## 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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