The Derived Model Panel

The Pisces Derived Model Panel.

To aid in the explanation, we will refer to the coded model. The coded model is the model as it is originally defined. This determines the dimensions of the "coded domain," and the "coded range."

Let F denote the current model. The domain of the model is determined by the current state of the model panel and may be changed using the permutation panel.

The currently-implemented derived models are:

None

Find level sets of the current model. The range of the current model is the same as the coded range.

Fixed Points Map

Find level sets of the model f(x) = F(x)-x. In order for this derivation to make sense, the dimension of the coded domain of F must equal the dimension of the coded range of F. Naturally, in order to compute a level set of f in Pisces, the user must inflate the domain by at least one variable using the permutation panel.

Saddle Node Map

Find level sets of the system f1(x) = F(x)-x, f2(x) = det(DF(x)-I) where I is the identity matrix. In order for this derivation to make sense, the dimension of the coded domain of F must equal the dimension of the coded range of F. In order to compute a level set of f in Pisces, the user must inflate the domain by at least two variables using the permutation panel.

Fixed Points ODE

Find level sets of the model f(x) = F(x). (If we think of F as being a vector field, then we are finding equilibria.) In order for this derivation to make sense, the dimension of the coded domain of F must equal the dimension of the coded range of F. Naturally, in order to compute a level set of f in Pisces, the user must inflate the domain by at least one variable using the permutation panel.

Saddle Node ODE

Find level sets of the system f1(x) = F(x), f2(x) = det(DF(x)). In order for this derivation to make sense, the dimension of the coded domain of F must equal the dimension of the coded range of F. In order to compute a level set of f in Pisces, the user must inflate the domain by at least two variables using the permutation panel.

Planar_Singularity

This derivation only makes sense if F is a scalar-valued function of two variables. Then the derivation is f1(x)=F(x), f2(x)=grad(F). The solutions to this system are singular points on the level sets of F. In order to compute a level set of f in Pisces, the user must inflate the domain by at least two variables using the permutation panel.