Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stability-in-odes]

For questions concerning stability of equilibria and of other solutions of ordinary differential equations and their systems.

Stability of a solution of a differential equation means that a small perturbation of initial data will result in only small (or even vanishing) perturbation of solution at later times. See Stability theory on Wikipedia.

906 questions
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Is there an attractor that cannot be established using a Lyapunov function?

I was wondering whether there exists an ODE system that has a global attractor that cannot be established using a Lyapunov function. That is, is it true that the existence of a Lyapunov function is not a necessary condition for global attraction in…
Frank
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About a theorem on stability of sytems of autonomous ODEs

I have the following definition and theorem are taken from [1], which I like as text book for ODE theory. However, I think that there is a problem below. Definition 8.1.2. Let $C$ be a critical point for the system $X^{\prime}=F(X)$. The point $C$…
bkarpuz
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Problem with stability of non linear second order difference equation

I have the following non linear difference equation: $$ \rho_{{t}}=- \left( -4+ \left( 2+\rho_{{t-2}} \right) \rho_{{t-1}} \right) ^{-1} $$ I found the three equilibrium points, which are: $$ \hat \rho^1 = 1 $$ $$ \hat \rho^2 = -3/2 + 1/2\,\sqrt…
Marco
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stability analysis of an ODE

I need help on how to linearize the following ODE equation so that I am able to do Stability analysis for the equation. Thanks for the help. $\frac{dQ}{dz} = 2aM^{1/2}$ $\frac{dM}{dz} = \frac{QF}{M}$ $\frac{dF}{dz} = bQ$ Where a and b are…
Dereje
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Asymptotic stability of a scalar autonomous equation

What is the proper definition for the Asymptotic stability of a scalar autonomous equation. For eg. $x'=f(x),$ where $x \in \mathbb{R},$ (say) the equilibrium is at a point $x_0.$ So $f(x_0)=0.$ And how do we go and determine whether it is…
user358174
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Critical points of Hamiltonian systems

How to prove that any nondegenerate critical point of a 2D Hamilotonian system is either a saddle or a center ? By definition a critical point of an autonomous system is nondegenerate if the Jacobian evaluated at this point is non-zero. Also system…
user358174