Discrete dynamical systems in solving H-equations

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Virginia Tech


Three discrete dynamical models are used to solve the Chandrasekhar H-equation with a positive or negative characteristic function. Two of them produce series of continuous functions which converge to the solution of the H-equation. An iteration model of the nth approximation for the H-equation is discussed. This is a nonlinear n-dimensional dynamical system. We study not only the solutions of the nth approximation for the H-equation but also the mathematical structure and behavior of the orbits with respect to the parameter function, i.e. characteristic function. The dynamical system is controlled by a manifold. For n=2, stability of the fixed points is studied. The stable and unstable manifolds passing through the hyperbolically fixed point are obtained. Globally, the bounded orbits region is given. For parameter c in some region a periodic orbit of one dimension will cause periodic orbits in the higher dimensional system. For changing parameter c, the bifurcation points are discussed. For c ∈ (-5.6049, 1] the system has a series of double bifurcation points. For c ∈ (-8, -5.6049] chaos appears. For c in a window contained the chaos region, a new bifurcation phenomenon is found. For c ≤ -7 any periodic orbits appear. For c in the chaos region the behavior of attractor is discussed. Chaos occurs in the n-dimensional dynamical system.



Chandrasekhar H-equation