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For N-dimensional dynamical systems, if we know there only exist one stable equilibrium (but whether it is global or local is not yet proved), can we infer, the stable equilibrium is globally stable?

Thank you in advance for any response!

chloe
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  • You will need extra conditions on the to infer the global stability such as the existence of a global Lyapunov function or that the system is contractive. – KBS May 13 '22 at 10:51
  • @KBS Thanks a lot! About inferring global stability via contractive property, could you recommend some materials? – chloe May 13 '22 at 13:16
  • @KBS And I think for gradient system, global stability can be inferred by the unique existence of stable equilibrium point. – chloe May 13 '22 at 13:17
  • Gradient systems are quite restrictive in general so unless you specifically work with them, this will not be very useful. For contraction analysis, you can check the seminal works by Lohmiller and Slotine. And more recently the works on incremental stability and p-dominance analysis (see e.g. Sepulchre and Forni). – KBS May 13 '22 at 13:24
  • @KBS Thanks a lot for sharing! – chloe May 13 '22 at 14:24

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Nope. There is a counterexample in this answer: $$ \dot x = x \, \frac{3 x^2 y^2 - 1}{x^2 y^2 + 1} ,\qquad \dot y = -y . $$ The only equilibrium is $(x,y)=(0,0)$, and it's locally but not globally stable. (See that other answer for details.)

Hans Lundmark
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