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Imperfect transcritical and pitchfork bifurcations. (English) Zbl 1139.47042

Let \(X\) and \(Y\) be Banach spaces, \(F: \mathbb{R} \times X \to Y\) a nonlinear differentiable map. The authors study the bifurcations of solutions for the equation \[ F(\lambda, u) = 0 \] in a neighborhood of a point \((\lambda_0, u_0)\) under the following assumptions:
(F1) \(\dim N\left(F_u\left(\lambda_0,u_0\right)\right) = \text{codim}\,R \left(F_u\left(\lambda_0,u_0\right)\right) = 1\);
(F2’) \(F_\lambda (\lambda_0,u_0) \in R \left(F_u\left(\lambda_0,u_0\right)\right)\).
It is shown that the solution set near \((\lambda_0,u_0)\) is either an isolated single point or a pair of transversally intersecting curves. In the second part of the paper, the authors consider the imperfect bifurcations of bifurcation diagram in \((\lambda,u)\) space for the equation \[ F(\varepsilon,\lambda, u) = 0 \] under small perturbations \(\varepsilon\). Some results concerning the symmetry breaking of transcritical and pitchfork bifurcations are presented.

MSC:

47J15 Abstract bifurcation theory involving nonlinear operators
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
47N20 Applications of operator theory to differential and integral equations
74G60 Bifurcation and buckling
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