Fernando Blesa

Breaking the limits: The Taylor series method

Abstract This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge–Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.

Fractal structures in the Hénon-Heiles Hamiltonian

During the past few years, several papers (Aguirre J., Vallejo J. C. and Sanjuan M. A. F., Phys. Rev. E, 64 (2001) 066208; de Moura A. P. S. and Letelier P. S., Phys. Lett. A, 256 (1999) 362; Seoane J. M., Sanjuan M. A. F. and Lai Y.-C., Phys. Rev. E, 76 (2007) 061208) have detected the presence of fractal escape basins in Henon-Heiles potentials in the unbounded range. Upon fixing the energy value, these basins are detected on the (x, y) and planes. In this paper, we explore the appearance of different kinds of fractal structures. We present an analysis of the fractal structures on the escape basins of the (x, y) and (y, E) planes (allowing the energy value E to change and studying the fat-fractal exponent); later, we present these structures on the KAM tori for low energy values, on small regular islands inside the chaotic sea close to the critical energy level on the (y, E)-plane, and most interestingly, on small regular regions inside the escape region. These small regions of bounded motion and regular behavior appear after the critical escape energy, when most of the orbits are escape orbits.

Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this package is that more and more scientific disciplines need very high precision solution of ODEs, and a standard ODE method is not able to reach these precision levels. The TIDES package combines a preprocessor step in Mathematica that generates Fortran or C programs with a library in C. Another capability of TIDES is the direct solution of sensitivities of the solution of ODE systems, which means that we can compute the solution of variational equations up to any order without formulating them explicitly. Different options of the software are discussed, and finally it is compared with other well-known available methods, as well as with different options of TIDES. From the numerical tests, TIDES is competitive, both in speed and accuracy, with standard methods, but it also provides new capabilities.

TO ESCAPE OR NOT TO ESCAPE, THAT IS THE QUESTION — PERTURBING THE HÉNON–HEILES HAMILTONIAN

In this work, we study the Henon–Heiles Hamiltonian, as a paradigm of open Hamiltonian systems, in the presence of different kinds of perturbations as dissipation, noise and periodic forcing, which are very typical in different physical situations. We focus our work on both the effects of these perturbations on the escaping dynamics and on the basins associated to the phase space and to the physical space. We have also found, in presence of a periodic forcing, an exponential-like decay law for the survival probability of the particles in the scattering region where the frequency of the forcing plays a crucial role. In the bounded regions, the use of the OFLI2 chaos indicator has allowed us to characterize the orbits. We have compared these results with the previous ones obtained for the dissipative and noisy case. Finally, we expect this work to be useful for a better understanding of the escapes in open Hamiltonian systems in the presence of different kinds of perturbations.

BIFURCATIONS AND CHAOS IN HAMILTONIAN SYSTEMS

This paper deals with the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom of Hamiltonian systems. These numerical methods are based on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and the skeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems are studied: the Copenhagen and the (n + 1)-body ring problems and the Henon–Heiles Hamiltonian. All the numerical integrations have been done by using the state-of-the-art numerical library TIDES based on the extended Taylor series method.

Qualitative and numerical analysis of the Rössler model: Bifurcations of equilibria

In this paper, we show the combined use of analytical and numerical techniques in the study of bifurcations of equilibria of low-dimensional chaotic problems. We study in detail different aspects of the paradigmatic Rossler model. We provide analytical formulas for the stability of the equilibria as well as some of their codimension one, two, and three bifurcations. In particular, we carry out a complete study of the Andronov-Hopf bifurcation, establishing explicit formulas for its location and studying its character numerically, determining a curve of generalized-Hopf bifurcation, where the Hopf bifurcation changes from subcritical to supercritical. We also briefly study some routes among the different Andronov-Hopf bifurcation curves and how these routes are influenced by the local and global bifurcations of limit cycles. Finally, we show the U-shape of the homoclinic bifurcation curve at the studied parameter values.

