mathematics//dynamical systems
Dynamical systems are mathematical models of a state that evolves in time under a local law of change, used to work out how that law becomes collective behavior and how that behavior can be observed, modified, or used to organize data. They are the mathematical machinery under systems theory, which studies wholes and emergence: a state, a rate law and what follows from them.
Dynamical systems are mathematical models of a state that evolves in time under a local law of change, used to work out how that law becomes collective behavior and how that behavior can be observed, modified, or used to organize data. They are the mathematical machinery under systems theory, which studies wholes and emergence: a state, a rate law and what follows from them.
A model states a rule for the rate of change of a state, x˙=f(x,u,t;θ)\dot x=f(x,u,t;\theta)x˙=f(x,u,t;θ). Everything else (trajectories, equilibria, modes, controllers) is derived from that rule plus the hypotheses around it; control adds a chosen input uuu, and the plant, the controller and the closed loop are then three different objects with three different spectra. See model and representation.
The bridge from one tank to a network: a balance gives the rate, a proportional loss gives exponential decay, several coupled rates give coupling and feedback, and a delayed response needs its own vocabulary (delay and lag).
What must be remembered to continue the evolution is the sufficient state. Markov chains, state-space control (controllability and observability) and graph diffusion all rest on that idea. Near an equilibrium the rule is replaced by its linearization, a Lyapunov function certifies where imbalance is lost, and discretization separates the model from the algorithm that simulates it.
The rule itself, how it is written from mechanisms and what it needs to become a concrete trajectory, is the subject of differential equations: time as the parameter of the path, the initial condition that selects one solution, the constraints a field must keep tangent, and the split between ODE and PDE. Whether the field is one global pattern or deforms with position is linear field; keeping only its slow modes is model reduction.
Notation used across these notes: FFF is the dynamics matrix in x˙=Fx+Bu\dot x=Fx+Bux˙=Fx+Bu, MMM nodal storage, WWW graph weights, DDD degrees, LLL the Laplacian, PPP a Markov transition matrix and Σ\SigmaΣ a covariance. The usual x˙=Ax+Bu\dot x=Ax+Bux˙=Ax+Bu is equivalent; FFF is used so it is never confused with a tank cross-section AAA.
Distributions are columns and transition matrices are row-stochastic, so pk+1=PTpkp_{k+1}=P^{\mathsf T}p_kpk+1=PTpk.