# Geometrical Dynamics of Complex Systems: A Unified Modelling by Vladimir G. Ivancevic

By Vladimir G. Ivancevic

This quantity provides a complete creation into rigorous geometrical dynamics of complicated structures of varied natures. by way of "complex systems", during this publication are intended high-dimensional nonlinear structures, which are (but now not inevitably are) adaptive. This monograph proposes a unified geometrical method of dynamics of complicated platforms of varied types: engineering, actual, biophysical, psychophysical, sociophysical, econophysical, and so forth. As their names recommend, some of these multi-input multi-output (MIMO) structures have anything in universal: the underlying physics. utilizing refined equipment composed of differential geometry, topology and direction integrals, this publication proposes a unified method of complicated dynamics – of predictive energy a lot more than the at present renowned "soft-science" method of advanced platforms. the most target of this ebook is to teach that high-dimensional nonlinear structures in "real existence" could be modeled and analyzed utilizing rigorous arithmetic, which permits their whole predictability and controllability, as though they have been linear platforms. The e-book has chapters and an appendix. the 1st bankruptcy develops the geometrical equipment in either an intuitive and rigorous demeanour. the second one bankruptcy applies this geometrical equipment to a few examples of advanced platforms, together with mechanical, actual, regulate, biomechanical, robot, neurodynamical and psycho-social-economical platforms. The appendix provides all of the helpful historical past for entire interpreting of this ebook.

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**Sample text**

D ˙ dt (X + Y ) = X d ˙ dt (f X) = f X + d ˙ dt (X · Y ) = XY + Y˙ ; ˙ and f X; + X Y˙ . A geodesic in M is a parameterized curve γ : I → M whose acceleration γ¨ ⊥ is everywhere orthogonal to M ; that is, γ¨ (t) ∈ Mα(t) for all t ∈ I ⊂ R. Thus a geodesic is a curve in M which always goes ‘straight ahead’ in the surface. Its acceleration serves only to keep it in the surface. It has no component of acceleration tangent to the surface. Therefore, it also has a constant speed γ(t). ˙ Let v ∈ Mm be a vector on M .

This space is called the ﬁbre of the cotangent bundle over the point m ∈ M . , the following diagram commutes: T ∗M T ∗ϕ π ∗M ✲ T ∗N π ∗N ❄ M ϕ ❄ ✲N All cotangent bundles and their cotangent maps form the category T ∗ B. The category T ∗ B is the natural stage for Hamiltonian dynamics. Now, we can formulate the dual version of the global chain rule. If ϕ : M → N and ψ : N → P are two smooth maps, then we have T ∗ (ψ ◦ϕ) = T ∗ ψ ◦T ∗ ϕ. 5 Tensor Fields and Bundles of a Smooth Manifold A tensor bundle T associated to a smooth n−manifold M is deﬁned as a tensor product of tangent and cotangent bundles: q T = p T ∗M ⊗ q times p times T M = T M ⊗ ...

S = {dz − ydx}). Now, a simple EDS is a triple (S, Ω, M ), where S is a system on M , and Ω is an independence condition: either a decomposable k−form or a system of k−forms on M . An EDS is a list of simple EDS objects where the various coframings are all disjoint. 2 Smooth Manifolds 33 An integral element of an exterior system (S, Ω, M ) is a subspace P ⊂ Tm M of the tangent space at some point m ∈ M such that all forms in S vanish when evaluated on vectors from P . Alternatively, an integral element ∗ P ⊂ Tm M can be represented by its annihilator P ⊥ ⊂ Tm M , comprising those 1−forms at m which annul every vector in P .