a [[discrete time]] dynamical system is a space $X$ paired with a map $T : X \to X$ (also denoted $f$) $ x_{t+1} := T(x_{t}) $ We seek to understand all [[orbit]]s of the system. see [[differential equation]]s for continuous time. also can consider discrete time [[stochastic process]]es eg computer [[interpret]]ers; [[Turing machine]]s # basics [[relation|function]] [[continuous|continuity]] - [[homeomorphisms and diffeomorphisms]] [[linear difference equation]]s [[orbit]]s - [[fixed and periodic points]] - [[attractors and repellers]] - [[hyperbolicity|the derivative of the transition map at a periodic point determines whether it is attracting or repelling]] [[phase transition|bifurcation / phase transition]] [[population dynamics|logistic / quadratic map / population dynamics]] [[topological conjugacy]] [[chaos]] [[continuous optimization]] [[structural stability]] ## examples [[Newtons method]] # sources [[1999BenderOrszagAdvancedMathematicalMethods]] ch 2