## Stability in autonomous systems |

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appendix asymptotically stable relative attractor relative autonomous system bounded neighborhood characteristic roots Consider the system constant continuously differentiable Corollary Discuss the stability displacement dU/dy equilibrium is unstable equilibrium point example Figure I.3.a Figure II.2 Figure l.2.b following theorem form V(x function U(x globally asymptotically stable Hamiltonian homogeneous functions implies inf x-m J(sx)x ds KcA(M kinetic energy largest invariant set Let V(x Lyapunov function Lyapunov stability Lyapunov's Direct Method Matematicas Mathematics matrix MCRn Method of Lyapunov negative definite negative semi-definite neutral equilibrium Non-linear open set Ordinary Differential Equations Particle may oscillate pendulum periodic orbit phase plane positive definite positive semi-definite positive semi-orbit potential energy Problem Proof real numbers relative to G sense of Lyapunov Stability of Motion Stability Theory stable equilibrium stable focus stable set system x Theorem III.5 total energy turning points uniform attractor V(xQ w-limit set w(y+ x'Qx xg(x xoe G(M xQeG y+(xQ