BEHAVIOR PATTERNS IN MULTIPARAMETRIC DYNAMICAL SYSTEMS: LORENZ MODEL

In experimental and theoretical studies of Dynamical Systems, there are usually several parameters that govern the models. Thus, a detailed study of the global parametric phase space is not easy and normally unachievable. In this paper, we show that a careful selection of one straight line (or other 1D manifold) permits us to obtain a global idea of the evolution of the system in some circumstances. We illustrate this fact with the paradigmatic example of the Lorenz model, based on a global study, changing all three parameters. Besides, searching in other regions, for all the detected behavior patterns in one straight line, we have been able to see that missing topological structures of the chaotic attractors may be found on the chaotic-saddles.

Unbounded dynamics in dissipative flows: Rössler model

Transient chaos and unbounded dynamics are two outstanding phenomena that dominate in chaotic systems with large regions of positive and negative divergences. Here, we investigate the mechanism that leads the unbounded dynamics to be the dominant behavior in a dissipative flow. We describe in detail the particular case of boundary crisis related to the generation of unbounded dynamics. The mechanism of the creation of this crisis in flows is related to the existence of an unstable focus-node (or a saddle-focus) equilibrium point and the crossing of a chaotic invariant set of the system with the weak-(un)stable manifold of the equilibrium point. This behavior is illustrated in the well-known Rössler model. The numerical analysis of the system combines different techniques as chaos indicators, the numerical computation of the bounded regions, and bifurcation analysis. For large values of the parameters, the system is studied by means of Fenichels theory, providing formulas for computing the slow manifold which influences the evolution of the first stages of the orbit.

OpenCL parallel integration of ordinary differential equations: Applications in computational dynamics ☆

Abstract In many physical problems the use of numerical simulations presents the only path to obtain insight into the behavior and evolution of the system of interest. GPU, CPU and MIC technologies are frequently employed for simulations on computational dynamics and we present results comparing different schemes for the numerical integration of ordinary differential systems (ODEs) in these architectures. The use of adapted methods with low memory storage (Low storage Runge–Kutta methods) gives good results for low precision studies, whereas the Taylor series method provides a powerful technique for high precision. We show how the computation of several dynamics indicators, such as a fast chaos indicator (FLI) or a phase shift indicator in small neuron networks (Central Pattern Generators), can be efficiently computed on these architectures by means of the numerical ODE methods executed through OpenCL. This high computational time reduction allows real-time simulations or generating video media. Program summary Program title: OCL-TIDES, OCL-RK Catalogue identifier: AEVW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEVW_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1836 No. of bytes in distributed program, including test data, etc.: 10,617 Distribution format: tar.gz Programming language: C and OpenCL. Computer: Any computer with a CPU or a GPU or a Xeon Phi. Operating system: Linux, MacOS, Windows. Has the code been vectorized or parallelized?: Yes, using the parallelization methods from OpenCL. RAM: Problem dependent Supplementary material: A video with a Fast Lyapunov indicator simulation is available (see Appendix A ). Classification: 4.3, 6.5. External routines: OpenCL version 1.2 Nature of problem: Solution of ODE problems in generic OpenCL environment oriented to large scale independent sets of initial conditions. Solution method: OpenCL parallel integrator based in Runge–Kutta or Taylor Series Method suitable for large sets of independent sets of initial conditions. Both are able to run either on CPU or in a GPU, using the parallelization methods from OpenCL. Running time: Problem dependent, sample tests take less than a minute to run.

Computer-assisted proof of skeletons of periodic orbits

Abstract When obtaining numerically invariants that describe the dynamics of a system, we are never sure about the real existence of that numerical objects. We propose to go further in the numerical search of periodic orbits, by performing a systematic computer-assisted proof of large sets of periodic orbits. First of all, we adapt the periodicity condition to a zero-finding criterion, so that we can apply validated numerical tools. This allows us to apply that criterion to a huge set of initial conditions (numerically calculated), to transform each numerical approach into a rigorous value. We show the figures representing periodic orbits, before and after validation, showing that these techniques allow to give, not only numerically calculated invariants that describe a system ( skeleton of periodic orbits), but a rigorous result beyond its numerical description. Finally, we also show how to obtain a computer-assisted proof of the linear stability of the orbits.