Nonperiodic Flow in the Numerical Integration of a Nonlinear Differential Equation of Fluid Dynamics

Viscous incompressible fluid flow along a flat plate is modeled by the Navier-Stokes equations with appropriate boundary conditions. A series solution is assumed and a set of three nonlinear ordinary differential equations is derived by truncating the series. The Reynolds number appears in these three equations as a parameter. These equations are solved by numerical integration. We show that these solutions exhibit qualitatively different behavior for different values of the Reynolds number of the fluid. The various modes include an asymptotic approach to a time-independent state, laminar (periodic) flow, and turbulence. We give several computer-generated pictures of the various modes

Routh-Hurwitz Criterion in the Examination of Eigenvalues of a System of Nonlinear Ordinary Differential Equations

In stability analysis of nonlinear systems, the character of the eigenvalues of the Jacobian matrix (i.e., whether the real part is positive, negative, or zero) is needed, while the actual value of the eigenvalue is not required. We present a simple algebraic procedure, based on the Routh-Hurwitz criterion, for determining the character of the eigenvalues without the need for evaluating the eigenvalues explicitly. This procedure is illustrated for a system of nonlinear ordinary differential equations we have studied previously. This procedure is simple enough to be used in computer code, and, more importantly, makes the analysis possible even for those cases where the secular equation cannot be solved

Periodic and Aperiodic Orbits in the Hamiltonian Formulation of a Model Magnetic System

Classical dynamics and quantum dynamics have influenced each other since the idea of a quantum mechanics originated. Classical dynamics came first, so its influence on quantum theory almost goes without saying. Quantum mechanics grew out of classical mechanics. The converse influence is often referred to in the abstract, but rarely in detail. One finds statements [1] roughly to the effect that the classical theory was developed more fully in order to use it to further elucidate the corresponding quantum dynamics. But specific examples of classical calculations, which were suggested by quantum results or ideas, are not common. One of these rare examples [2] is ‘rotators’ or ‘classical spins,’ and that is the subject considered here. This study makes contact with optimization theory at two places. First, the spin problem is initially expressed as an application of Hamiltonian dynamics; that is, it is simply an explicit particular example of the principle of least action. In the course of solution, we uncover two qualitatively different types of behavior, viz ‘regular’ and ‘chaotic,’ whose occurrence depends on the value of a (control) parameter. The chaotic solutions, moreover, are not equally chaotic; there is a more-or-less smooth progression into and back out of chaos as the parameter changes. The second contact with control theory then is a question; can the ‘intensity’ of the chaos be quantified, and if so, is there a value of the parameter for which the system is maximally chaotic? The paper has four sections. Section 2 is a review of classical dynamics, including a description of numerical techniques for distinguishing regular from chaotic motion. Section 3 describes how the quantum mechanical form of a classical dynamics problem is produced. Section 4 discusses the exchange-type interactions relevant to classical spins and presents results of numerical integration for one specific such model

Regular and Chaotic Time Evolution in Spin Clusters

We calculate spin-autocorrelation functions (as time averages over chaotic trajectories) and their intensity spectra for clusters of two classical spins, interacting via a nonintegrable Hamiltonian. The long-time behaviour observed includes both power-law decay and persistent oscillatory components, resulting in an intensity spectrum with power-law singularities and discrete lines