MATCONT:

Continuation toolbox for ODEs

in Matlab

W. Govaerts, Yu. A. Kuznetsov, H.G.E. Meijer,


B. Al-Hdaibat, V. De Witte, A. Dhooge, W. Mestrom, N. Neirynck, A.M. Riet and B. Sautois

January 2018, adapted for version MatCont 6.10.

Belgium

Utrecht University The Netherlands

The Netherlands

• Contents
• Mathematical aspects of numerical continuation and handling of singularities
  • Prediction
  • Correction
    • Pseudo-arclength continuation
    • Moore-Penrose continuation
  • Stepsize control
  • Singularity handling
    • Test functions
    • Multiple test functions
    • Singularity matrix
    • User location
  
  
• General software aspects of MatCont
  • System definition
  • Continuation and output
  • Curve file
  • Options
    • The options-structure
    • Derivatives of the defining system of the curve
    • Singularities and test functions
    • Locators
    • User functions
    • Defaultprocessor
    • Special processors
    • Workspace
    • Adaptation
    • Tangent search order
    • Summary
  • Failure handling
  • General remarks on the data flow
  • Directories
  • Global Structures
  
  
• The odefile of a dynamical system
  • Structure and construction of an odefile
  • Handling auxiliary functions in the construction of the odefile
  • Access to function and Jacobian values
  • Time integration using the odefile
  
  
• Time integration and Poincaré maps
  • Time integration
    • Solver output properties
    • Jacobian matrices
    • The Steinmetz-Larter example
  • Poincaré section and Poincaré map
    • Poincaré maps in CL/SMALL>_MATCONT
  
  
• Equilibrium continuation
  • Mathematical definition
  • Initialization by time integration
  • Bifurcations and their normal form coefficients
    • Branch point locator
  • Equilibrium initialization
  • Bratu example
  
  
• Continuation of limit cycles
  • Mathematical definition
  • Discretization of a limit cycle
  • Plotting the output of a continuation of limit cycles
  • Initialization by time integration
  • Bifurcations of limit cycles
    • Branch Point Locator
    • Normal form coefficients
  • Limitcycle initialization
  • Example
  • The phase response curve
  
  
• Continuation of codim 1 bifurcations
  • Fold Continuation
    • Mathematical definition
    • Bifurcations along a fold curve
    • Fold initialization
    • Adaptation
    • Example: a catalytic oscillator
  • Hopf Continuation
    • Mathematical definition
    • Bifurcations along a Hopf curve
    • Hopf initialization
    • Adaptation
    • Example
  • Period Doubling
    • Mathematical definition
    • Output of a continuation of period doubling bifurcation points
    • Bifurcations along a flip curve
    • Period doubling initialization
    • Example
  • Continuation of fold bifurcation of limit cycles
    • Mathematical definition
    • Bifurcations along a fold of cycles curve
    • Fold of cycles initialization
    • Example: the fast Morris-Lecar equations
  • Continuation of torus bifurcation of limit cycles
    • Mathematical definition
    • Bifurcations along a Neimark-Sacker curve
    • Torus bifurcation initialization
    • Example: an autonomous electronic circuit
  
  
• Continuation of codim 2 bifurcations
  • Branch Point Continuation
    • Mathematical definition
    • Bifurcations
    • Branch Point initialization
    • Example
  • Branch Point of Cycles Continuation
    • Mathematical Definition
    • Bifurcations
    • Branch Point of Cycles initialization
    • Example
  
  
• Continuation of homoclinic and heteroclinic orbits
  • Homoclinic orbits: Mathematical definition
    • Homoclinic-to-Hyperbolic-Saddle Orbits
    • Homoclinic-to-Saddle-Node Orbits
  • Bifurcations
  • Homoclinic initialization
  • Examples
    • CL_MATCONT: the MLFast example
    • MATCONT: the Koper example
  • Heteroclinic orbits and further literature
  
  
• Continuer example: a curve object
• The Brusselator example: Continuation of a solution to a boundary value problem in a free parameter
• Bibliography
• About this document ...