We consider a Hamiltonian system that models the scattering of helium atoms off a copper surface. The interaction between the He and the Cu atoms is described by a corrugated Morse potential. Using corrugation coefficients values in the potential obtained by fitting to experimental values, we prove that, provided some coefficient of an auxiliary function is different from 0, there are regions of the phase space, corresponding to sufficiently large energy of the incident atom, where the scattering is chaotic. Furthermore, we prove that the system has oscillatory motions.

Replicating systems sustained by error-prone enzymatic amplification can undergo critical transitions between persistence and extinction. In RNA viruses, such transitions are classically governed by mutation rates and fitness landscapes, giving rise to error thresholds and lethal mutagenesis. Motivated by experimental evidence that polymerase-targeting antivirals constrain replication, we analyze replicating systems with explicit delays in replication-enzyme availability. We identify a lag-induced (dynamical) critical transition driven by the loss of temporal coordination between genome translation and replication. At a fixed mutation rate and replicative fitness landscape, populations cross an extinction threshold solely due to time delays. Within the quasispecies framework, replication-translation timing emerges as an independent control parameter, defining a distinct dynamical route to extinction and suggesting new antiviral strategies based on modulating replicase availability. More generally, we propose that the pathway to collapse described in this article can be understood as lag-time-induced tipping (τ-tipping).

. We consider the secant method S-p applied to a real polynomial p of degree d + 1 as a discrete dynamical system on R-2. If the polynomial p has a local extremum at a point alpha then the discrete dynamical system generated by the iterates of the secant map exhibits a critical periodic orbit of period 3 or three-cycle at the point (alpha, alpha). We propose a simple model map T-a,T-d having a unique fixed point at the origin which encodes the dynamical behaviour of Sp3 at the critical three-cycle. The main goal of the paper is to describe the geometry and topology of the basin of attraction of the origin of T-a,T-d as well as its boundary. Our results concern global, rather than local, dynamical behaviour. They include that the boundary of the basin of attraction is the stable manifold of a fixed point or contains the stable manifold of a two-cycle, depending on the values of the parameters of d (even or odd) and a is an element of R (positive or negative).

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the $(n+2)$-body problem in which the distances among the first $n$ bodies remain bounded for all time, while the relative distances between the first $n$-bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first $n$ bodies corresponds to a KAM torus of the $n$-body problem.

In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of N-symbols for any integer N≥ 2 or infinity.

We consider analytic maps and vector fields defined in R2×Td, having a d-dimensional invariant torus T. The map (resp. vector field) restricted to T defines a rotation of Diophantine frequency vector ω∈Rd, and its derivative restricted to transversal directions to T does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which T possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and a posteriori theorems that can be used to validate numerical computations. Moreover, we present some applications of the results.

We consider the classical approximation of a realistic model for the scattering of He atoms from Cu surfaces. For this problem, modeled by a two-degrees-of-freedom Hamiltonian system, the existence of chaos has been proven analytically very recently for sufficiently large values of the energy, if some quantity, known as the Stokes constant, is non-zero (Borondo et al., 2024). Taking two different and independent approaches, this paper provides numerical evidence that this is indeed the case. Both approaches provide the same value of the non-zero Stokes constant.

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point [Formula: see text] while, simultaneously, two fixed points collide with [Formula: see text] in a transcritical bifurcation. Moreover, all points of a line containing [Formula: see text] become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation.

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.

We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times \mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map (resp. vector field) restricted to $\mathcal{T}$ defines a rotation of frequency $\omega$, and its derivative restricted to transversal directions to $\mathcal{T}$ does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which $\mathcal{T}$ possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and present some applications of the results.

Critical slowing down arises close to bifurcations and involves long transients. Despite slowing down phenomena have been widely studied in local bifurcations i.e., bifurcations of equilibrium points, less is known about transient delay phenomena close to global bifurcations. In this paper, we identify a novel mechanism of slowing down arising in the vicinity of a global bifurcation i.e., zip bifurcation, identified in a mathematical model of the dynamics of an autocatalytic replicator with an obligate parasite. Three different dynamical scenarios are first described, depending on the replication rate of cooperators, (L), and of parasites, (K). If KL the system is monostable and both species become extinct. In the case K=L coexistence of both species takes place in a Curve of Quasi-Neutral Equilibria (CQNE). The novel slowing down mechanism identified is due to an underlying ghost CQNE for the cases K≲L and K≳L. We show, both analytically and numerically, that the delays caused by the ghost CQNE follow scaling laws of the form τ∼|K−L|−1 for both K≲L and K≳L. We propose the ghost CQNE as a novel transient-generator mechanism in ecological systems.

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $F$, there exist invariant curves of class $C^r$ away from the fixed point, and that they are analytic when $F$ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of $F$ on them.

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.

Research on hypercycles focuses on cooperative interactions among replicating species, including the emergence of catalytic parasites and catalytic shortcircuits. Further interactions may be expected to arise in cooperative systems. For instance, molecular replicators are subject to mutational processes and ecological species to behavioural shifts due to environmental and ecological changes. Such changes could involve switches from cooperative to antagonistic interactions, in what we call a functional shift. In this article, we investigate a model for a two-member hypercycle model, considering that one species performs a functional shift. First, we introduce the model dynamics without functional shifts to illustrate the dynamics only considering obligate and facultative cooperation. Then, two more cases maintaining cross-catalysis are considered: (i) a model describing the dynamics of ribozymes where a fraction of the population of one replicator degrades the other molecular species while the other fraction still receives catalytic aid; and (ii) a system in which a given fraction of the population predates on the cooperating species while the rest of the population still receives aid. We have characterised the key bifurcation parameters determining extinction, survival, and coexistence of species. We show that predation, regardless of the fraction that benefits from it, does not significantly change dynamics with respect to the degradative case (i), thus conserving dynamics and bifurcations. Their biological significance is interpreted, and their potential implications for the dynamics of early replicators and ecological species are outlined.

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.

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 [4]. Assuming a sufficiently good approximation is found, here we provide an “a posteriori” result which gives a true invariant manifold close to the approximate 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.

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 four-member 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-Neimark–Sacker 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 co-extinctions and out-competition processes in both ecosystems and RNA-based systems.

The planar $$(n+1)$$-body problem models the motion of $$n+1$$ bodies in the plane under their mutual Newtonian gravitational attraction forces. When $$n\ge 3$$, 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).

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, Invariant manifolds of parabolic fixed points (I), these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided.

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 the invariant 2-dimensional stable/unstable manifolds. We analyse the different changes of dominant harmonic in the splitting functions. We describe how the dominant harmonics depend on the quotients of the continuous fraction expansion of the periodic forcing frequency. We have considered different frequencies including quadratic irrationals, frequencies having continuous fraction expansion with bounded quotients and frequencies with unbounded quotients. The methodology used is general enough to systematically deal with all these frequency types. All together allow us to get a detailed description of the asymptotic splitting behaviour for the concrete perturbation considered.