Find the approximate linear ODE system

  • #1

Main Question or Discussion Point

dx/dt = x-y^2 dy/dt= x^2 -xy -2x
For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it.

I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
 

Answers and Replies

  • #2
S.G. Janssens
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Just calculate the Jacobian matrices of the vector field at the three critical points. Your approximate linear ODE system at the critical point ##(x_c,y_c)##. is going to be of the form ##\dot{u}(t) = A(x_c,y_c)u(t)## where ##A(x_c,y_c)## is the Jacobian matrix.

To check that this system is valid in a small neighborhood, verify that ##A(x_c,y_c)## has no spectrum on the imaginary axis. Which theorem do you use here?
 
  • #3
I'm not sure about the theorem
 
  • #4
S.G. Janssens
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I'm not sure about the theorem
It's the Hartman–Grobman theorem. You should consider looking it up, it's worthwhile.
 
  • #5
HallsofIvy
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To "linearize" an equation simply means to replace any non-linear function by a linear approximation. But the only linear approximations to [itex]x^2[/itex] and [itex]xy[/itex] are "0". At (0, 0), dx/dt= x- y^2 linearizes to dx/dt= x and dy/dt= x^2- xy- 2x to dy/dt= -2x.

About (4, 2), let u= x- 4 and v= y- 2 so that x= u+ 4, y= v+ 2, dx/dt= du/dt, and dy/dt= dv/dt. The equations become du/dt= u+4- (v+2)^2= u+ 4- v^2- 4v- 4 which linearizes to du/dt= u- 4v and dv/dt= (u+ 4)^2- (u+ 4)(v+ 2)- 2(u+ 4)= u^2+ 8u+ 16- uv- 4v- 2u- 8- 2u- 8= u^2- uv+ 4u- 6v- uv which linearizes to dv/dt= 4u- 6v.
 
  • #6
S.G. Janssens
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To "linearize" an equation simply means to replace any non-linear function by a linear approximation.
Why (and when, and when not) the linearisation says something meaningful about the original system in a neighborhood of the critical point, is less simple, though.
 
  • #7
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dx/dt = x-y^2 dy/dt= x^2 -xy -2x
For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it.

I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
(4, -2) is NOT a critical point.

Also, you posted a question in another forum section about the critical points of this system. Your other question was posted in the right section (Calculus & Beyond under Homework & Coursework Questions). Please take care to post homework questions there, not here in the technical math sections.

Also, don't post essentially the same question in multiple forum sections.
 

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