MAT244-2014F > MAT244 Math--Lectures

Type and Stability of Origin

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Chenxi Lai:
Hi Pro,
1)What's the difference between an asymptotic stable point and stable point?
2)What is an intermediate point?
3)Why the type of origin sometimes depends on the linearization of the system of equation?

Victor Ivrii:

--- Quote from: Chenxi Lai on November 25, 2014, 11:40:15 PM ---Hi Pro,
1)What's the difference between an asymptotic stable point and stable point?
2)What is an intermediate point?
3)Why the type of origin sometimes depends on the linearization of the system of equation?

--- End quote ---

Hi Bro,
1) Stationary point $\mathbf{x}_0$ is stable if for arbitrary small vicinity $U$ of it there is such small vicinity $V$ such that every solution $\mathbf{x}(t)$ with $\mathbf{x}(0)\in V$ remains in $U$ for all $t>0$.

Stationary point $\mathbf{x}_0$ is asymptotically stable if it is stable and  every solution $\mathbf{x}(t)$ with $\mathbf{x}(0)\in V$ tends to $\mathbf{x}_0$ as $t\to +\infty$.

2) No idea where you got this

3) No idea what do you mean

Kelly Yang:
Hi Chenxi,

For question 2, I think you mean 'Indeterminate'. If I recall correctly, it occurs when the stability cannot be determined in the Locally Linear System, when the eigenvalues are complex with zero real value (i.e. +- i b)