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2013
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# Carles Simó

Gran Via de les Corts Catalanes, 585. 08007 Barcelona, Spain
University of Barcelona

## Publications:

 Fontich E., Simó C., Vieiro A. On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena 2018, vol. 23, no. 6, pp.  638-653 Abstract 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. Keywords: quasi-periodic splitting, dominant harmonics, hidden harmonics, irrational numbers properties Citation: Fontich E., Simó C., Vieiro A.,  On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 638-653 DOI:10.1134/S1560354718060011
 Martinez R., Simó C. Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion 2014, vol. 19, no. 6, pp.  745-765 Abstract Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant $C$ large enough are compared to direct numerical computations showing improved agreement when $C$ increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing $C$. Several anomalous phenomena are described. Keywords: invariant rotational curves, separatrix maps, splitting function, restricted three-body problem Citation: Martinez R., Simó C.,  Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 745-765 DOI:10.1134/S1560354714060112
 Miguel N., Simó C., Vieiro A. From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values 2013, vol. 18, no. 5, pp.  469-489 Abstract In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense. First, we present a study of some dynamical properties of orientation-preserving and orientationreversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects. Then the phase space of the standard map, for large values of the parameter k, is studied. There are some stable orbits which appear periodically in $k$ and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as $k \to \infty$. Elementary considerations about diffusion properties of the standard map are also presented. Keywords: Hénon maps, measure of regular and chaotic dynamics domains, islands in the standard map for large parameter, accelerator modes Citation: Miguel N., Simó C., Vieiro A.,  From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 469-489 DOI:10.1134/S1560354713050018
 Puig J., Simó C. Resonance tongues in the quasi-periodic Hill–Schrödinger equation with three frequencies 2011, vol. 16, no. 1-2, pp.  61-78 Abstract In this paper we investigate numerically the following Hill’s equation $x'' + (a + bq(t))x = 0$ where $q(t) = cost + \cos\sqrt{2}t + \cos\sqrt{3}t$ is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential. Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of $b$ the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of $a$, for large $b$, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case. Keywords: quasi-periodic Schrödinger operators, quasi-periodic cocycles and skew-products, spectral gaps, resonance tongues, rotation number, Lyapunov exponent, numerical explorations Citation: Puig J., Simó C.,  Resonance tongues in the quasi-periodic Hill–Schrödinger equation with three frequencies, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 61-78 DOI:10.1134/S1560354710520047
 Vitolo R., Broer H. W., Simó C. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems 2011, vol. 16, no. 1-2, pp.  154-184 Abstract This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type. Keywords: bifurcations, invariant tori, resonances, KAM theory Citation: Vitolo R., Broer H. W., Simó C.,  Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 154-184 DOI:10.1134/S1560354711010060
 Martinez R., Simó C. Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples 2009, vol. 14, no. 3, pp.  323-348 Abstract This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear springpendulum problem for the values of the parameter that can not be decided using first order variational equations. Keywords: non-integrability criteria, differential Galois theory, higher order variationals, springpendulum system Citation: Martinez R., Simó C.,  Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples, Regular and Chaotic Dynamics, 2009, vol. 14, no. 3, pp. 323-348 DOI:10.1134/S1560354709030010
 Simó C. Invariant curves of analytic perturbed nontwist area preserving maps 1998, vol. 3, no. 3, pp.  180-195 Abstract Area preserving maps close to integrable but not satisfying the twist condition are studied. The existence of invariant curves is proved, but they are no longer graphs with respect to the angular variable. Beyond the generic, codimension 1 case, several higher codimension cases are studied. Meandering curves, higher order meandering and labyrinthic curves show up. Several examples illustrate that this behavior occurs in very simple families of maps. Citation: Simó C.,  Invariant curves of analytic perturbed nontwist area preserving maps, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 180-195 DOI:10.1070/RD1998v003n03ABEH000088
 Giorgilli A., Lazutkin V. F., Simó C. Visualization of a Hyperbolic Structure in Area Preserving Maps 1997, vol. 2, nos. 3-4, pp.  47-61 Abstract We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. The method is illustrated for the case of the Carleson standard map. As it follows from our experiments, the structure of the chaotic zone for the standard map is different from the one found for the systems of Anosov type. Citation: Giorgilli A., Lazutkin V. F., Simó C.,  Visualization of a Hyperbolic Structure in Area Preserving Maps, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 47-61 DOI:10.1070/RD1997v002n03ABEH000047