Link to our rather neat Maple worksheet.
The CSTR model is defined by the first order non-linear autonomous system:
Set u' = 0 and solve uniquely for v:
Then substitute into the v' equation to decouple the system at the steady states, and set v' = 0 to yield the steady state equation:
Limit point bifurcation analysis:
We can differentiate the right hand side of f(u) with respect to u to obtain:
Take the the derivative of (4) with respect to u:
This will only equal zero if u = 1/2 (and fix k = 4).
The derivative of f(u) with respect to a is:
This never equals zero (k > 0), so there are limit point bifurcations for all u except when u = 1/2. There are no transcritical bifurcations in this model since (6) never equals zero (a is our bifurcation parameter).
Next, we can simultaneously solve f(u) and df(u)/du for a and b and find closed form solutions for these parameters.
With these equations for the parameters, we can essentially pick a value of u and determine a and b and a v with (2) such that a limit point bifurcation point is present with those values.
We can plot b/(1+k) vs. u to get some information on the the number of limit point bifurcations for a given value of b (k = 4 is fixed). This graph shows that for values of b/(1+k) > 4, there are two limit point bifurcations, when b/(1+k) = 4 there is one limit point bifurcation, and no limit point bifurcations when b/(1+k) < 4. However, this "critical" value when b/(1+k) = 4 corresponds to u = 1/2, which we already said is not a limit point bifurcation. Thus, something interesting is going on here for u = 1/2.
Stability of steady states analysis:
We now define values for a and b (k=4) such that we can get one specific example for each case shown immediately above. (here, b=30, then b=20, then b=1.) It seems appropriate to generalize findings for each example to the entire case for the given value of b.
We found that b > 20 has two limit point bifurcations with respect to a. With increasing a, we go from one steady state, to three (one sink, one saddle, and one source), and then back to one steady state.
For b=20, we found there is always one steady state (a sink). There is one "exception", in that for a value of a near 0.14, their appears to be a vertical tangent to the bifurcation diagram. Since we determined above that there are no limit bifurcations for this case, then we can conclude that this vertical tangent is representative of this "critical point" for the number of bifurcations for arbitrary b. It follows that the stability of the steady state does not change for this case, so all fixed points are sinks for a system where b = 20.
For b < 20, there is always one steady state (a sink).
Stability of a given steady state is determined by using the Jacobian of the system:
Substitute in any values of b, k, u, v, a, that completely determine a steady state (information that is directly available from these plots and steady state formulas above). The well known theorem gives conditions that state whether or not a fixed point is stable based on the eigenvalues of the Jacobian evaluated at a point. If both have positive real part, then the fixed point is a source (unstable). If one eigenvalue has positive real part while the other has negative real part, the fixed point is a saddle (unstable). If both have negative real part, the fixed point is a sink (stable). Of course, there are only two eigenvalues of the Jacobian since this is only a two dimensional system.
Hopf Bifurcation Analysis:
A Hopf bifurcation can occur at a given fixed point if the trace of the Jacobian is zero:
and the determinant of the Jacobian is greater than zero:
Substitute v for (2) in (8). Next, simultaneously solve the steady state equation (3) and the (8) equation for a and b to obtain closed form equations for these parameters in terms of u and k. We fixed k = 4:
and then plotted these two functions separately versus u. The b versus u plot shows that for a Hopf bifurcation to occur, b > 11.89 (approximately) is required. This is the minimum value of b on the domain 0 < u < 1.
Here is the a versus u plot.
In general, a set of parameters with can be determined (with a given u) that will fully define a Hopf bifurcation point using the above equations for a and b
Since we have determined that Hopf bifurcations do exist, then there must be some set of parameters such that we have periodic steady states. Phase portraits were made using the xrk program of several different sets of parameters:
a = 0.215, b = 7, k = 4: xrk There are no periodic steady states here. (0 < u < 1 is on the x-axis, 0 < v < 10 is on the y-axis)
a = 0.215, b = 11.9, k = 4: xrk There should be a periodic steady state here. See conclusions.
a = 0.215, b = 15, k = 4: xrk There is a periodic steady state here. Check out that banana!
[ Index ] [ Introduction ]
[ Background ] [ Conclusions ]
Date: October 15, 1998
Time: 12:50 PM