ops : Optimization of Periodic Solutions.

This demo illustrates the method of successive continuation for the optimization of periodic solutions. For a detailed description of the basic method see DoKeKe:91b DoKeKe:91b. The illustrative system of autonomous ODEs, taken from Alej:91 Alej:91, is

$$\begin{array}{cl} x'(t) & = [-\lambda_4(x^3/3-x) + (z-x)/\lam... ...y'(t) &= x-\lambda_3, \\ z'(t) &= -(z-x)/\lambda_2, \end{array}$$ (17.1)

with objective functional

$$\omega = \int_0^{1} g(x,y,z;\lambda_1,\lambda_2,\lambda_3,\lambda_4) ~ dt,$$

where $g(x,y,z;\lambda_1,\lambda_2,\lambda_3,\lambda_4) \equiv \lambda_3$. Thus, in this application, a one-parameter extremum of $g$ corresponds to a fold with respect to the problem parameter $\lambda_3$, and multi-parameter extrema correspond to generalized folds. Note that, in general, the objective functional is an integral along the periodic orbit, so that a variety of optimization problems can be addressed.

For the case of periodic solutions, the extended optimality system can be generated automatically, i.e., one need only define the vector field and the objective functional, as in done in the file ops.f. For reference purpose it is convenient here to write down the full extended system in its general form :

$$\begin{array}{cl} &u'(t) = T f \bigl( u(t),\lambda \bigr) , \... ...au_i \in {\rm R}, \quad i=1, \cdots, n_{\lambda}.\\ \end{array}$$ (17.2)

Above $u_0$ is a reference solution, namely, the previous solution along a solution family.

In the computations below, the two preliminary runs, with IPS=1 and IPS=2, respectively, locate periodic solutions. The subsequent runs are with IPS=15 and hence use the automatically generated extended system.

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Run 1.  Locate a Hopf bifurcation. The free system parameter is $\lambda_3$.

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Run 2.  Compute a family of periodic solutions from the Hopf bifurcation.

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Run 3.  This run retraces part of the periodic solution family, using the full optimality system, but with all adjoint variables, $w(\cdot), \kappa, \gamma$, and hence $\alpha$, equal to zero. The optimality parameters $\tau_0$ and $\tau_3$ are zero throughout. An extremum of the objective functional with respect to $\lambda_3$ is located. Such a point corresponds to a branch point of the extended system. Given the choice of objective functional in this demo, this extremum is also a fold with respect to $\lambda_3$.

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Run 4.  Branch switching at the above-found branch point yields nonzero values of the adjoint variables. Any point on the bifurcating family away from the branch point can serve as starting solution for the next run. In fact, the branch-switching can be viewed as generating a nonzero eigenvector in an eigenvalue-eigenvector relation. Apart from the adjoint variables, all other variables remain unchanged along the bifurcating family.

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Run 5.  The above-found starting solution is continued in two system parameters, here $\lambda_3$ and $\lambda_2$; i.e., a two-parameter family of extrema with respect to $\lambda_3$ is computed. Along this family the value of the optimality parameter $\tau_2$ is monitored, i.e., the value of the functional that vanishes at an extremum with respect to the system parameter $\lambda_2$. Such a zero of $\tau_2$ is, in fact, located, and hence an extremum of the objective functional with respect to both $\lambda_2$ and $\lambda_3$ has been found. Note that, in general, $\tau_i$ is the value of the functional that vanishes at an extremum with respect to the system parameter $\lambda_i$.

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Run 6.  In the final run, the above-found two-parameter extremum is continued in three system parameters, here $\lambda_1$, $\lambda_2$, and $\lambda_3$, toward $\lambda_1=0$. Again, given the particular choice of objective functional, this final continuation has an alternate significance here : it also represents a three-parameter family of transcritical secondary periodic bifurcations points.

