Stable manifolds and homoclinic points near resonances in the restricted three-body problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem

Coupling between feedback loops in autoregulatory networks affects bistability range, open-loop gain and switching times

Biochemical regulatory networks governing diverse cellular processes such as stress-response, differentiation and cell cycle often contain coupled feedback loops. We aim at understanding how features of feedback architecture, such as the number of loops, the sign of the loops and the type of their coupling, affect network dynamical performance. Specifically, we investigate how bistability range, maximum open-loop gain and switching times of a network with transcriptional positive feedback are affected by additive or multiplicative coupling with another positive- or negative-feedback loop. We show that a network's bistability range is positively correlated with its maximum open-loop gain and that both quantities depend on the sign of the feedback loops and the type of feedback coupling. Moreover, we find that the addition of positive feedback could decrease the bistability range if we control the basal level in the signal-response curves of the two systems. Furthermore, the addition of negative feedback has the capacity to increase the bistability range if its dissociation constant is much lower than that of the positive feedback. We also find that the addition of a positive feedback to a bistable network increases the robustness of its bistability range, whereas the addition of a negative feedback decreases it. Finally, we show that the switching time for a transition from a high to a low steady state increases with the effective fold change in gene regulation. In summary, we show that the effect of coupled feedback loops on the bistability range and switching times depends on the underlying mechanistic details

Tracing KAM tori in presymplectic dynamical systems

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $\omega$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $\omega$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.Comment: 33 pages and one figur

Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes

The scaling of the time delay near a "bottleneck" of a generic saddle-node bifurcation is well-known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate $C^0$ vector fields. Our main result is a new phenomenon in two-parameter families having a saddle-node bifurcation upon changing the first parameter. We find distinct scalings for different values of the second parameter ranging from power laws with exponents in (0,1) to scalings given by O(1). We illustrate this rapid quantitative change of the scaling law by a an overdamped pendulum with varying length.Comment: preprint version - for final version see journal referenc

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, f, s) = p2/2+ cos q - 1 + I2/2 + h(q, f, s; e) — proving that for any small periodic perturbation of the form h(q, f, s; e) = e cos q (a00 + a10 cosf + a01 cos s) (a10a01 ¿ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ p/2µ, µ = a10/a01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any µ). The bifurcations of the scattering map are also studied as a function of µ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.Peer ReviewedPostprint (published version

Ghost-patterning and non-patterning in a draining film model

Patterns can form when the uniform state of any system is unstable so that some non-uniform motif grows in amplitude. Here, we identify an alternative way to form non-trivial structures, which we call “ghost-patterns”. Ghost-patterns emerge from noisy initial conditions when all non-uniform modes decay in amplitude except for one non-trivial motif which fails to decay. Hence, in seeking structured states, it is not necessary to find positive growth rates. We demonstrate ghost-patterns in an idealized non-equilibrium model intended to emulate draining thin-film suspensions

A λ-lemma for Normally Hyperbolic Invariant Manifolds

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Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems