Symmetry subspaces, slices and phase-space geometry

Computing in an invariant subspace

If the equations are equivariant under a reflection \(S\) (for example \(x \to -x\) with \(u\) odd and \(T\) even), the solutions with \(S u = u\) form an invariant subspace \(V_S\). Conatus can compute entirely in \(V_S\): the user declares the symmetry and the parity of each scalar field (the parities of vector components follow: the component normal to the mirror changes sign), and the library

  1. checks that the equations and the boundary conditions are equivariant, and refuses otherwise, naming the offending term;

  2. keeps only the independent coefficients: along a reflected Chebyshev direction the polynomials of the right parity, along a Fourier direction cosines or sines, and only the taus and boundary conditions that are not mirror images of each other;

  3. runs the transforms, the nonlinear terms on the folded grid, and the banded solves in \(V_S\), which halves each group;

  4. provides the embedding \(E: V_S \to V\) and the projection \(P: V \to V_S\) (\(PE = I\)) to move a symmetric solution into the full space and back.

In curvilinear geometries each symmetry (azimuthal rotation by \(2\pi/n\), equatorial reflection, axial reflection) acts on coefficients as a signed permutation; the subspace keeps one representative per orbit. Regularity at the axis is not a symmetry choice: it is always imposed by the basis.

What a subspace hides

A computation in \(V_S\) cannot see instabilities that break the symmetry. explain() always says so, and symmetry_breaking_check computes the leading eigenvalue (or Floquet multiplier) in each of the other isotypic components, reusing the same discretisation with the opposite parities. A symmetric solution is declared stable only after this check.

Slices

A continuous symmetry (translation along a periodic direction) is reduced by a slice. In the first-Fourier-mode slice, each state is shifted so that the phase of its first Fourier mode is zero; the shift is reconstructed by integrating the phase velocity. Relative periodic orbits become periodic orbits of the reduced flow. States where the first mode vanishes (the slice border) are detected and reported.

Trajectories, sections and manifolds

  • Poincaré sections: crossings of a hypersurface in a declared direction are located on the dense output of the time stepper (or exactly, by a partial step of the discrete map); successive crossings give the first-return map.

  • Unstable manifolds of equilibria: by the parametrisation method, a polynomial expansion in the unstable directions, then globalised by time integration.

  • Connections: heteroclinic orbits between invariant solutions by shooting with projection boundary conditions onto the stable and unstable subspaces.

References

  • M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser (2002).

  • N. B. Budanur, P. Cvitanović, R. L. Davidchack and E. Siminos, Reduction of SO(2) symmetry for spatially extended dynamical systems, Phys. Rev. Lett. 114 (2015) 084102.

  • X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds, Indiana Univ. Math. J. 52 (2003).

  • W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal. 10 (1990) 379–405.

  • P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum, ChaosBook.org.