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Register a new userVariational attraction of the KAM torus for the conformally symplectic system
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For the conformally symplectic system [ left{ begin{aligned} dot{q}&=H_p(q,p),quad(q,p)in T^*mathbb{T}^n dot p&=-H_q(q,p)-lambda p, quad lambda>0 end{aligned} right. ] with a positive definite Hamiltonian, we discuss the variational significance of invariant Lagrangian graphs and explain how the presence of the KAM torus dominates the $C^1-$convergence speed of the Lax-Oleinik semigroup.
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We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $cin H^1(mathbb T,mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(sigma)in H^1(mathbb T,mathbb R)$ with $c(sigma)$ continuous and monotonic and [ partial_tu_c+H(x,partial_x u_c+c,t)=alpha(c),quad text{a.e. } (x,t)inmathbb T^2, ] such that sequence of weak KAM solutions ${u_c}_{cin H^1(mathbb T,mathbb R)}$ is $1/2-$Holder continuity of parameter $sigmain mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving [ left{ begin{aligned} &dot{x}(s)in text{co} Big[partial_pHBig(x(s),c+D^+u_cbig(x(s),s+tbig),s+tBig)Big], & &x(0)=x_0,quad (x_0,t)inmathbb T^2,& end{aligned} right. ] we evaluate it by a uniquely identified rotational number $omega(c)in H_1(mathbb T,mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
We study the planetary system of $upsilon$~Andromed{ae}, considering the three-body problem formed by the central star and the two largest planets, $upsilon$~And~emph{c} and $upsilon$~And~emph{d}. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e., a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the normal form related to a KAM torus, whose shape is in excellent agreement with the orbits of the secular model. We rigorously prove that the algorithm constructing the final KAM invariant torus is convergent, by adopting a suitable technique based on a computer-assisted proof.
This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorovs invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II.
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic similarity problems.
Lectures given on KAM theory at the University of Ouargla (Algeria). I present a functional analytic treatment of the subject which includes KAM theory into the general framework of deformations and singularity theory.
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