11.4 Gradient systems
Definition 11.4.1. A gradient system is \(x' = -\nabla \Phi (x)\) for a \(C^2\) function \(\Phi \), the potential.
Theorem 11.4.2. For a gradient system:
- (i)
- \(\dot \Phi = -\|\nabla \Phi \|^2 \leq 0\), with equality only at equilibria, which are the critical points of \(\Phi \);
- (ii)
- there are no periodic orbits;
- (iii)
- the Jacobian at an equilibrium is \(-D^2\Phi \), symmetric, so its eigenvalues are real: there are no spirals or centres;
- (iv)
- an isolated local minimum of \(\Phi \) is asymptotically stable, and every bounded orbit tends to the set of equilibria.
Proof. (i) \(\dot \Phi = \nabla \Phi \cdot (-\nabla \Phi )\). (ii) On a periodic orbit \(\Phi \) would return to its initial value while strictly decreasing, unless the orbit were an equilibrium. (iii) The Jacobian of \(-\nabla \Phi \) is \(-D^2\Phi \), a symmetric matrix. (iv) \(V = \Phi - \Phi (x^*)\) is positive definite near an isolated minimum \(x^*\) and \(\dot V < 0\) away from \(x^*\) near it, so Theorem 11.1.2 applies; for bounded orbits, LaSalle’s theorem with \(E = \{\nabla \Phi = 0\}\), the set of equilibria, gives the last statement. □
Gradient flows are continuous-time versions of the method of steepest descent, and the theorem says that they behave as one expects of a ball rolling downhill in thick syrup: it comes to rest in a valley, never cycles, and never spirals.
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