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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2008 , Volume 10 , Issue 2&3, September

Special Issue dedicated to Carles Simo on the occasion of his 60th anniversary

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2008, 10(2&3, September): i-iii
doi: 10.3934/dcdsb.2008.10.2i

*+*[Abstract](134)*+*[PDF](54.3KB)**Abstract:**

This volume is dedicated to Carles Simó on the occasion of his 60th anniversary and contains papers of the participants of the conference “International Conference on Dynamical Systems, Carles Simó Fest”, celebrated in S’Agaró, near Barcelona, from May 29th to June 3rd, 2006 and organized by former PhD students of him.

&nbs Carles Simó was born in Barcelona in 1946. He studied Industrial Engineering and Mathematics at once, earning his Ph.D. in Mathematics in 1974.

For the full preface, please click the Full Text "PDF" button above.

2008, 10(2&3, September): 265-293
doi: 10.3934/dcdsb.2008.10.265

*+*[Abstract](177)*+*[PDF](327.0KB)**Abstract:**

In this paper we give a mechanism to compute the families of clas- sical hamiltonians of two degrees of freedom with an invariant plane and normal variational equations of Hill-Schrödinger type selected in a suitable way. In particular we deeply study the case of these equations with polynomial or trigonometrical potentials, analyzing their integrability in the Picard-Vessiot sense using Kovacic’s algorithm and introducing an algebraic method (algebrization) that transforms equations with transcendental coefficients in equations with rational coefficients without changing the Galoisian structure of the equation. We compute all Galois groups of Hill-Schrödinger type equations with polynomial and trigonometric (Mathieu equation) potentials, obtaining Galoisian obstructions to integrability of hamiltonian systems by means of meromorphic or rational first integrals via Morales-Ramis theory.

2008, 10(2&3, September): 295-322
doi: 10.3934/dcdsb.2008.10.295

*+*[Abstract](143)*+*[PDF](328.3KB)**Abstract:**

In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of one dimensional parabolic manifolds of degenerate fixed points, i.e. invariant manifolds tangent to the eigenspace of 1, which is assumed to be a simple eigenvalue. We use the parameterization method, reducing the dynamics on the parabolic manifold to a polynomial. We prove that the asymptotic expansions of the parabolic manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence $C^\infty$.

2008, 10(2&3, September): 323-347
doi: 10.3934/dcdsb.2008.10.323

*+*[Abstract](192)*+*[PDF](374.7KB)**Abstract:**

A classical problem in the study of the (conservative) unfoldings of the so called Hopf-zero bifurcation, is the computation of the splitting of a heteroclinic connection which exists in the symmetric normal form along the z-axis. In this paper we derive the inner system associated to this singular problem, which is independent on the unfolding parameter. We prove the existence of two solutions of this system related with the stable and unstable manifolds of the unfolding, and we give an asymptotic formula for their difference. We check that the results in this work agree with the ones obtained in the regular case by the authors.

2008, 10(2&3, September): 349-375
doi: 10.3934/dcdsb.2008.10.349

*+*[Abstract](179)*+*[PDF](373.5KB)**Abstract:**

Vortex modeling has a long history. Descartes (1644) used it as a model for the solar system. J.J. Thomson (1883) used it as a model for the atom. We consider point-vortex systems, which can be regarded as “discrete” solutions of the Euler equation. Their dynamics is described by a Hamiltonian system of equations. In particular we are interested in vortex dynamics on simply connected surfaces of constant curvature $K$, i.e. a plane, spheres and hyperbolic surfaces. It is known that polygonal configurations of $N$ point-vortices are relative equilibria of the system. We study the stability of such polygonal configurations, and, more specifically, how stability depends upon the number of vortices $N$ and the curvature $K$ of the surface. To address such a question we have to formulate the problem in a unified geometrical way. The fact that the surfaces of interest can be viewed as Kähler manifolds greatly simplify our task. Nonlinear stability is then studied by making use of the Dirichlet Criterion. Stability ranges are the $K$-intervals for which the Hessian of the Hamiltonian is positive or negative definite, when evaluated at the equilibrium configuration.

