MHB What Determines the Local Extrema in Dynamical Systems?

  • Thread starter Thread starter mt91
  • Start date Start date
AI Thread Summary
The discussion focuses on determining local extrema in dynamical systems, particularly through the analysis of a separable differential equation. Key points include identifying steady states at u = -a, 0, and 1, with stability assessments indicating that u = -a and u = 1 are local maxima, while u = 0 is a local minimum. The parameter 'a' is crucial, with different cases (a < 1, a = 1, a > 1) affecting the stability and behavior of the system. The integration of the equation using partial fractions is also highlighted as a method for analysis. Overall, the conversation emphasizes the importance of understanding stability and equilibrium points in dynamical systems.
mt91
Messages
13
Reaction score
0
1596153990630.png


Any help would be appreciated
 
Mathematics news on Phys.org
what do you need help with? Do you know what "steady state" means? Do you know what stability is? An interesting, non-mathematical, phrase here is "biological relevance". What do you think that means? There is the parameter, a, that is undefined. You might want to consider the cases a<1, a= 1, and a> 1 separately.

Do you know what a "separable differential equation" is? This equation is "separable"- it can be written $\frac{du}{u(1- u)(a+ u)}= dt$. The left side can be integrated using "partial fractions".
 
Last edited:
Country Boy said:
what do you need help with? Do you know what "steady state" means? Do you know what stability is? An interesting, non-mathematical, phrase here is "biological relevance". What do you think that means? There is the parameter, a, that is undefined. You might want to consider the cases a<1, a= 1, and a> 1 separately.

Do you know what a "separable differential equation" is? This equation is "separable"- it can be written $\frac{du}{u(1- u)(a+ u)}= dt$. The right side can be integrated using "partial fractions".
I got 3 steady states, as u=0, u=1 and u=-a?

and then by plotting u(1-u)(a+u), indicated from the graph that u=-a is stable, u=0 is unstable and u=1 is stable. Not sure what you mean by looking at the cases for a, so are you ok to explain that?
 
That was because I had misread the problem and was looking at u= -1, 0, and -a!

Yes, the equilibrium points are at u= -a, 0, and 1. We can write u'= (u+ a)u(1-u)= (-1)(u+ a)u(u- 1). if u< -a then all four of those are negative so the product, and so u', is positive. u is increasing up to u(-a). If -a< u< 0 then -1, u, and u-1 are negative but u+ a is positive so the product, and so u', is negative. u goes down from u(-a) to u(0). If 0< u< 1, both u+ a and u are positive while -1 and u are negative so the product, and so u', is positive. u goes up to u(1). Finally, for u> 1, all except -1 are negative so u' is negative. u goes down from u(1).

That is, u goes up to u(-a) then down after that so u(-a) is a local maximum. u goes down to u(0) then up so u(0) is a local minimum. u goes up to u(1) then down so u(1) is a local maximum.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
6
Views
4K
Replies
3
Views
1K
Replies
1
Views
2K
Replies
2
Views
5K
Replies
2
Views
2K
Replies
9
Views
2K
Back
Top