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Finding a Hopf bifurcation from a cube equation

September 27, 2014 1 comment

Let me first state what I found out this week. Forgive me if this result is trivial for you, but this result indeed helped me in my research.

Lemma 1
Suppose we have a cube equation x^3 + ax^2 + bx + c=0. The roots of this equation are a couple of purely imaginary roots and a single real root if and only if the following conditions are satisfied b>0 and c=ab.

The proof is very easy and I am not going to prove this. Apparently, the conditions b>0 and c=ab are a necessary and sufficient condition for the cube equations to have a single real root and a couple of purely imaginary roots.

Okay, now let us move to the beginning why I bother with cube equations in the first place.

At the moment I am studying a mathematical model on HIV-1 virus with cell mediated immunity. I just started so I still don’t know all the representation of this model in the real life. But here is the model.

\dot{x} = \lambda - dx - \beta x v
\dot{y} = \beta x v - ay - p y z
\dot{v} = k y- u v
\dot{w} = c y w - c q y w - b w
\dot{z} = c q y w - h z

Yes, you are right, it is a five dimensional model. The variables x, y, v, w, z are the state variables that are representing the density of uninfected cells, the infected cells, the density of virus, the CTL response cells and the CTL effector cells respectively. Why it is five dimensional, what are the parameters representing and other modeling questions, I still don’t know the answer. At the moment I am interested to find an equilibrium in this system that can undergo a Hopf bifurcation. An equilibrium can have a Hopf if the Jacobian of the system evaluated at this equilibrium has a purely imaginary eigenvalue. Actually this is the first time I handle a five dimensional system so it will be interesting.

Now we know why I need Lemma 1.

In my case, the situation is greatly simplified as all parameters are numbers except two which are d and b. From the paper that I read, I found that there are three equilibria in the system, so called E_0, E_1, E_2. At the moment I am interested in the equilibrium E_1 because I suspected that this equilibrium can have a Hopf bifurcation and because the coordinate of this equilibrium has the following conditon w=z=0. Let us have a look at the Jacobian matrix evaluated at E_1=(x_1,y_1,v_1,0,0).

J = \left( \begin{array}{ccccc} -d-\beta v_1 & 0 & -\beta x_1 & 0 & 0\\\beta v_1 & -a & \beta x_1 & 0 & -p y_1 \\ 0 & k & -u & 0 & 0 \\ 0 & 0 & 0 & cy_1-c q y_1 - b & 0 \\ 0 & 0 & 0 & c q y_1 & -h \end{array} \right)

As I said before, all the parameters are numbers except d and b. Also E_1 will be an expression that depends on d and b. As a result the matrix above will be depending on d,b.

We are interested in what value of d,b such that the matrix above has a purely imaginary eigenvalues. Even though the above matrix is not a block matrix but when we compute the eigenvalues, we can really separate the matrix above. We can obtain the eigenvalues of the above matrix by computing the eigenvalues of the following two matrices:

A = \left( \begin{array}{ccccc} -d-\beta v_1 & 0 & -\beta x_1 \\ \beta v_1 & -a & \beta x_1 \\ 0 & k & -u \\ \end{array} \right) and

B = \left( \begin{array}{ccccc} - cy_1-c q y_1 - b & 0 \\ c q y_1 & -h \end{array} \right).

The eigenvalues of matrix B is easy and it is real. So the problem really is in the computing the eigenvalue of matrix A. Consider the matrix A in a simpler notation.

A = \left( \begin{array}{ccccc} a & b & c \\ d & e & f \\ g & h & i \\ \end{array} \right).

The characteristic polynomial of the above matrix is

p(\lambda) = \lambda^3 - (a+d+g) \lambda^2 + (bd-ae+fh-ei+cg-ai) \lambda + \det(A)

Therefore to find the value d,b such that we have a Hopf bifurcation, we only need to solve the following conditions:
1. make sure that (bd-ae+fh-ei+cg-ai) is positive and
2. solve - (a+d+g) \times (bd-ae+fh-ei+cg-ai) = \det(A).

I created this note (which is a part of our paper) so that I won’t forget what I had done. We don’t usually show this kind of computation but for me this little computation will be useful. Even though I uses software to compute and investigate the existence of Hopf bifurcation but it does not show the exact value of the parameter.  Using algebraic approach I found the exact value.

Reference of the model
Yu, Huang, and Jiang, Dynamics of an HIV-1 infection model with cell mediated immunity, Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19.

Additional note: (Two dimensional case) Suppose the Jacobian matrix evaluated at the equilibrium is given below

B = \left( \begin{array}{ccccc} p & q \\ r & s \end{array} \right).

Then the equilibrium has a Hopf bifurcation if the following conditions are satisfied:
1. \det(B) > 0 and
2. trace(B) = 0.

 

 

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Math on the web 2

September 11, 2014 Leave a comment

This is a second edition of Math on the web, in which I try to list mathematical articles around the internet. I found these articles from googling, from social media networks and many more sources. I will write website links that relate to mathematics and its applications that I found from May 2014 until now. I hope you find them interesting because these are really very subjective.

  1. @TopologyFact : Notes on hyperbolic geometry http://ow.ly/wsX8D
    The above link will bring you to a kind of ebook about hyperbolic geometry. To be honest, I have not read it. I just downloaded it and it now sits in my computer hard drive. I got this because in case I want to learn about hyperbolic geometry.
  2. http://paper.li/nick_chater/1317156219
    paper.li is a webpage that (I think) is similar to the (late) google reader. It can bring you the latest information you are interested in and even it can bring you information from social media like Facebook, Google Plus and Twitter. The link above is a paper created by Nick Chater that brings us information about Mathematical Education. I believe the information will be very useful for mathematics educators like me and you.
  3. @approx_normal: A few awesome bumper stickers here http://visualizingmath.tumblr.com/post/86363962441/i-mathematics
    The link will bring you to various bumper stickers that are very cool when you are mathematicians. I would buy it if there is someone selling these, especially with the Lorrentz Attractor one.
  4. @kdnuggets: Numeric matrix manipulation: cheat sheet for MATLAB, Python NumPy, R, and Julia #DataScience http://buff.ly/1ig68eg I am a fan of mathematical softwares. I could sit in my office for hours trying to understand new language of mathematical softwares. This link will bring us the cheat sheet of some most popular languanges. It is not a complete cheat sheet though as it is only discussing the basic operation of matrices., which can be very useful for people who just begin to learn the language.
  5. @pickover Teachers, introduce your students to the wonders of Hilbert Curves at this awesome website. http://www.datagenetics.com/blog/march22013/index.html
    @pickover is a twitter acount of a mathematician named Cliff Pickover . He is a well known author who has published more than 50 books. Cited from his website, his books explore topics ranging from computers and creativity to art, mathematics, parallel universes, Einstein, time travel, alien life, religion, dimethyltryptamine elves, and the nature of human genius.
    The link above will bring a very introductive article about Hilbert Curve.
Categories: Teachings