2008, 10(2&3, September): 377-400
doi: 10.3934/dcdsb.2008.10.377

*+*[Abstract](167)*+*[PDF](287.8KB)**Abstract:**

We give sufficient conditions on the spectrum at the equilibrium point such that a Gevrey-$s$ family can be Gevrey-$s$ conjugated to a simplified form, for $0\le s\le 1$. Local analytic results (i.e. $s=0$) are obtained as a special case, including the classical Poincaré theorems and the analytic stable and unstable manifold theorem. As another special case we show that certain center manifolds of analytic vector fields are of Gevrey-$1$ type. We finally study the asymptotic properties of the conjugacy on a polysector with opening angles smaller than $s\pi$ by considering a Borel-Laplace summation.

2008, 10(2&3, September): 401-419
doi: 10.3934/dcdsb.2008.10.401

*+*[Abstract](222)*+*[PDF](2102.9KB)**Abstract:**

The understanding of atmospheric and oceanic low-frequency variability is an old problem having both theoretical interest and practical importance, e.g., for the assessment of climate change. In this paper possible relations with dynamical systems theory are given, in particular through bifurcation theory. Firstly, a specific type of oceanic low frequency variability is described, the so-called Atlantic Multidecadal Oscillation (AMO). Then recent work is reviewed, that investigates bifurcations as these occur in a few low-order models of the atmospheric circulation. It is shown that the Shil′nikov bifurcation in the Hopf-saddle-node bifurcation scenario takes place in each of the above atmospheric models. Related strange attractors and intermittency behavior are also found, both in agreement with the theoretical expectations and with qualitative aspects of the climate variability, like the AMO. It is discussed how the latter connection may be consolidated in higher dimensional and in PDE models.

2008, 10(2&3, September): 421-438
doi: 10.3934/dcdsb.2008.10.421

*+*[Abstract](206)*+*[PDF](260.3KB)**Abstract:**

From a normal form analysis near the Lagrange equilateral relative equilibrium, we deduce that, up to the action of similarities and time shifts, the only relative periodic solutions which bifurcate from this solution are the (planar) homographic family and the (spatial) $P_{12}$ family with its twelfth-order symmetry (see [13, 5]). After reduction by the rotation symmetry of the Lagrange solution and restriction to a center manifold, our proof of the local existence and uniqueness of $P_{12}$ follows that of Hill's orbits in the planar circular restricted three-body problem in [7, 1]. Indeed, near the Lagrange solution, the restrictions of constant energy levels of the reduced flow to a center manifold (actually unique) turn out to be three-spheres. In an annulus of section bounded by relative periodic solutions of each family, the normal resonance along the homographic family entails that the Poincaré return map is the identity on the corresponding connected component of the boundary. Using the reflexion symmetry with respect to the plane of the relative equilibrium, we prove that, close enough to the Lagrange solution, the return map is a monotone twist map.

2008, 10(2&3, September): 439-454
doi: 10.3934/dcdsb.2008.10.439

*+*[Abstract](134)*+*[PDF](1985.8KB)**Abstract:**

In the present paper we study the global dynamics corresponding to a realistic model of self-consistent triaxial galactic system. We extend a previous work [17] where the authors investigate 3,472 orbits in this model at different energy levels, using Lyapunov exponents to measure chaoticity and frequency analysis to classify regular orbits. Here we first display the main properties of that potential and then focus our attention on the global dynamical features of the box domain for nine energy surfaces. Using the MEGNO as a fast dynamical indicator, we gain insight in the resonance structure at different energy levels, the way in which relatively large chaotic domains arise due to overlapping as well as crossings of resonances and we measure the fraction of chaotic motion in the energy space. It is interesting to notice that the flatness of the model varies over a rather wide range, namely from ~ $0.5$ to ~ $1$, and the fraction of chaotic motion ranges from ~ $0.15$ at small energies up to ~ $0.75$ at moderate values of the energy, decreasing then again down to values close to ~ $0.4$ where the system becomes nearly spherical.

