Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

In the relativistic and the nonrelativistic theoretical treatment of moderate and high-power laser-matter interaction, the generalized Bessel function occurs naturally when a Schrödinger-Volkov and Dirac-Volkov solution is expanded into plane waves. For the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field, it is often necessary to evaluate large arrays of generalized Bessel functions, of arbitrary index but with fixed arguments. We show that the generalized Bessel function can be evaluated, in a numerically stable way, by utilizing a recurrence relation and a normalization condition only, without having to compute any initial value. We demonstrate the utility of the method by illustrating the quantum-classical correspondence of the Dirac-Volkov solutions via numerical calculations.

Efficiency of trapping processes in regular and disordered networks

We use a Markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks. The efficiency is gauged by the mean absorption time (average of the mean number of steps performed by the random walker before being absorbed by the trap). We compute this quantity in terms of different control parameters, namely, the length of the walker jumps, the mobility of the trap, and the degree of spatial disorder of the network. For a proper choice of the system size, we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter. We thus arrive at the conclusion that, despite the decrease of the effective system size underlying the increase of the control parameter, the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap. This somewhat anti-intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter. For the case of a ring lattice, results for the mean absorption time in systems of arbitrary size are given in terms of a two-parameter scaling function. For the case of a mobile trap, we deal with both trapping via a single channel (walker-trap overlap) and via two channels (walker-trap overlap and walker-trap crossing), thereby generalizing previous work. As for the disordered case, our analysis concerns small world networks, for which we see several crossovers of the absorption time as a function of the control parameter and the system size. The methodology used may be well suited to exploring characteristic time scales of encounter-controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles.