Brief Intro to LOCBIF

LOCBIF is distributed as part of the Windows version of XPP thanks to the kind contribution of Alex Khibnik who parted with the source code. LOCBIF was originally a stand-alone DOS program and a recent improved version, CONTENT . The version here is somewhat old but still seems to work OK. AUTO is the bifurcation package that I originally included in XPP but was unable to get the port to work in Windows. Anyone who'd like to try should contact me.

LOCBIF vs AUTO

LOCBIF is easier to use and handles bifurcations of equilibria and fixed points of maps better than AUTO. On the other hand, AUTO handles boundary value problems and continuation of periodic orbits better than LOCBIF. If you really need AUTO, you can get the native X version for Windows of XPPAUT or switch to Linux (the recommended solution).

Introduction

LOCBIF can continue and find bifurcations for the following four types of problems depending on parameters p with state variables x .
  1. Equilibria of ODEs :

    dx/dt = f(x,p)

  2. Fixed points of maps :

    x(n+1) = f(x(n),p)

  3. Periodic solutions of forced systems :

    dx/dt = f(x,p,t) where f(x,p,t+T)=f(x,p,t)

  4. Autonomous oscillations of ODEs:

    dx/dt = f(x,p)

NOTE The last three types of curves are essentially the same since the periodically forced system is equivalent to a map taking the phase-space at t=0 to the phase-space at t=T . In the case of an autonomous oscillation, the one forms the Poincare map and looks at that map. Fixed points of the latter two systems correspond to periodic solutions. To invoke LOCBIF, you must click on File Locbif from the main WINPP menu. You will get a window like this:

The basic objects that LOCBIF computes are curves in the product space of the parameters and the phase space. For example, it will compute curves of equilibria F(x,p)=0 for ODEs. Along these curves, LOCBIF keeps track of special points where bifurcations occur. These special points are obtained by evaluating certain bifurcation functions the zeros of which tell you something about the system. For example, a zero determinant tells you that there is a zero eigenvalue. The equations for these functions along with the ODE/MAP form the curve-defining system. So, if you want to follow the set of equilibria where there is a zero eigenvalue, you need two parameters since you must solve the n+1 equations:

F(x,p1,p2) = 0

det A(p1,p2) = 0

along some curve in (x,p1,p2) space. There are many different types bifurcations and I will refer the reader to a comprehensive book such as Wiggins , Guckenheimer and Holmes or Kuznetsov.

File menu

Nothing here works yet.

Numerics

This invokes the numerics menu. There are many numerical parameters and their suggested settings. I will briefly describe them but a comprehensive description is best obtained from the hardcopy documentation for the DOS version lbf.ps

Curve parameters

These are parameters for the continuation in phase-space.


Orbit parameters

These are parameters for the LOCBIF integrators.

Plots

These items let you redraw the diagram and set the plot parameters. Only two-dimensional plots are allowed at present. The Autoscale check box will scale the plot to fit the diagram.

Edit

Note that by defining different curves, you can look at a system as one parameter varies, then as a different one varies and keep them separated.

Run

Equilibria

This command lets you find equilibria and other curves as one or more parameters varies. To start, you must have an equilibriaum point as the initial condition in WinPP or you must start on a point you have grabbed from a prior calculation. This opens the dialog box:

There are many different choices that you can pick here. For ordinary equilibria, choose Equilibrium and LOCBIF will track the equilibria as a parameter varies. Note the number in the parenthese in each point is the number of parameters you must choose for that curve. Suppose after computing a curve of equilibria, you find a Hopf bifurcation. Then you can do a two-parameter continuation finding all equilibria that are points of a Hopf bifurcation. Once you have chosen the type of curve to follow, choose the parameters and the maximum and minimum ranges. Finally choose a direction (+/- 1) and click on OK. If all goes well, the diagram will be computed. You can terminate the calculation by clicking on ABORT.

Fixed points of maps

This command lets you find fixed points of maps as a function of some parameters and lets you trace multi-parameters curves of

Examples

These examples are from the LOCBIF manual, adopted for WINPP. Click on the ecology model, eco.ode . This is a modification of the Lotka-Volterra model:

x'= x(1-beta x) - x y/(1 + alpha x)

y'= -y(1+delta y) + x y/(1+alpha x)

Traversing the Diagram

Clicking on the Traverse button once you have computed a curve, allows you to move a little cross around and information about the points is given in the window above the graph. You can use the mouse to move through the diagram by clicking on

Then click on Grab to load this point into the initial conditions and to set the parameters at the value corresponding to the point. Use this to start new curves and extend old ones. Click on Esc or type Esc to end traversing the diagram without grabbing a point.

Bibliography

  1. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, NY, 1990
  2. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations, Springer-Verlag, NY, 1983.
  3. Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NY, 1995 (1998, New Edition).