2008, 10(2&3, September): 455-483
doi: 10.3934/dcdsb.2008.10.455

*+*[Abstract](260)*+*[PDF](619.2KB)**Abstract:**

Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.

We recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map.

In the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariant manifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.

In the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.

Scattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.

2008, 10(2&3, September): 485-494
doi: 10.3934/dcdsb.2008.10.485

*+*[Abstract](180)*+*[PDF](178.8KB)**Abstract:**

In this work we study the Liénard systems that can be transformed into Riccati differential equations, using changes of variables more general than the ones used by the classical Lie theory.

2008, 10(2&3, September): 495-509
doi: 10.3934/dcdsb.2008.10.495

*+*[Abstract](168)*+*[PDF](188.3KB)**Abstract:**

We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.

2008, 10(2&3, September): 511-536
doi: 10.3934/dcdsb.2008.10.511

*+*[Abstract](164)*+*[PDF](537.5KB)**Abstract:**

We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Hénon maps.

2008, 10(2&3, September): 537-567
doi: 10.3934/dcdsb.2008.10.537

*+*[Abstract](219)*+*[PDF](602.8KB)**Abstract:**

We introduce a scenario for the fractalization of invariant curves for a special class of quasi-periodically forced 1-D maps. In this situation, a smooth invariant curve becomes increasingly wrinkled when its Lyapunov exponent goes to zero, but it keeps being smooth as long as its exponent is negative. It is remarkable that the curve becomes so wrinkled that numerical simulations may not distinguish the curve from a strange attracting set.

Moreover, we show that a nonreducible invariant curve with a positive Lyapunov exponent is not persistent in a general quasi-periodically forced 1-D map. We also derive some new results on the behaviour of the Lyapunov exponent of an invariant curve w.r.t. parameters.

The paper contains some numerical examples. One of them is based on the quasi-periodically forced logistic map, where we show numerically that the fractalization of an invariant curve of this system may fit into our scenario.

2008, 10(2&3, September): 569-595
doi: 10.3934/dcdsb.2008.10.569

*+*[Abstract](181)*+*[PDF](341.9KB)**Abstract:**

We investigate the planar three-body problem in the range where one mass, say the ‘sun’ is very far from the other two, call them ‘earth’ and ‘moon’. We show that “stutters” : two consecutive eclipses in which the moon lies on the line between the earth and sun, occur for an open set of initial conditions. In these motions the moon reverses its sense of rotation about the earth. The mechanism is a kind of tidal torque (see the ‘key equation’). The motivation is to better understand the limits of variational methods. The methods of proof are classical estimates and bounds in this asymptotic regime.

2008, 10(2&3, September): 597-608
doi: 10.3934/dcdsb.2008.10.597

*+*[Abstract](154)*+*[PDF](565.0KB)**Abstract:**

In this paper we consider the family of circle maps $f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when written mod 1 are of the form $f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$ and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while the maximum with respect to $\alpha$ is larger. In the case of the average the difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families of expanding Blaschke products depending on rotations where the averages are equal and for which the inequality for averages goes in the other direction when the expanding property does not hold, see [4]. A striking fact for both results is that the maximum of the entropies is greater than or equal to $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with [3], where similar questions are considered for a family of diffeomorphisms of the two sphere.

2008, 10(2&3, September): 609-620
doi: 10.3934/dcdsb.2008.10.609

*+*[Abstract](232)*+*[PDF](264.3KB)**Abstract:**

A topological existence proof is presented for certain symmetrical periodic orbits of the collinear three-body problem with two equal masses, called Schubart orbits. The proof is based on the construction of a Wazewski set $W$ in the phase space. The periodic orbits are found by a shooting argument in which symmetrical initial conditions entering $W$ are followed under the flow until they exit $W$. Topological considerations show that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.

2008, 10(2&3, September): 621-650
doi: 10.3934/dcdsb.2008.10.621

*+*[Abstract](187)*+*[PDF](14437.4KB)**Abstract:**

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom corresponding to fast motion and the other one corresponding to slow motion. The Hamiltonian function is the sum of potential and kinetic energies, the kinetic energy being a weighted sum of squared momenta. The ratio of time derivatives of slow and fast variables is of order $\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition we prove the existence of many, of order $1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order $\epsilon$. Thus, the total measure of the stability islands is estimated from below by a value independent of $\epsilon$. We find the location of stable periodic trajectories and an asymptotic formula for the number of these trajectories. As an example, we consider the problem of motion of a charged particle in the parabolic model of magnetic field in the Earth magnetotail.

2008, 10(2&3, September): 651-659
doi: 10.3934/dcdsb.2008.10.651

*+*[Abstract](170)*+*[PDF](155.6KB)**Abstract:**

A homeomorphism of the plane $h$ has trivial dynamics if every positive orbit $\{ h^n (p)\}_{n\geq 0}$ is either convergent (to a fixed point) or divergent (to infinity). The main result of this paper says that the property of trivial dynamics can be decided by computing the topological degree of $id -h$. In this result it is assumed that $h$ is analytic in the real sense. Some applications to difference equations and to periodic Newtonian differential equations are obtained.

2008, 10(2&3, September): 661-679
doi: 10.3934/dcdsb.2008.10.661

*+*[Abstract](151)*+*[PDF](319.0KB)**Abstract:**

We show that the set of $C^{\infty}$ mechanical Lagrangians $L(p,v)$ in $T^{2}$ without continuous invariant graphs in all supercritical energy levels is dense in the $C^{1}$ topology. A mechanical Lagrangian $L: T$$T^{2} \rightarrow \mathbb R$ is a function in the tangent space of the torus $T$$T^{2}$ given by $L(p,v)=\frac{1}{2}g(v,v)-U(p)$, where $g$ is a Riemannian metric and $U$ is a smooth potential.

2008, 10(2&3, September): 681-698
doi: 10.3934/dcdsb.2008.10.681

*+*[Abstract](146)*+*[PDF](3328.6KB)**Abstract:**

We discuss the problem of existence of elliptic periodic trajectories inside lobes bounded by segments of stable and unstable separatrices of a hyperbolic fixed point. We show that such trajectories generically exist in symplectic maps arbitrary close to integrable ones. Elliptic periodic trajectories as a rule, generate stability islands. The area of such an island is of the same order as the lobe area, but the quotient of areas can be very small. Numerical examples are included.

2008, 10(2&3, September): 699-717
doi: 10.3934/dcdsb.2008.10.699

*+*[Abstract](293)*+*[PDF](302.2KB)**Abstract:**

Weconsiderthecollinearthree-bodyproblemwithtwoequalmasses for the Newtonian potential $1/r$. We give a rigorous proof of the existence of a symmetric periodic solution with two collisions per period. This solution has been discovered numerically in 1956 by J. von Schubart (see [12]). Our proof is based on the direct method in Calculus of Variations, which consists in the minimization of the action on a well chosen set of periodic loops. The main difficulty is to show that the minimizer has only two collisions per period.

2008, 10(2&3, September): 719-731
doi: 10.3934/dcdsb.2008.10.719

*+*[Abstract](191)*+*[PDF](776.0KB)**Abstract:**

We discuss iterates of random circle diffeomorphisms with identically distributed noise, where the noise is bounded and absolutely continuous. Using arguments of B. Deroin, V.A. Kleptsyn and A. Navas, we provide precise conditions under which random attracting fixed points or random attracting periodic orbits exist. Bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle are treated. We show that this typically involves a transition from a unique random attracting periodic orbit to a unique random attracting fixed point.

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