# Toronto Math Forum

## MAT244-2014F => MAT244 Math--Lectures => Topic started by: Chenxi Lai on November 25, 2014, 11:40:15 PM

Title: Type and Stability of Origin
Post by: 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?
Title: Re: Type and Stability of Origin
Post by: Victor Ivrii on November 26, 2014, 05:48:55 AM
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?

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
Title: Re: Type and Stability of Origin
Post by: Kelly Yang on December 02, 2014, 12:45:23 PM
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)