By Oded Regev
The self-discipline of nonlinear dynamics has built explosively in all parts of physics over the past 20 years. This accomplished primer summarizes the most advancements within the mathematical concept of dynamical platforms, chaos, trend formation and complexity. An creation to mathematical suggestions and methods is given within the first a part of the publication, prior to being utilized to stellar, interstellar, galactic and massive scale complicated phenomena within the Universe. Oded Regev demonstrates the prospective software of principles together with unusual attractors, Poincar? sections, fractals, bifurcations, and intricate spatial styles, to express astrophysical difficulties.
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Extra resources for Chaos and Complexity in Astrophysics
Example text
Unstable ones (unstable periodic orbits) appeared at curve intersections. When the value of the parameter was increased to E = 1/8, still below any possibility of escape from the potential, a surprising result was obtained. 7. There are still some closed curves (around the stable fixed points), but they no longer fill the whole surface. The gaps between the periodic orbits are full of isolated points, indicating ergodic three-dimensional motion. ) strengthens this conclusion. H´enon and Heiles also pointed out that the five little loops around the large closed curve, in the right portion of the figure, belong in fact to one orbit.
This phenomenon is called attraction. In contrast, when D > 0, phase volumes expand and the solutions diverge for large times. This last observation is usually not very useful in understanding the flow without fully solving the equation, but the diagnosis in itself is often important. Of particular importance are systems for which D = 0 exactly. These are the conservative or volume-preserving systems. Liouville’s theorem guarantees that Hamiltonian dynamical systems, like the ones describing the dynamics of particles in classical mechanics, are volume-preserving.
Another manifestation of complexity in nature is that of form, or spatial structure. In this case we are interested in spatial patterns apparent in the system and we try to understand their origin and behaviour. Systems exhibiting spatial patterns cannot, by definition, be homogeneous. They must consist of some spatially varying characteristic fields defining the pattern. A particularly simple example is a system composed of just two different phases, each one being locally uniform. The shape of the boundaries separating such different regions define the pattern in this case.