Although not illustrated here, one can restart an ordinary continuation of periodic solutions, using IPS=2 or IPS=3, from a labeled solution point on a family computed with IPS=15.

The free scalar variables specified in the AUTO constants-files for Run 3 and Run 4 are shown in Table 17.2.

Table 17.2: Runs 3 and 4  (files c.ops.3 and c.ops.4).

Index	3	11	12	22	-22	-23	-31
Variable	$\lambda_3$	$T$	$\alpha$	$\tau_2$	$[\lambda_2]$	$[\lambda_3]$	$[T]$

The parameter $\alpha$, which is the norm of the adjoint variables, becomes nonzero after branch switching in Run 4. The negative indices (-22, -23, and -31) set the active optimality functionals, namely for $\lambda_2$, $\lambda_3$, and $T$, respectively, with corresponding variables $\tau_2$, $\tau_3$, and $\tau_0$, respectively. These should be set in the first run with IPS=15 and remain unchanged in all subsequent runs.

Table 17.3: Run 5  (file c.ops.5).

Index	3	2	11	22	-22	-23	-31
Variable	$\lambda_3$	$\lambda_2$	$T$	$\tau_2$	$[\lambda_2]$	$[\lambda_3]$	$[T]$

In Run 5 the parameter $\alpha$, which has been replaced by $\lambda_2$, remains fixed and nonzero. The variable $\tau_2$ monitors the value of the optimality functional associated with $\lambda_2$. The zero of $\tau_2$ located in this run signals an extremum with respect to $\lambda_2$.

Table 17.4: Run 6  (file c.ops.6).

Index	3	2	1	11	-22	-23	-31
Variable	$\lambda_3$	$\lambda_2$	$\lambda_1$	$T$	$[\lambda_2]$	$[\lambda_3]$	$[T]$

In Run 6 $\tau_2$, which has been replaced by $\lambda_1$, remains zero.

Note that $\tau_0$ and $\tau_3$ are not used as variables in any of the runs; in fact, their values remain zero throughout. Also note that the optimality functionals corresponding to $\tau_0$ and $\tau_3$ (or, equivalently, to $T$ and $\lambda_3$) are active in all runs. This set-up allows the detection of the extremum of the objective functional, with $T$ and $\lambda_3$ as scalar equation parameters, as a bifurcation in the third run.

The parameter $\lambda_4$, and its corresponding optimality variable $\tau_4$, are not used in this demo. Also, $\lambda_1$ is used in the last run only, and its corresponding optimality variable $\tau_1$ is never used.

Table 17.5: Commands for running demo ops.

AUTO -COMMAND	ACTION
! mkdir ops	create an empty work directory
cd ops	change directory
demo('ops')	copy the demo files to the work directory
run(c='ops.1')	locate a Hopf bifurcation
sv('0')	save output-files as b.0, s.0, d.0
run(c='ops.2',s='0')	compute a family of periodic solutions; restart from s.0. Constants changed : IPS, IRS, NMX, NUZR
ap('0')	append the output-files to b.0, s.0, d.0
run(c='ops.3',s='0')	locate a 1-parameter extremum as a bifurcation; restart from s.0. Constants changed : IPS, IRS, ICP, $\cdots$
sv('1')	save the output-files as b.1, s.1, d.1
run(c='ops.4',s='1')	switch branches to generate optimality starting data; restart from s.1. Constants changed : IRS, ISP, ISW, NMX
ap('1')	append the output-files to b.1, s.1, d.1
run(c='ops.5',s='1')	compute 2-parameter family of 1-parameter extrema; restart from s.1. Constants changed : IRS, ISW, ICP, ISW, $\cdots$
sv('2')	save the output-files as b.2, s.2, d.2
run(c='ops.6',s='2')	compute 3-parameter family of 2-parameter extrema; restart from s.2. Constants changed : IRS, ICP, EPSL, EPSU, NUZR
sv('3')	save the output-files as b.3, s.3, d.3