Bifurcation, symmetry, and cosymmetry in differential equations unresolved with respect to the derivative with variational branching equations (Q845155)

The authors consider first order autonomous differential equations in real Banach spaces, not necessarily resolved in terms of the derivative; that is, they consider \(u \in E_1\) and, for \(p = d u / dt \in T E_1\) and \(\varepsilon \in R\) a perturbation parameter, an equation of the general form \(F(p,x,\varepsilon) = 0\), with \(F : T E_1 \times E_1 \times R \to E_2\). Moreover, \(x_0\) is a solution for all \(\varepsilon\), \(F(0,x_0,\varepsilon) = 0\). They assume \(F\) is smooth, and that \(A := (\partial F / \partial p) (0,x_0,0)\) and \(B = - (\partial F / \partial x) (0,x_0,0) \) are Fredholm operators. Thus the equation can be written, in a vicinity of \(x_{0}\), as \[ A \;dx/dt \;= \;B \;(x-x_0) \;- \;R ( x_0 , dx/dt,x-x_0,\varepsilon), \] where of course \( R ( x_0 , dx/dt,x-x_0,\varepsilon) = F (p,x,\varepsilon) ' A p + B (x-x_0)\). Finally, they make bifurcation assumptions, i.e. that the spectrum of the operator \(B\) decomposes into a part strictly contained in the left half complex plane and two eigenvalues \(\pm i \alpha\) of multiplicity \(n\) which cross the imaginary axis. Under these assumptions, the authors study the branching equation (obtained via a Lyapounov-Schmidt reduction), and study the case where the original problem has a symmetry -- which will be inherited by the branching equation. In this case, there will be a relation between Lie symmetries of the branching equation (see e.g. Olver's or Ovsjannikov's books on symmetries of differential equations [\textit{P. J. Olver}, Applications of Lie groups to differential equations. Graduate Texts in Mathematics, 107. New York etc.: Springer-Verlag. (1986; Zbl 0588.22001)] [\textit{L. V. Ovsjannikov}, Group analysis of differential equations. Moscow: ``Nauka''. 399 p. R. 1.90 (1978; Zbl 0484.58001)] and Yudovich's cosymmetry properties [\textit{V. I. Yudovich}, J. Appl. Math. Mech. 36, 424--432 (1972); translation from Prikl. Mat. Mekh. 36, 450--459 (1972; Zbl 0257.34040)]. Concrete applications are considered, including the rotation groups SO(2), SH(2) and H(2).