The Fixed Points Window

Figure 2.24: The Fixed Points window.

Window title:

DsTool: Fixed Points

Function:

The Fixed Points window allows the user to find equilibria and to compute one dimensional stable and unstable manifolds.

Description:

The Fixed Points window is opened by selecting the Fixed Points option from the Panels menu button located in the Command window. The window contains parameters for controlling the computation of stable and unstable manifolds.

Color coding is performed according to the stability type of objects: saddle points are marked as green crosses, sinks are blue triangles, and sources are red squares. Furthermore, unstable manifolds are plotted in red and stable manifolds are plotted in blue.

Panel items:

• Fixed point algorithm stack setting: Allows the user to choose an algorithm to use for root-finding. The current options are the Secant method and Newton's method. The default is Newton's method.
• Seed algorithm stack setting: Allows the user to choose the method of providing an initial guess (``seed'') to the root-finding algorithm. The choices are:

  Monte Carlo:

  The seed is a pseudo-random point chosen from the volume in phase space defined by the coordinate values in the Defaults window. This is the default setting.

  Selected Point:

  The seed is the currently selected point. The currently selected point is displayed in the Selected Point window.

• Monte Carlo points numeric field: Displays the number of guesses taken by the Monte Carlo routine. If the Seed algorithm setting is Selected Point, then this field is not relevant. The default value is 10.
• Period numeric field: Displays the integer length of periodic orbits to search for. This field is only shown if the current dynamical system is a mapping. Note that points with lower periods which divide the specified period may also be located. The default value is 1. Setting this field to a positive number should be used to search for periodic points (for mappings), instead of using the Periodic Orbits window.
• Found read-only text field: Displays the number of periodic orbits or equilibrium points found by DsTool's fixed point algorithms. This may be different than the total number of fixed points stored in memory, since fixed points may be loaded into memory from a data file. For a mapping, this field displays the number of orbits found, not the total number of points found.
• Fixed point algorithm parameters message: Identifies the text items which follow as parameters associated with the finding of fixed points.
• maximum iterations numeric field: Displays the maximum number of iterations allowed while trying to converge to a periodic point. The default value is 30.
• convergence criterion read-write text field: Displays the criterion used to determine when Newton's method has converged. An iterative sequence is said to converge to a periodic point (resp. equilibrium) if the norm of $f(x_i)-x_i$ (resp. $f(x_i)$) is less than the value of this field. Here $f$ is some iterate of the mapping (resp. $f$ defines the vector field). The default value is $10^{-8}$.
• minimum step read-write text field: Displays the minimum step to take in Newton's method. Periodic points (resp. equilibria) are found via an iterative process. If the difference between successive points is less than the value in this field, then no further steps will be taken and the final iterative point will not be regarded as a periodic point. The default value is $10^{-8}$.
• duplicate criterion read-write text field: Displays the distance used to determine when two points should be considered the same. Two periodic points (resp. equilibria) are considered to be identical if the norm of their difference is less than the entry in this field. The default value is $10^{-6}$.
• finite difference step read-write text field: Displays the stepsize to use when using finite differencing in the DsTool fixed point calculations. The default value is $10^{-6}$.
• One-D manifold algorithm parameters message: Identifies the text items which follow as parameters associated with the calculation of one dimensional stable and unstable manifolds.
• stable manifold steps numeric field: Displays the number of time steps to take in computing the manifold. The length of the stable manifold displayed depends on the strength of the eigenvalues and the number of time steps taken along the manifolds. The default value is 200. This number should be increased for manifolds associated with small eigenvalues.
• (stable manifold) points numeric field: Displays the number of interpolating points between steps. Given a stable manifold and its pre-image, the manifold may be better visualized by interpolating additional points along the manifold. The default value is 1, but for manifolds which oscillate wildly (e.g., the transverse intersection of homoclinic orbits) this number can be chosen much larger.
• unstable manifold steps numeric field: Displays the number of time steps to take in computing the manifold. The length of the unstable manifold displayed depends on the strength of the eigenvalues and the number of time steps taken along the manifolds. The default value is 200. This number should be increased for manifolds associated with large eigenvalues.
• (unstable manifold) points numeric field: Displays the number of interpolating points between steps. Given an unstable manifold and its pre-image, the manifold may be better visualized by interpolating additional points along the manifold. The default value is 1, but for manifolds which oscillate wildly (e.g., the transverse intersection of homoclinic orbits) this number can be chosen much larger.
• Initial stepsize read-write text field: Displays the distance from the periodic point (resp. equilibrium) along the eigenvector to choose the initial point of the stable or unstable manifold calculation. The default value is $10^{-6}$. This number should be decreased for highly nonlinear systems, i.e., when the linearization at a periodic point (resp. equilibria) is only a good approximation to the (nonlinear) mapping (resp. vector field) within a tiny neighborhood of the periodic point (resp. equilibria).
• Find fixed points command button: Initiates the search for equilibria, using the current window settings.
• Compute 1-d manifolds command button: Initiates the calculation and display of one dimensional stable and unstable manifolds.
• Clear points command button: Erases the periodic points in memory and resets the Found text field to 0.
• Clear manifolds command button: Erases the manifolds computed by the Fixed Points window.
• Dismiss command button: Closes the Fixed Points window.