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Mathematical aspects of numerical
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ManualJan2018
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ManualJan2018
Contents
Mathematical aspects of numerical continuation and handling of singularities
Prediction
Correction
Stepsize control
Singularity handling
General software aspects of MatCont
System definition
Continuation and output
Curve file
Options
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
Poincaré section and Poincaré map
Equilibrium continuation
Mathematical definition
Initialization by time integration
Bifurcations and their normal form coefficients
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
Limitcycle initialization
Example
The phase response curve
Continuation of codim 1 bifurcations
Fold Continuation
Hopf Continuation
Period Doubling
Continuation of fold bifurcation of limit cycles
Continuation of torus bifurcation of limit cycles
Continuation of codim 2 bifurcations
Branch Point Continuation
Branch Point of Cycles Continuation
Continuation of homoclinic and heteroclinic orbits
Homoclinic orbits: Mathematical definition
Bifurcations
Homoclinic initialization
Examples
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