How to Quickly Converge to a Limit Cycle in Higher Order Systems?

[bor0000]

Help. If anybody knows them beyond introductory level...

i have a higher order system with a single fixed point. i chose such parameters that the fixed point is unstable. so it should either oscillate to infinity, or in this case it actually has a limit cycle. when running this system on the computer, it does get to that limit cycle for most initial conditions, i.e. max amplitude of 23.8 and and min at .8 while the steady state is at 3.9. but the problem is that if i choose all of the initial conditions to the same value, then instead of the limit cycle it just converges to that single value of 3.9. The other problem is that when i pick initial conditions close to that 3.9 steady state, it actually takes a very long time to converge to the limit cycle(it takes more periods than if i had taken some random values not even close to the limit cycle values), while i want it to converge to the limit cycle very quickly.

And what i'd like to know is if there are ways to get to a limit cycle very quickly!? Because I'm trying to think of a strategy of what system would be more resistant to oscillations. I figure if limit cycles take your initial conditions all over the places, then it's no good, and i'll have more correlation with just a regular stable spiral.

 

[saltydog]

i have a higher order system with a single fixed point.

You mind posting the equations for the system? Some plots too would be nice if you save them to jpeg and then attach them to the post.

 

[bor0000]

thanks! of course the part where they converge to the steady state no longer confuses me, because the equations are the same. but i am still very much looking for strategies/explanations for which init. conditions it takes a lot of periods and for which it takes few, to reach the limit cycle. i don't have soln curves on this computer, i might post them later from another computer, but equations are:
mi'=-mi+f(pj)+alpha0
pi'=-Beta(pi-mi)
where i=1,2,3 and j=2,3,1
f(pj)=alpha/(1+pj^n)
and parameters are beta=5, alpha0=120*5*.0001, alpha=60, n=2

also, i wanted to know if there was a way to easily construct conservative systems(have multiple nonlinear centres at the bifurcation point) in 3-d? And if you have such a system, what happens to it once you perturb it in either direction away from the bifurcation point?

and also if you have a system like the one above, is there any theory on how the nonlinear terms(i.e. f(p) ) influence how the limit cycle grows or shrinks based on small perturbations(away from the fixed point)?
thanks!

 

[saltydog]

thanks! of course the part where they converge to the steady state no longer confuses me, because the equations are the same. but i am still very much looking for strategies/explanations for which init. conditions it takes a lot of periods and for which it takes few, to reach the limit cycle. i don't have soln curves on this computer, i might post them later from another computer, but equations are:
mi'=-mi+f(pj)+alpha0
pi'=-Beta(pi-mi)
where i=1,2,3 and j=2,3,1
f(pj)=alpha/(1+pj^n)
and parameters are beta=5, alpha0=120*5*.0001, alpha=60, n=2

So is this it:

$$m_1^{'}=-m_1+\frac{\alpha}{1+p_2^2}+\alpha_0$$

$$m_2^{'}=-m_2+\frac{\alpha}{1+p_3^2}+\alpha_0$$

$$m_3^{'}=-m_3+\frac{\alpha}{1+p_1^2}+\alpha_0$$

$$p_1^{'}=-\beta(p_1-m_1)$$

$$p_2^{'}=-\beta(p_2-m_2)$$

$$p_3^{'}=-\beta(p_3-m_3)$$

with:

$$\beta=5;\quad \alpha_0=0.06;\quad \alpha=60$$

So suppose I start them all at 1 and run them through Runge-Kutta. Since there are 6 degrees of freedom, how are you modeling the dynamics in 3-D?

 

[bor0000]

So is this it:
$$m_1^{'}=-m_1+\frac{\alpha}{1+p_2^2}+\alpha_0$$
$$m_2^{'}=-m_2+\frac{\alpha}{1+p_3^2}+\alpha_0$$
$$m_3^{'}=-m_3+\frac{\alpha}{1+p_1^2}+\alpha_0$$
$$p_1^{'}=-\beta(p_1-m_1)$$
$$p_2^{'}=-\beta(p_2-m_2)$$
$$p_3^{'}=-\beta(p_3-m_3)$$
with:
$$\beta=5;\quad \alpha_0=0.06;\quad \alpha=60$$
So suppose I start them all at 1 and run them through Runge-Kutta. Since there are 6 degrees of freedom, how are you modeling the dynamics in 3-D?

yes, that is it. if you start them all at 1, you'll get all of them to be 3.9 or something. sorry, i don't model them in 3-d, i just meant to say that i don't even know anything about 3-d equations, i only learned about 2-d(i assume 3-d or 6-d is same thing in terms of theory).

