Consider the ellipsoid V(x, y,z) = 2rx2 + ?y2 + ?(z ? 3r) 2 = c > 0. (a) Calculate dV dt = ?V ?x dx dt + ?V ?y dy dt + ?
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Consider the ellipsoid V(x, y,z) = 2rx2 + ?y2 + ?(z ? 3r) 2 = c > 0. (a) Calculate dV dt = ?V ?x dx dt + ?V ?y dy dt + ?V ?z dz dt along trajectories of the Lorenz equations (1). (b) Determine a sufficient condition on c so that every trajectory crossing V(x, y, z) = c is directed inward. (c) Evaluate the condition found in part (b) for the case ? = 10, b = 8 3 , r = 28. In each of Problems 5 through 7, carry out the indicated investigations of the Lorenz equations.
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