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The Equilibrium Continuation Window


  
Figure 2.26: The Continuation window.
48#48

Window title:
DsTool: Equilibrium Continuation
Function:
The Equilibrium Continuation window allows the user to compute curves of bifurcations.
Description:
The Equilibrium Continuation window is opened by selecting the Equilibrium Continuation option from the Panels menu button located in the Command window. The window allows the user to compute curves of equilibrium points with one varying parameter and curves of saddle-node and Hopf bifurcations with two varying parameters. A brief tutorial on continuation calculations is at the end of this section. The different Hopf bifurcation algorithms are described in [2] and [3].

Color coding  for hyperbolic equlibrium points is performed according to the value of the Monitor switch setting. For the bifurcation types, color coding is performed according to the choice of colormap, as shown in the Continuation Colors window. In the default colormap, Hopf bifurcation points are displayed in magenta, saddle-node bifurcation points are displayed in green, degenerate Hopf points are displayed in orange, and resonant saddle-node points are displayed in sea green.

Panel items:



Appendix: Continuation Calculations

One parameter continuation is a systematic strategy for computing curves of solutions to l equations 51#51 in l+1 variables. The mathematical foundation for continuation algorithms is the implicit function theorem. If G is differentiable and the Jacobian of G is surjective at a point 52#52 where G vanishes, then the solutions form a curve in a neighborhood of x. Moreover, the implicit function theorem gives a formula for the tangent vector to the solution curve. Continuation algorithms exploit this information. They use an initial value solver for ordinary differential equations (often just the Euler method) to step along an approximation to the solution curve. They then appply a root finding algorithm to significantly improve the approximation. The alternation of root finding with numerical integration steps distinguish continuation methods. Choices of how to parametrize the solution curve, choose time steps and restrict the equations to a hyperplane to obtain a square system of equations with a unique solution (locally) need to be made. Various continuation packages take different approaches to these matters: the continuation ``engine'' used by DsTool is PITCON (Rheinboldt ...).

If 53#53 is a k parameter family of vector fields, then we assemble systems of equations G for varied calculations. To do so we restrict ourselves to a submanifold of dimension 54#54. Most frequently this submanifold is obtained by fixing k-s of the parameters of f (called inactive parameters) and varying s active parameters. In the simplest case, G=f, the number of equations is n, s=1 and the continuation calculation computes curves of equilibria with a single active parameter. To compute curves of bifurcations, one adds defining equations to the system of equations f=0 to produce an augmented system. The number of independent defining equations added to f=0 is the codimension of the bifurcation. (In some circumstances, the augmented system uses additional auxiliary variables such as eigenvalues of the Jacobian and has a corresponding number of additional equations.)

The continuation window of DsTool includes calculations for saddle-node and Hopf bifurcations. These are the codimension one bifurcations of equilibrium points. For saddle-node bifurcations, the defining equation is 55#55or an algorithm that computes another scalar quantity, such as the smallest singular value, that vanishes precisely when Df is singular. For Hopf bifurcations, the defining equation(s) compute where the matrix Df has a pair of pure imaginary eigenvalues. Explicit expressions in the coefficients of Df that compute whether Df has a pair of complex eigenvalues are very complicated. Alternate ways of performing this computation are discussed in [3]. There are also other algorithms that perform the calculation by introducing auxiliary variables for the pure imaginary eigenvalues, and in some cases the eigenvectors of Df.


next up previous contents
Next: The Continuation State Window Up: Attributes of Interface Windows Previous: The Periodic Orbits Window
John Lapeyre
1998-09-04