##
**CSTR**

Original source of the model: A. Uppal, W.H. Ray, A.B. Poore: On the dynamic behavior
of continuous stirred tank reactors. Chem. Eng. Sci. **29** (1974) 967-985.
A branching diagram is shown in Figure 1. We see a branch
of stationary solutions with two Hopf bifurcation points, for which
the parameter values and initial periods are

The two Hopf points are connected by a branch of periodic
orbits (indicated by two parts in Figure 1). This example exhibits both subcritical and
supercritical bifurcation. At the right Hopf point there is a soft
generation of limit cycles (for decreasing ), whereas at the left
Hopf point the loss of stability is hard. The branch of stationary solutions has
two turning points close to each other with a difference in
of
.
As is hardly seen in Figure 1,
the lower turning point is above the left Hopf point. The hysteresis part
of the branching diagram thus lies fully in the unstable range
between the Hopf points.
In Figures 2 and 3, simulations are performed with
.
The soft loss of stability in Figure 3 *looks* like a hard loss. The relatively steep transition is due to a slow reaction of the system.
Note that the
scaling that corresponds to
in Figure 2 is the same as in Figure 1.
The phase condition of the calculations in Figure 1 is
,
here fixing the maximum of *y*_{1}(*t*). Hence there is an immediate correspondence between Figure 1 and Figure 2 in size, scaling, and dynamical behavior. This is similar in Figure 3, except for the reverse time; flip Figure 3 and it matches Figure 1 and 2.

``Continuous" refers to a continuous flow entering (and
leaving) the reactor--that is, a CSTR is an open system.
Human beings and other living organisms that have input of
reactants (nutrients) and output of products (wastes) are
complex examples of CSTRs.
The exponential term in the equations reflects an infinite activation
energy.

Figure 1

Branching diagram *y*_{1}(0) versus Da. One branch of stationary solutions, and two branches of periodic orbits branching off at two Hopf bifurcations.

Figure 2

*y*_{1}(*t*) simulation for
,
,
with initial values
*y*_{1}(0)=0.1644,
*y*_{2}(0)=0.6658

Figure 3

*y*_{1}(*t*) simulation for
,
,
with initial values
*y*_{1}(0)=0.9279,
*y*_{2}(0)=3.76

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Postscript-Files for better printing results:

This Example

Figure 1

Figure 2

Figure 3