Periodic orbit numerical methods

Numerical Results. - We apply the new methods to the well known almost periodic orbit problem studied by Stiefel and Bettis n. 

A stability analysis of the new method is also presented. Numerical results from its application to well-known periodic orbital problems show the efficiency of the new methods. 

In 40 the authors present a new explicit Runge-Kutta method with algebraic order four, minimum error of the fifth algebraic order (the limit of the error is zero, when the step-size tends to zero), infinite order of dispersion and eighth order of dissipation i.e. they present an optimized explicit Runge-Kutta method of fourth order. The efficiency of the newly developed method is shown through the numerical illustrations of a wide range of methods when these are applied to well-known periodic orbital problems. 

Numerical methods are described for locating resonant quasiperiodic and periodic orbits in the 3D F+H2 reaction with J=0. A number of plots of both types of resonant orbit are presented. This is the first time that resonant orbits have been found for a non-collin-ear reaction. These orbits are then used in the arbitrary trajectory semlclasslcal quantization scheme of DeLeon and Heller. The lowest resonance energy predicted using this procedure is in good agreement with all available quantal and adiabatic semlclasslcal results. 

Numerical Methods for Locating Quaslperiodic and Periodic Orbits... 

Using the numerical method described earlier, a number of periodic resonant orbits have been computec. for 3D One of them, at E=>0.4... 

Numerical methods were described for locating both quaslperlodlc and periodic resonant orbits in the noncollinear F+H2 reaction with J=0. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

Notice that the variation of the semimajor axis depends on both the radial and the torque, but because the orbit can be taken in the two-body problem as planar, there is no dependence on W. Changes in co and are equivalent to orbital precession. All of these may be periodic or secular, depending on the details of 91. For a given disturbing function, this system of equations can be well explored using numerical methods. 

Numerically integrate the Rossler system for a = 0.4, 6=2, c = 4, and obtain a long time series for x(f). Then use the attractor-reconstruction method for various values of the delay and plot (x(f), x(f + t)). Find a value of t for which the reconstructed attractor looks similar to the actual Rossler attractor. How does that T compare to typical orbital periods of the system ... 

Linear dependencies of Gaussian-type orbital basis sets employed in the framework of the HF SCF method for periodic structures, which occur when diffuse basis functions are included in a basis set in an uncontrolled manner, were investigated [468]. The basis sets constructed avoid numerical linear dependences and were optimized for a number of periodic structures. The numerical AO basis sets for solids were generated in [469] by confining atoms within spheres and smoothing the orbitals so that the first and second derivatives go to zero at the boundary. This forms small atomic-like basis sets that can be applied to solid-state problems and are efficient for treating large systems. 

The preceding methods were applied to a number of periodic polymers with a small unit cell. Many of these calculations used different all-valence electron semiempirical crystal-orbital methods (CNDO/2, — INDO, — MINDO, and MNDO CO methods). At this point we shall discuss only some of the not very numerous ab initio vibrational calculations and also an MNDO CO work on fra/is-polyacetylene (PA), because in this investigation disorder in PA due to soliton and polaron formation was also taken into account. ... 

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