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Register a new userA Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
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Let $[A]: Y=AY$ with $Ain mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $Rin {cal M}_{n}(bar{k})$ is a {em reduced form} of $[A]$ if $Rin mathfrak{g}(bar{k})$ and there exists $Pin GL_n (bar{k})$ such that $R=P^{-1}(AP-P)in mathfrak{g}(bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.
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In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincare rank 1 at the origin over a fixed neighborhood $D_r$. The unfolding parameter $epsilon $ is taken in a sector S pointed at the origin of opening larger than $2 pi$ in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value in S, we cover $D_r$ with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the analytic invariants linked to the monodromy of the chosen basis around the singular points. The analytic invariants give a complete geometric interpretation to the well-known Stokes matrices at $epsilon =0$: this includes the link (existing at least for the generic cases) between the divergence of the solutions at $epsilon =0$ and the presence of logarithmic terms in the solutions for resonance values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.
We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay) solutions for these systems.
We propose an effective and lightweight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At the heart of our algorithm is a novel neural network architecture consisting of two sub-networks. Both are embedded with terms in the form of Taylor series expansion designed with symmetric structure. The key mechanism underpinning our infrastructure is the strong expressiveness and special symmetric property of the Taylor series expansion, which naturally accommodate the numerical fitting process of the gradients of the Hamiltonian with respect to the generalized coordinates as well as preserve its symplectic structure. We further incorporate a fourth-order symplectic integrator in conjunction with neural ODEs framework into our Taylor-net architecture to learn the continuous-time evolution of the target systems while simultaneously preserving their symplectic structures. We demonstrated the efficacy of our Taylor-net in predicting a broad spectrum of Hamiltonian dynamic systems, including the pendulum, the Lotka--Volterra, the Kepler, and the Henon--Heiles systems. Our model exhibits unique computational merits by outperforming previous methods to a great extent regarding the prediction accuracy, the convergence rate, and the robustness despite using extremely small training data with a short training period (6000 times shorter than the predicting period), small sample sizes, and no intermediate data to train the networks.
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small.
A class of two-dimensional linear differential systems is considered. The box-counting dimension of the graphs of solution curves is calculated. Criteria to obtain the box-counting dimension of spirals are also established.
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