Chetayev instability theorem

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The Chetayev instability theorem for dynamical systems states that if there exists for the system \dot{\textbf{x}} = X(\textbf{x}) a function V(x) such that

  1. in any arbitrarily small neighborhood of the origin there is a region D1 in which V(x) > 0 and on whose boundaries V(x) = 0;
  2. at all points of the region in which V(x) > 0 the total time derivative \dot{V}(\textbf{x}) assumes positive values along every trajectory of \dot{\textbf{x}} = X(\textbf{x})
  3. the origin is a boundary point of D1;

then the trivial solution is unstable.

This theorem is somewhat less restrictive than the Lyapunov instability theorems, since a complete sphere (circle) around the origin for which V and \dot{V} both are of the same sign does not have to be produced..