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Per Wikipedia:

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic; in other words, for a given time interval only one future state follows from the current state however, some systems are stochastic, in that random events also affect the evolution of the state variables.

Is there any result on dynamical system (i.e. solving dynamical equations, finding asymptotic states to a system, dynamics in chaotic system) with complexity theory, hardness for finding a solution, etc.?

Thanks

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This is a well-researched area. For a representative result, see Kawamura's proof that solving ODEs is difficult.

A different line of works studies the hardness of computing Nash equilibria and related problems. See for example the recent breakthrough of Bitansky, Paneth and Rosen, who base hardness of cryptographic assumptions; earlier work based it on complexity theoretic assumptions.

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