 

[saltydog]

yes, that is it. if you start them all at 1, you'll get all of them to be 3.9 or something. sorry, i don't model them in 3-d, i just meant to say that i don't even know anything about 3-d equations, i only learned about 2-d(i assume 3-d or 6-d is same thing in terms of theory).

Ok. We could just pick 3 of them and plot them in 3-D and observe how the dynamics change as the parameters are varied, say the m ones. You are aware of the Lorenz attractor right? That's one in 3-D. Rossler too. How about Peitgen's book "Chaos and Fractals"? And also Julien Sprott wrote a nice book on strange attractors, "Strange Attractors: Creating Patterns in Chaos".

I'll set up your system in Mathematica tonight and turn the crank.

 

[HallsofIvy]

yes, that is it. if you start them all at 1, you'll get all of them to be 3.9 or something. sorry, i don't model them in 3-d, i just meant to say that i don't even know anything about 3-d equations, i only learned about 2-d(i assume 3-d or 6-d is same thing in terms of theory).

Actually, no. The topological situation in 3-d is very different from 2-d.
In the plane, since orbits can cross only at equilibrium points, there are a limited number of possibilities. In 3 dimensional space, orbits can be very complicated- look at Lorenze's butterfly!

 

[bor0000]

thanks. i will run it more on xpp later. in my class we use "nonlinear dynamics&chaos" by strogatz. there is a chapter on lorenz equations and following chapters on other 3d systems. we'll start those next week but they won't be on the exam, and I'm not sure whether i'll continue to study those. this equation is for a project in another class. the project is about "noise", and i wondered if i could show that 1 system(this one) is more or less susceptible to changes from small perturbations, then it would be more "robust". and i made a proposal that once the system is in limit cycle, it may have a relatively constant period and amplitude or not, depending on what fixed point the limit cycle came from. but now i see that it may take a long time to even reach a limit cycle, depending on initial conditions, and then it will be susceptible to most of the noise before even converging to the limit cycle.

yeah, i meant 3-d and 6-d is the same to me, that is very different from 2-d. in 2-d you can obviously start on some radius near the limit cycle and it will only get closer to it...

 

[saltydog]

I've looked at it somewhat in Mathematica.

The point 3.85073 is an equilibrium point of the system (for the parameters given above). Just set up a set of 6 equations in 6 unknowns for the right hand side and equate each to zero). Since this is an autonomous system (independent of t), then once it gets to that point is stays there for all t, thus the point is most likely a sink.

Really, if I were analyzing this system, I would approach it in the same manner as the approach used by Devaney and crew in "Differential Equations" by Blanchard, Devaney, and Hall: First linearize it, determine the Jacobian matrix, calculate the characteristic polynomial, eigenvalues, and then depending on the nature of the eigenvalues, one then characterizes the dynamics close to the equlibrium point.

I'd first of course do a few 2-D ones, then a few 3-D ones like the Lorenz attractor, following Devaney's analysis, then go on to work on this 6-D one or some others like it or even do a 4-D and a 5-D first.

Also, a good reference is "Perspectives of Non-linear dynamics" (vol 1 and 2) by E. Atlee Jackson.

 

[Physics Monkey]

Modeling some transcriptional regulators are we, bor0000? In this case, the symmetry of the problem makes it possible to calculate the eigenvalues of the linearized problem analytically. Moreover, you can get the stability of the fixed point even easier by using the Routh-Hurwitz criterion. Information about the speed of escape from the unstable fixed point would be contained in the eigenvalues of the linearized problem.

In this case, the system can't really explode to infinity no matter the initial condition because of the first order degradation. When the fixed point is unstable, a limit cycle is (roughly speaking) unavoidable. There are theorems to this effect for two dimensional systems, though I don't know to what extent they can be modified for higher dimensions.

 

[bor0000]

thanks! yeah of course 3.8-3.9 is the steady state. but it's unstable and leading to the limit cycle unless you pick all the initial values equal to each other.

yes, i am supposed to critique a few articles, in one of them this model was used. and i need to propose something new for the class. yeah, i got the eigenvalues and one of them always happens to be negative. it's a constant beta times the slop of that function f(A). but i never knew that the system must have limit cycle because of the degradation term. what if the 2 imaginary roots happen to have very large positive real parts? But if you claim that the negative term always forces the system to have a limit cycle, i'll take that. Also you say that this works if the degradation power is larger than f(a), i.e. it wouldn't work if f(A) were something like A^2? either way, in the article they proposed this model but their experiment had growing oscillations, wonder if that means their model is not representative of the experiment whatsoever
Yeah i must use the routh hurwitz condition to go over another article(i never tried using it before). but i think i'll just take their derivations as a given, what interests me is just the hopf bifurcation point.