WIAS Preprint No. 1573, (2010)

A permutation characterization of Sturm attractors of Hamiltonian type


  • Fiedler, Bernold
  • Rocha, Carlos
  • Wolfrum, Matthias
    ORCID: 0000-0002-4278-2675

2010 Mathematics Subject Classification

  • 35B41 34C29


  • Global attractor, Sturm permutation




We consider Neumann boundary value problems of the form $u_t = u_xx + f $ on the interval $0 leq x leq pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^primeprime + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation.

Appeared in

  • J. Differential Equations, 252 (2012) pp. 588-623.

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