We noted in the previous section that were three different steady-state
solutions to the case 2 parameter set. Here we wish to study the
dynamic behavior under this same parameter set. Recall that numerical
integration techniques were presented in chapter 4.

The m-file to integrate the modeling equations is cstr_dyn.m,
shown in Appendix 2. The command to integrate the equations is

[t,x] = ode45('cstr_dyn',t0,tf,x0);

where t0 is the inital time (usually
0), tf is the final time, x0
is the initial condition vector. t is
the time vector and x is the state
variable solution vector. Before performing the integration it
is necessary to define the global parameter vector CSTR_PAR.
To plot only concentration or temperature as a function of time,
use plot(t,x(:,1)) and
plot(t,x(:,2)), respectively.

You may also run a JAVA based dynamic behaviour simulator to check out the different steady state solutions ! |

**Initial condition 1
**

Here we use initial conditions that are close to the low temperature steady-state. The initial condition vector is [conc, temp] = [9,300]. The curves plotted in Figure 2 show that the state variables converge to the low temperature steady-state.

**Initial condition 2
**

Here we use initial conditions that are close to the intermediate
temperature steady-state. The initial condition vector for the
solid curve in Figure 3 is [conc, temp] = [5,350], which converges
to the high temperature steady-state. The inital condition vector
for the dotted curve in Figure 3 is [conc, temp] = [5,325], which
converges to the low temperature steady-state.

If we perform many simulations with initial conditions close to
the intermediate temperature steady-state, we find that the temperature
always converges to either the low temperature or high temperature
steady-states, but not the intermediate temperature steady-state.
This indicates to us that the intermediate temperature steady-state
is *unstable*. This will be shown clearly by the stability
analysis in section 5.

**Initial condition 3
**

Here we use initial conditions that are close to the high temperature
steady-state. The initial condition vector is [conc, temp] = [1,400].
The curves plotted in Figure 4 show that the state variables converge
to the high temperature steady-state.

In this section we have performed several simulations and presented
several plots. In section 6 we will show how these solutions can
be compared on the same ìphase planeî plot.