abc : The A $\to$ B $\to$ C Reaction.

This demo illustrates the computation of stationary solutions, Hopf bifurcations and periodic solutions in the A $\to$ B $\to$ C reaction (DoHe:83 DoHe:83).

$$\begin{array}{cl} u_1 ' &= -u_1 + p_1 (1-u_1) e^{u_3}, \\ u_2... ..._3 u_3 + p_1 p_4 e^{u_3} ( 1-u_1 + p_2 p_5 u_2 ),\\ \end{array}$$ (14.2)

with $p_2=1$, $p_3=1.55$, $p_4=8$, and $p_5=0.04$. The free parameter is $p_1$.

The equations, as programmed in the equations-file abc.f, appear in Table 14.2. The starting point, an equilibrium of the equations, is also defined in the equations-file abc.f, as shown in Table 14.3. (The equations-file abc.f also contains the skeletons of some other routines, which must be supplied, but which are not used in this application.)

In the constants-file (c.abc.1) for the first run, as shown in Table 14.4, we note the following:

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IPS=1 : a family of stationary solutions is computed.

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IRS=0 : the starting point defined in STPNT is to be used (see Table 14.3).

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ICP(1)=1 : the continuation parameter is PAR(1)

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NUZR=1 : there is one user output point, namely at PAR(1)=0.4. Moreover, since the index ("-1") in the last line of the constants-file c.abc.1 is negative, the calculation will terminate when the calculation reaches the value PAR(1)=0.4..

In the constants-file (c.abc.2) for the second run, as shown in Table 14.5, we note that:

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IPS=2 : a family of periodic solutions is computed.

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IRS=2 : the starting point is the solution with label 2, (a Hopf bifurcation point), to be read from the solutions-file (here s.abc).

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NICP=2 : there are two continuation parameters (namely PAR(1), and the period, PAR(11)).

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NUZR=1 : there is one user output point, now at PAR(1)=0.25, where the calculation is to terminate, since the index ("-1") is negative.

Table 14.2: The equations for demo abc, as defined in the equations-file abc.f.

Table 14.3: The starting solution for demo abc, as defined in the equations-file abc.f.

Table 14.4: The constants-file c.abc.1 for Run 1 (stationary solutions) of demo abc.

Table 14.5: The constants-file c.abc.2 for Run 2 (periodic orbits) of demo abc.

Table 14.6: Unix Commands for running demo abc.

COMMAND	ACTION
mkdir abc	create an empty work directory
cd abc	change directory
@dm abc	copy the demo files to the work directory
@R abc 1	compute the stationary solution family with four Hopf bifurcations
@sv abc	save output-files as b.abc, s.abc, d.abc
@R abc 2	compute a family of periodic solutions from the first Hopf point
@ap abc	append the output-files to b.abc, s.abc, d.abc
@R abc 3	compute a family of periodic solutions from the second Hopf point
@ap abc	append the output-files to b.abc, s.abc, d.abc
@R abc 4	compute a family of periodic solutions from the third Hopf point
@ap abc	append the output-files to b.abc, s.abc, d.abc
@R abc 5	compute a family of periodic solutions from the fourth Hopf point
@ap abc	append the output-files to b.abc, s.abc, d.abc

Table 14.7: Python Commands for running demo abc.

Table 14.8: Python Program for running demo abc.