NORMALIZATION OF CLASSICAL HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM

В работе исследовано семейство гамильтоновых систем с двумя степенями свободы. Расчетами сечений Пуанкаре показано, что при произвольных значениях параметров функции Гамильтона система является неинтегрируемой и в ней реализуется динамический хаос. Найдено, что для трех наборов параметров рассматриваемая система является интегрируемой, однако в одном интегрируемом случае при этих же значениях параметров на поверхности потенциальной энергии имеется область с отрицательной гауссовой кривизной, в то же время в двух других случаях интегрируемости при соответствующих значениях параметров областей с отрицательной гауссовой кривизной не имеется. Таким образом, наличие областей с отрицательной гауссовой кривизной на поверхности потенциальной энергии не достаточно для развития в системе глобального хаоса. Получена классическая нормальная форма для произвольных значений параметров. The family of the Hamiltonian systems with two degrees of freedom was investigated. The calculations of the Poincaré sections show that with arbitrary values of the parameters of the Hamilton function, the system is non-integrable and dynamic chaos is realized in it. For the three parameter sets, the system in question was found to be integrable, but shows that in one integrable case on the potential energy surface (PES) there are regions with the negative Gaussian curvature. It was found that in one integrable case for the same values of the parameters, the potential energy surface has a region with the negative Gaussian curvature. At the same time, in the other two cases, the domains with negative Gaussian curvature are not integrable for the corresponding values of the parameters. Thus, the presence of regions with negative Gaussian curvature on the potential energy surface is not enough for the development of the global chaos in the system. The classical normal form for arbitrary parameter values is obtained.

The identity G(D)f = F for a linear partial differential operator G(D). Lusin type and structure results in the non-integrable case

We prove a Lusin type theorem for a certain class of linear partial differential operators G(D), reducing to [1, Theorem 1] when G(D) is the gradient. Moreover, we describe the structure of the set {G(D)f = F}, under assumptions of non-integrability on F, in terms of lower dimensional rectifiability and superdensity. Applications to Maxwell type system and to multivariable Cauchy–Riemann system are provided.

Extensive Numerical Results for Integrable Case of Standard Map

In recent years, conservative dynamical systems have become a vivid area of research from the statistical mechanical characterization viewpoint. With this respect, several areapreserving maps have been studied. It has been numerically shown that the probability distribution of the sum of the suitable random variable of these systems can be well approximated by a Gaussian (q-Gaussian) when the initial conditions are randomly selected from the chaotic sea (region of stability islands) in the available phase space. In this study, we will summarize these results and discuss a special case for the standard map, a paradigmatic example of area-preserving maps, for which the map is totally integrable.

High precision numerical approach for Davey–Stewartson II type equations for Schwartz class initial data

We present an efficient high-precision numerical approach for Davey–Stewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll’s composite Runge–Kutta method for the time dependence. Since DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of 10 −6 or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for DS II via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C. Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.

A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits

The entanglement in operator space is a well established measure for the complexity of quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study the dynamics of local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of ``completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a ``phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits.