Dynamical Systems Theorycharacterize the asymptotic behavior, as t → ±∞ of every solution, or, of almost every solution, or, of almost every solution of almost every f , or, • • • ω-limit set of solution x(t): {y : y = lim m→∞ x(t m ), t m → ∞} Goal Reached for n = 1, 2; Little known for n ≥ 3. Restrict to special systems: Hamiltonian, Gradient,etc. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, 2012 3 / 31Dynamical Systems Theorycharacterize the asymptotic behavior, as t → ±∞ of every solution, or, of almost every solution, or, of almost every solution of almost every f , or, • • • ω-limit set of solution x(t): {y : y = lim m→∞ x(t m ), t m → ∞} Goal Reached for n = 1, 2; Little known for n ≥ 3. Restrict to special systems: Hamiltonian, Gradient,etc.