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Suppose we have an autonomous ode $$\dot{x} = f(x)$$ which corresponds to a $\underline{gradient-like}$ dynamical system, $f: \mathbb{R}^n \to \mathbb{R}^n$. Is it true that the set of initial conditions that have in their limit set an unstable fixed point (i.e fixed point whose Jacobian has an eigenvalue with positive real part) is measure zero? Thanks

  • By a gradient system, I assume that you mean that $f$ arises as the gradient of some function $g:{\mathbb R}^n \rightarrow \mathbb R$, ie. $f=\nabla g$. So, is the Lorenz system a gradient system? – Mark McClure Feb 25 '14 at 14:11
  • Yes that's right. Lorenz is not. But this is not my question... For example in Lorenz is not measure zero. I am asking if it is in gradient systems. Thanks –  Feb 25 '14 at 14:59
  • No. Take $\dot x=x$, this is a gradient system with potential $x^2/2$, and for any initial condition its $\alpha$-limit set is the unstable equilibrium $\hat{x}=0$, or am I missing something in your questions? – Artem Mar 05 '14 at 02:44
  • I consider unstable, a fixed point whose Jacobian has an eigenvalue with positive real part (inside parenthesis) –  Mar 05 '14 at 06:09

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