I provide a procedure to construct a specified invariant manifold for a specified system of ordinary differential equations or delay differential equations. The invariant manifold may be any of a centre manifold, a slow manifold, an un/stable manifold, a sub-centre manifold, a nonlinear normal form, or any spectral submanifold. Thus the procedure may be used to analyse pitchfork bifurcations, or oscillatory Hopf bifurcations, or any more complicated superposition. In the cases when the neglected spectral modes all decay, the constructed invariant manifold supplies a faithful, large time, emergent, model of the dynamics of the differential equations. Further, in the case of a slow manifold, this procedure now derives vectors defining the projection onto the invariant manifold along the isochrons: this projection is needed for initial conditions, forcing, system modifications, and uncertainty quantification. The procedure now also empowers one to account for sinusoidal time dependence in the ODEs, such as to derive spectral sub-manifold models of forced nonlinear normal modes. Execute on your computer? The procedure uses computer algebra, the package Reduce, to construct approximations to the invariant manifolds.
- So download and install Reduce, and then download InvariantManifold.zip.
- Startup Reduce in the unzipped InvariantManifold folder.
- Execute in_tex "invariantManifold.tex"$ to load the procedure.
- Test by executing exampleslowman(); and confirm the output is as in invariantManifold.pdf
- See examples in allExamples.pdf and then try for systems of your interest.
Non-autonomous ODEs?Systems with slowly-varying time-dependence, or with sinusoidal time-dependence may be analysed here. Systems with more general time-dependence are significantly more difficult, but are analysed via the web page Normal form of stochastic or deterministic multiscale differential equations.
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