# Questions tagged [lyapunov-functions]

Lyapunov functions are scalar fields that may be used to prove the stability of an equilibrium point of an ODE.

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### Counterexample of Lyapunov' Stability Theorem for equilbrium

I know that thanks to Lyapunov' Stability Theorem one can study the stability of an equilibrium by finding a Lyapunov function associated to that equilibrium. But, is the converse true? Is it true ...
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### Inequality manipulation for a vector norm

How can I manipulate the following inequality to reach from $$\dot{V}\leq -4x_1^2 +4x_1x_2 -2x_2^2$$ to $$\dot{V}\leq -(3-\sqrt{5}) \|x\|^2$$ where $x=[x_1 \;x_2]^T$ is a 2D vector and $\|x\|$ is ...
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### How is Grönwall's inequality applied here?

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ Now, let $(X_t)_{t\ge0}$ be the unique strong ...
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### If the solutions to Lorentz equation diverge exponentially, how can they be confined in a strange attractor? [duplicate]

In the book of Chaosbook, at the beginning of chapter 6, it is given that [...]if the attractor is strange, any two trajectories $x(t) = f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ ...
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### How to solve an ODE of this form $\dot{P}(t)=A^TP(t)+P(t)A$?

Background If $\dot{x}(t)=A \,x(t)$, then we know the solution is $x(t)=e^{At}x(0)$. Question Now let $\dot{P}(t)=A^TP(t)+P(t)A$, what is $P(t)$? Attempt If we take $P(t)$ as common factor (I'm ...
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