Non-linear dynamics approach to a manifold of a saddle point using power series

[iratxo_flores]

Homework Statement

Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos
I need to solve a problem that is similar to problem 6.1.14
Basically it consist in the following

You have a saddle node at (T_s,Z_s) which is (1,1). Consider curves passing through this saddle point :

Z= Z_s +h(T-T_s)

where h(s) is a power series:

h(s)= a₁s+a₂s²+a₃s³+...

Find the coefficients for a₁, a₂ and a₃+ for both stable and unstable manifolds of the saddle node at (T_s,Z_s)

Homework Equations

I do know the following about the Unstable manifold (0,2) and (1,1) and (5,0) and for the stable manifold (0,0) and (1,1). all (T,Z) I am supposed to use the equation of above and figure out the coefficients.

The Attempt at a Solution

I already try to substitute the equation with the values that i know of but I am 1 equation short

any ideas?

 

[HallsofIvy]

Homework Statement

Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos
I need to solve a problem that is similar to problem 6.1.14
Basically it consist in the following

You have a saddle node at (T_s,Z_s) which is (1,1). Consider curves passing through this saddle point :

Z= Z_s +h(T-T_s)

where h(s) is a power series:

h(s)= a₁s+a₂s²+a₃s³+...

Find the coefficients for a₁, a₂ and a₃+ for both stable and unstable manifolds of the saddle node at (T_s,Z_s)

Homework Equations

I do know the following about the Unstable manifold (0,2) and (1,1) and (5,0) and for the stable manifold (0,0) and (1,1). all (T,Z) I am supposed to use the equation of above and figure out the coefficients.

I have no idea what you mean by this! For the unstable manifold, you "know" (0, 2), (1, 1), and (5, 0). What do you mean you "know" them? Are they points? vectors?

The Attempt at a Solution

I already try to substitute the equation with the values that i know of but I am 1 equation short

any ideas?

 

[iratxo_flores]

I have no idea what you mean by this! For the unstable manifold, you "know" (0, 2), (1, 1), and (5, 0). What do you mean you "know" them? Are they points? vectors?

they are points

http://img502.imageshack.us/img502/3485/manifold.png

thats a sketch of the overall qualitative behaviour of the system, (0,2) and (5,0) are stable nodes and (0,0) is unstable. Of course i don't know the exact function of the manifolds, but i think i can make an approximation by using Z=Z_s+h(T-T_s), which are the curves that pass through the saddle node (1,1)What I am being asked to answer, and don't know how to do.. is to determine the coefficients for both the unstable and stable manifolds,

 

[iratxo_flores]

any ideas?