Re: Finding Fixed Points for a Nonlinear System of equations
- To: mathgroup at smc.vnet.net
- Subject: [mg42893] Re: [mg42862] Finding Fixed Points for a Nonlinear System of equations
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Fri, 1 Aug 2003 01:26:03 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
In principle, here's what you need to do:
eqns = {vs*(Ki^nexp/(Ki^nexp + Pn[t]^nexp)) -
vm*(M[t]/(Km + M[t])) == 0,
ks*M[t] - Vone*(Pzero[t]/(Kone + Pzero[t])) +
Vtwo*(Pone[t]/(Pone[t] + Ktwo)) == 0,
Vone*(Pzero[t]/(Kone + Pzero[t])) -
Vtwo*(Pone[t]/(Ktwo + Pone[t])) -
Vthree*(Pone[t]/(Kthree + Pone[t])) +
Vfour*(Ptwo[t]/(Kfour + Ptwo[t])) == 0,
Vthree*(Pone[t]/(Kthree + Pone[t])) -
Vfour*(Ptwo[t]/(Kfour + Ptwo[t])) - kone*Ptwo[t] +
ktwo*Pn[t] - vd*(Ptwo[t]/(Kd + Ptwo[t])) == 0,
kone*Ptwo[t] - ktwo*Pn[t] == 0}/.{(f_)[t]-> f} //Simplify
Solve[eqns, {M, Pzero, Pone, Ptwo, Pn}]
But it's not going to work (in any reasonable length of time) unless
you provide values for at least some of the parameters.
The key parameter is nexp. With nexp:=1 and nexp:=2, I was able to make
the following work.
First eliminate M, Pzero, and Pn:
sol1 = First[Solve[eqns[[1]], M]]
sol2 = First[Solve[eqns[[2]], Pzero]]
sol5 = First[Solve[eqns[[5]], Pn]]
neweqns= eqns //. Flatten[{sol1, sol2, sol5}] // Simplify
Then solve the reduced system:
Solve[neweqns, {Pone, Ptwo}]
This approach *might* work with other values of nexp, but I don't think
you're even going to like what you see of the solution with nexp:=1.
-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis
On Thursday, July 31, 2003, at 10:19 AM, Katherine Gurdziel wrote:
> This is what they look like:
>
> vs*((Ki^nexp)/((Ki^nexp) + (Pn[t]^nexp))) - vm*(M[t]/(Km + M[t])) ==
> M'[t],
>
> ks*M[t] - Vone*(Pzero[t]/(Kone + Pzero[t])) + Vtwo*(Pone[t]/(Pone[t] +
> Ktwo)) == Pzero'[t],
>
> Vone*(Pzero[t]/(Kone + Pzero[t])) - Vtwo*(Pone[t]/(Ktwo + Pone[t])) -
> Vthree*(Pone[t]/(Kthree + Pone[t])) + Vfour*(Ptwo[t]/(Kfour +
> Ptwo[t]))
> ==
> Pone'[t],
>
> Vthree*(Pone[t]/(Kthree + Pone[t])) - Vfour*(Ptwo[t]/(Kfour +
> Ptwo[t])) -
> kone*Ptwo[t] + ktwo*Pn[t] - vd*(Ptwo[t]/(Kd + Ptwo[t])) ==
> Ptwo'[t],
>
> kone*Ptwo[t] - ktwo*Pn[t] == Pn'[t]
>
>
> Thank you for your help.
> Katherine
>
> -----Original Message-----
> From: Selwyn Hollis [mailto:selwynh at earthlink.net]
To: mathgroup at smc.vnet.net
> Sent: Thursday, July 31, 2003 10:16 AM
> To: Katherine Gurdziel
> Cc: mathgroup at smc.vnet.net
> Subject: [mg42893] Re: [mg42862] Finding Fixed Points for a Nonlinear System of
> equations
>
>
> It depends on the form of functions involved. If they are polynomials,
> use Solve or NSolve. Otherwise you may need FindRoot.
>
> -----
> Selwyn Hollis
> http://www.math.armstrong.edu/faculty/hollis
>
>
> On Thursday, July 31, 2003, at 08:02 AM, Katherine Gurdziel wrote:
>
>> Sorry I wasn't clear. I am looking for all of the solutions where the
>> system is equal to zero.
>>
>> [0 [equation for the derivatives dP1
>> 0 dP2
>> 0 = dP3
>> 0 dP4
>> 0] dP5]
>>
>> I hope this makes things clearer.
>>
>> Katherine
>>
>> -----Original Message-----
>> From: Selwyn Hollis [mailto:selwynh at earthlink.net]
To: mathgroup at smc.vnet.net
>> Sent: Wednesday, July 30, 2003 8:27 PM
>> To: Katherine Gurdziel
>> Cc: mathgroup at smc.vnet.net
>> Subject: [mg42893] Re: [mg42862] Finding Fixed Points for a Nonlinear System of
>> equations
>>
>>
>> Katherine,
>>
>> I think you need to be more specific about what you want to do. It is
>> not clear what you mean by "the fixed points" of a system of
>> differential equations. Perhaps you mean the equilibrium/critical
>> points??
>>
>> -----
>> Selwyn Hollis
>> http://www.math.armstrong.edu/faculty/hollis
>>
>>
>> On Wednesday, July 30, 2003, at 07:31 PM, Katherine Gurdziel wrote:
>>
>>> I am trying to isolate the fixed points for five differential
>>> equations that
>>> are dependent on each other. I have experimented with using NDSolve
>>> but am
>>> having problems finding the fixed points. Specifically, I need to be
>>> able
>>> to solve the system without setting initial conditions and need to
>>> find all
>>> of the fixed points.
>>> Could you make some suggestions about an approach that I could try to
>>> solve
>>> this problem?
>>>
>>> Thank you very much.
>>>
>>> Katherine
>>>
>>>
>>>
>>
>>
>>
>>
>
>
>
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