rest perform a bounded motion. These solutions are related to whiskered parabolic tori at infinity, that is, parabolic tori with stable and unstable invariant manifolds which lie at infinity. These parabolic tori appear in cylinders which can be considered 'normally parabolic'. The existence of these whiskered parabolic tori is a consequence of a general theorem on parabolic tori developed in this paper. Another application of our theorem is a conjugation result for a class of skew product maps with a parabolic torus with its normal form generalizing results of Takens (Ann Inst Fourier 23(2):163-195, 1973), and Voronin (Funktsional Anal i Prilozhen 15(1):1-17, 96, 1981)

Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions.

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper [BFM] these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided

Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters.

Abstract. In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable approximate solution of a functional equation. In the case of parabolic points, if the manifolds have dimension two or higher, in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in [BFM]. Assuming a sufficiently good approximation is found, here we provide an "a posteriori" result which gives a true invariant manifold close to the approximated one. In the differentiable case, in some cases, there is a loss of regularity. We also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters. Examples are provided. We apply our method to prove that in several situations, namely, related to the parabolic infinity in the elliptic spatial three body problem, these invariant manifolds exist and do have polynomial approximations

On the 'hidden' harmonics associated to best approximants due to quasi-periodicity in splitting phenomena

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics, that could be expected to be dominant in some ranges of the perturbation parameter, actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies

Splitting of the separatrices after a Hamiltonian-Hopf bifurcation under periodic forcing

We consider the effect of a non-autonomous periodic perturbation on a 2-dof autonomous system obtained as a truncation of the Hamiltonian-Hopf normal form. Our analysis focuses on the behaviour of the splitting of invariant 2D stable/unstable manifolds. Due to the interaction of the intrinsic angle and the periodic perturbation the splitting behaves quasi-periodically on two angles. We analyse the different changes of the dominant harmonic in the splitting functions when the unfolding parameter of the bifurcation varies. We describe how the dominant harmonics depend on the quotients of the continuous fraction expansion (CFE) of the periodic forcing frequency. We have considered different frequencies including quadratic irrationals, frequencies having CFE with bounded quotients and frequencies with unbounded quotients. The methodology combines analytical and numeric methods with heuristic estimates of the role of the non-dominant harmonics. The approach is general enough to systematically deal with all these frequency types. Together, this allows us to get a detailed description of the asymptotic splitting behaviour for the concrete perturbation considered

Dynamical effects of loss of cooperation in discrete-time hypercycles

Hypercycles' dynamics have been widely investigated in the context of origins of life, especially using time-continuous dynamical models. Different hypercycle architectures jeopardising their stability and persistence have been discussed and investigated, namely the catalytic parasites and the short-circuits. Here we address a different scenario considering RNA-based hypercycles in which cooperation is lost and catalysis shifts to density-dependent degradation processes due to the acquisition of cleaving activity by one hypercycle species. That is, we study the dynamical changes introduced by a functional shift. To do so we use a discrete-time model that can be approached to the time continuous limit by means of a temporal discretisation parameter, labelled $C$. We explore dynamical changes tied to the loss of cooperation in two-, three-, and fourmember hypercycles in this discrete-time setting. With cooperation, the all-species coexistence in two- and three-member hypercycles is governed by an internal stable fixed point. When one species shifts to directed degradation, a transcritical bifurcation takes place and the other hypercycle members go to extinction. The asymptotic dynamics of the four-member system is governed by an invariant curve in its cooperative regime. For this system, we have identified a simultaneous degenerate transcritical-NeimarkSacker bifurcation as cooperation switches to directed degradation. After these bifurcations, as we found for the other systems, all the cooperative species except the one performing degradation become extinct. Finally, we also found that the observed bifurcations and asymptotic dynamical behaviours are independent of $C$. Our results can help in understanding the impact of changes in ecological interactions (i.e., functional shifts) in multi-species systems and to determine the nature of the transitions tied to coextinctions and out-competition processes in both ecosystems and RNA-based systems

Habitat loss causes long extinction transients in small trophic chains

Transients in ecology are extremely important since they determine how equilibria are approached. The debate on the dynamic stability of ecosystems has been largely focused on equilibrium states. However, since ecosystems are constantly changing due to climate conditions or to perturbations driven by the climate crisis or anthropogenic actions (habitat destruction, deforestation, or defaunation), it is important to study how dynamics can proceed till equilibria. This article investigates the dynamics and transient phenomena in small food chains using mathematical models. We are interested in the impact of habitat loss in ecosystems with vegetation undergoing facilitation. We provide a dynamical study of a small food chain system given by three trophic levels: primary producers, i.e., vegetation, herbivores, and predators. Our models reveal how habitat loss pushes vegetation towards tipping points, how the presence of herbivores in small habitats could promote ecosystem's extinction (ecological meltdown), or how the loss of predators produce a cascade effect (trophic downgrading). Mathematically, these systems exhibit many of the possible local bifurcations: saddle-node, transcritical, Andronov-Hopf, together with a global bifurcation given by a heteroclinic bifurcation. The associated transients are discussed, from the ghost dynamics to the critical slowing down tied to the local and global bifurcations. Our work highlights how the increase of ecological complexity (trophic levels) can imply more complex transitions. This article shows how the pernicious effects of perturbations (i.e., habitat loss or hunting pressure) on ecosystems could not be immediate, producing extinction delays. These theoretical results suggest the possibility that some ecosystems could be currently trapped into the (extinction) ghost of their stable past