11 Lyapunov Functions, Gradient and Hamiltonian Systems
Linearisation decides stability only at hyperbolic equilibria, and only locally. Lyapunov (1892) found a method that needs neither condition: find a function that decreases along solutions, an abstract energy, and stability follows from its level sets, without solving the equation. The method proves stability where linearisation fails, estimates how large the basin of attraction is, and, through LaSalle’s invariance principle, handles the common case in which the energy only fails to increase. The section ends with the two great classes of systems defined by a scalar function: gradient systems, which descend a potential, and Hamiltonian systems, which conserve one.
Throughout, \(x' = f(x)\) with \(f\) continuously differentiable on an open set \(U\), \(f(0) = 0\) (after a translation), and for a \(C^1\) function \(V\) we write \[ \dot V(x) = \nabla V(x)\cdot f(x) , \] the derivative of \(V\) along solutions: \(\frac {d}{dt}V(\phi _t(x)) = \dot V(\phi _t(x))\).
11.2 Limit sets and LaSalle’s invariance principle
11.3 Basins of attraction and the Lyapunov equation
11.4 Gradient systems
11.5 Hamiltonian systems
11.6 Lyapunov methods in software
11.7 Practice Problems
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