Ivancevic_Applied-Diff-Geom

398 Applied Differential Geometry: A Modern IntroductionBesides the σ−Hamiltonian phase–flow φ σt(3.212), including N individualrandom–phase trajectories, we can also define (see [Elworthy (1982)])an average or mean 〈σ〉 − Hamiltonian flow 〈φ〉 σt〈φ〉 σt: G 1 × I ∗ M → I ∗ M : (〈p(0)〉 , 〈q(0)〉) ↦→ (〈p(t)〉 , 〈q(t)〉),(〈φ〉 σt◦ 〈φ〉 σs= 〈φ〉 σt+s, 〈φ〉 σ0= identity),which stochastically corresponds to the trajectory of the center ofmass in the human–like dynamics, approximatively lumbo–sacral spinalSO(3)−joint.The necessary conditions for existence of a unique non–anticipating solutionof the σ−Hamiltonian biodynamical system in a fixed time intervalare Lipschitz condition and growth condition (see [Elworthy (1982);Mayer (1981)]). For constructing an approximate solution a simple iterativeCauchy–Euler procedure could be used to calculate (qk+1 i , pk+1 i ) from theknowledge of (qk i , pk i ) on the mesh of time points tk , k = 1, . . . , s, by addingdiscrete δ–Hamiltonian drift–terms A i (qk i )∆tk and A i (p k i )∆tk ,as well as astochastic term B ij (qi k, tk )∆W j k .σ−Hamiltonian biodynamical system (3.210–3.211), with its σ−Hamiltonian phase–flow φ σt(3.212), i.e., the functorStoch[Can], represents our second, continual–discrete stochastic model forthe biodynamics functor machine F[S] with the nonlinear system behaviorF[r]. In the next section we generalize this model once more to includefuzzy SN.Fuzzy–Stochastic–Lie–Hamiltonian FunctorGenerally, a fuzzy differential equation model (FDE–model, for short)is a symbolic description expressing a state of incomplete knowledge of thecontinuous world, and is thus an abstraction of an infinite set of ODEsmodels. Qualitative simulation (see [Berleant and Kuipers (1992)]) predictsthe set of possible behaviors consistent with a FDE model and aninitial state. Specifically, as a FDE we consider an ordinary deterministic(i.e., crisp) differential equation (CDE) in which some of the parameters(i.e., coefficients) or initial conditions are fuzzy numbers, i.e., uncertain andrepresented in a possibilistic form. As a solution of a FDE we consider atime evolution of a fuzzy region of uncertainty in the system’s phase–space,which corresponds to its the possibility distribution.Recall that a fuzzy number is formally defined as a convex, normalizedfuzzy set [Dubois and Prade (1980); Cox (1992); Cox (1994)]. The concept