IMP Overview

A Matlab toolbox for rigorous and nonrigorous parameterization of invariant manifolds for ODEs. Invariant sets are parameterized as coefficient sequences in some analytic function space and a proof of existence and rigorous error bounds come in the form of a-posteriori analytic error estimates.

See REFS for further details and examples.

Classes

The basic usage revolves around several classes which efficiently perform and store important computations. Each class is briefly outlined below.

Scalar

The scalar class gives a finite approximation representation for analytic functions of the form, $f: D^d \to \mathbb{R}^n$ where $D^d$ is the unit ball in $\mathbb{C}^d$.

Properties

• Basis: Taylor, Fourier, Chebyshev
• Coefficient: A $d$-dimensional array of coefficients
• Dimension: The value of $d$ which is equivalent to the number of nontrivial dimensions in the coefficient array.
• NumericalClass: double or intval (requires IntLab toolbox for Matlab)
• Truncation: Integer vector of length $d$ denoting the number of coefficients in each direction.

Hidden Properties

• Weight: Allows specification of weights for alternate $\ell_1$ weights. This should only be set to 'ones' to optimize numerical stability.

Methods

• append
• bestfitdecay
• decay
• dot
• double
• dt
• eval
• exponent
• fixtime
• fouriertaylortimes
• imag
• intlabpoly
• intval
• intvaltimes
• inv
• leftmultiplicationoperator
• minus
• mtimes
• ndims
• norm
• plus
• real
• shift
• sqrt
• subdivide
• subsref
• tailratio
• uminus

Usage

Chart

Properties

• Coordinate
• Truncation: {M, [N1,N2,...,Nd]}
• MaterialCoordinate{TimeSpan,[s11,s12;s21,s22;...;sd1,sd2]}
• Tau
• ErrorBound
• TimeDirection

Hidden Properties

• Dimension
• Weight
• InitialData
• SubDivisionDepth
• SubDivisionTol
• CoefType
• BasisType %{'Taylor',etc}
• ParentId

Methods

Usage

Atlas

function x = solve_logistic_eqn(x0, N)
x = x_0
for n = 1:N-1
    x(n+1) = dot(x, fliplr(x))/(n+1);
end

Examples