Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

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Dynamics and bifurcations of nonsmooth systems: A survey

The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Lienard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonic...

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Canonical Discontinuous Planar Piecewise Linear Systems

We consider the cubic nonlinear Schrodinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$
 . We prove modified scattering and construct modified wave operators for small initial and final data respectively ( $1\leqslant d\leqslant 4$
 ). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geqslant 2$
 . As a consequence, we obtain global strong solutions (for $d\geqslant 2$
 ) with infinitely growing high Sobolev norms  $H^{s}$
 .

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Modified scattering for the cubic Schrödinger equation on product spaces and applications

Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences.

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Generic bifurcations of low codimension of planar Filippov Systems

We consider the cubic defocusing nonlinear Schr¨ odinger equation on the two-dimen- sional torus. Fixs > 1. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions withs-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c > 0 such that for any K 1 we find a solution u and a time T such thatku.T/kHs Kku.0/kHs . Moreover, the timeT satisfies the polynomial bound 0< T < K c .

https://www.math.umd.edu/~vkaloshi/papers/GrowthSobolev.pdf

Growth of Sobolev norms in the cubic defocusing nonlinear Schrodinger equation

In this paper, we analytically consider sliding bifurcations of periodic orbits in the dry-friction oscillator. The system depends on two parameters: F, which corresponds to the intensity of the friction, and $\omega$, the frequency of the forcing. We prove the existence of infinitely many codimension-2 bifurcation points and focus our attention on two of them: $A_1:=(\omega\ii, F)=(2, 1/3)$ and $B_1:=(\omega\ii, F)=(3,0)$. We derive analytic expressions in ($\omega\ii$, F) parameter space for the codimension-1 bifurcation curves that emanate from $A_1$ and $B_1$. Our results show excellent agreement with the numerical calculations of Kowalczyk and Piiroinen [Phys. D, 237 (2008), pp. 1053–1073].

An Analytical Approach to Codimension-2 Sliding Bifurcations in the Dry-Friction Oscillator

Growth of Sobolev norms for the analytic NLS on T2

We consider the cubic defocusing nonlinear Schr\"odinger equation in the two dimensional torus. Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions with $s$-Sobolev norm growing in time. 
We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\mathcal{K}\gg 1$ we find a solution $u$ and a time $T$ such that $\| u(T)\|_{H^s}\geq\mathcal{K} \| u(0)\|_{H^s}$. Moreover, time $T$ satisfies polynomial bound $0<T<\mathcal{K}^c$.

Marcel Guardia

Papers

Generic bifurcations of low codimension of planar Filippov Systems

Growth of Sobolev norms in the cubic defocusing nonlinear Schrodinger equation

An Analytical Approach to Codimension-2 Sliding Bifurcations in the Dry-Friction Oscillator

Growth of Sobolev norms for the analytic NLS on T2

Growth of Sobolev norms in the cubic defocusing nonlinear Schr\"odinger equation