Dynamical systems(1,9,10) as a field of study have been around since the time of Newton due to their great importance in the sciences. Only in rare instances can such systems be solved algebraically, with linear (time independent) systems and some Hamiltonian systems as exceptions. Usually we need computers to find the solution.
A dynamical system is usually presented in one of two ways:
The questions to be answered using dynamical systems are: What is the behavior of the solution as time tends to infinity? Does the solution converge to a constant (equilibrium/steady state) solution, a periodic one, or a chaotic one, and what do these outcomes depend on? How stable are the dynamics? What is turbulence? Can we define some average behavior?
A typical class in dynamical systems will devote itself to a few of these topics:
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