Separable Multiply Periodic Systems

Our next problem is to extend the results found for a system with one degree of freedom to systems with several degrees of freedom. 

In the ease of absolutely general systems there is no object in introducing angle and action variables, since these are associated with the existence of periodic properties. 

We consider first the simple case in which the Hamiltonian function of the system resolves into a sum of terms, each of which contains only one pair of variables qk, pk  

The Hamilton-Jacobi equation is solved by separation of the variables on putting 

Example Spatial Oscillator.—A massive particle is restrained by any set of forces in a position of stable equilibrium (t.g. a light atom in a molecule otherwise consisting of heavy, and therefore relatively immovable atoms). The potential eneigy is then, for small displacement, a positive definite quadratic function of the displacement components. The axes of the co-ordinate system (x, y, z) can always be chosen to lie along the principal axes of the ellipsoid corresponding to this quadratic form. The Hamiltonian function is then 

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While the coexistence between two limit cycles or between a limit cycle and a stable steady state is also shared by the two-variable models of fig. 12.1b and c, new modes of complex dynamic behaviour arise because of the presence of a third variable in the multiply regulated system. The coexistence between three simultaneously stable limit cycles, i.e. trirhythmicity, is the first of these. Moreover, the interaction between two instability-generating mechanisms allows the appearance of complex periodic oscillations, of the bursting type, as well as chaos. The system also displays the property of final state sensitivity (Grebogi et ai, 1983a) when two stable limit cycles are separated by a regime of unstable chaos. 

Sprinklers installed in duct systems shall be located at 12-foot intervals, and at changes in direction. The sprinklers shall be hydraulically designed to provide 0.5 GPM over an area derived by multiplying the distance between the sprinklers in a horizontal duct by the width of the duct. A separate indicating control valve shall be provided for sprinklers installed in ductwork. Drainage shall be provided to remove all sprinkler water discharged in ductwork. The sprinklers shall be accessible for periodic inspection and maintenance. (UMC 609.7, NFPA 318 2-1.2.6.1, NFPA 318 2-1.2.6.2, and NFPA 318 2-1.2.6.5). 

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