A delay can be approximated by using a finite-dimensional system of differential equations. This command provides the ability to automatically generate such an approximation from a differential delay equation.
The delay2ode command generates a new vector field file from an existing file that contains delays. The new vector field will be a system of ordinary differential equations in which the delays have been replaced by finite dimensional approximations.
Background.
Suppose
y_{1}(t + δ/N) | = | x(t) |
y_{2}(t + δ/N) | = | y_{1}(t) |
. | ||
. | ||
. | ||
y_{N}(t + δ/N) | = | y_{N-1}(t) |
When p=1, we obtain
When p=2, we obtain
y_{1,k}' | = | y_{2,k} |
y_{2,k}' | = | (2N/δ)(-y_{2,k} + (N/δ)(y_{1,k-1} - y_{1,k})) |
y_{1,k}' | = | y_{2,k} |
y_{2,k}' | = | y_{3,k} |
y_{3,k}' | = | (3N/δ) (-y_{3,k} - (2N/δ)(y_{2,k} - (N/δ)(y_{1,k-1} - y_{1,k}))) |
A new vector field file containing the definition of a system of ordinary differential equations in which all the delays in the DDE defined in vector_field_file.vf have been replaced by finite dimensional approximations is created with the command
The new vector field file is written to the console, so the command is generally used by redirecting the output to a file:
p |
Order of the approximation. As explained above, this is the order
of the Taylor series retained in the approximation of a ``small'' delay.
Only p=1, p=2, and p=3 are allowed.
Default: p=1 |
N |
Number of grid points in the approximation to the delayed
expression in the delay interval.
Default: N=10 |
We consider the equation
We create a delayed copy of x(t) by defining an expression with the formula delay(x,h) Here is the vector field file simpledelay.vf. We create a new vector field with the command
We can now apply VFGEN to this new vector field to create a solver. The following commands create and compile a solver written in C that uses the GSL library. (See the GSL section for more information about using the GSL command of VFGEN.)
The Mackey-Glass equation is
We create a new vector field with the command
The following commands create and compile a solver for this approximation to the delay equation:
(Another example in which the delay2ode command is used to approximate the Mackey-Glass equation with a finite dimensional system is given in the VFGEN auto command documentation.)