Home > bifurcation, Justincase > Finding a Hopf bifurcation from a cube equation

Finding a Hopf bifurcation from a cube equation

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. $trace(B) = 0$.