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Dynamical Systems (Definition)

The mathematical study of systems that evolve in time according to a fixed rule.
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The statement of the theorem

Let MM be a smooth manifold (the phase space) and let XX be a smooth vector field on MM. A dynamical system is formally defined by the flow ϕt:U→M\phi_t: U \to M, where U⊂MU \subset M is an open set and t∈Rt \in \mathbb{R} is time, such that ϕt\phi_t is the unique solution to the initial value problem (IVP) generated by XX: \n\n\frac{d}{dt} \big\|_t \big\phi_t(x_0) = X(\big\phi_t(x_0\big)\), \text{ with } \phi_0(x_0) = x_0 \n\nThis flow ϕt\phi_t maps points in UU to points in MM and describes the evolution of the state x(t)=ϕt(x0)x(t) = \phi_t(x_0) over time tt. The system is autonomous if the vector field XX does not explicitly depend on tt.