Bifurcations, symmetries and the notion of fixed subspace

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

The generation of non-ordinary state-based peridynamics by the weak form of the peridynamic method

A weak form of the peridynamic (PD) method derived from the classical Galerkin framework by substituting the traditional derivatives into the PD differential operators is proposed. The attractive features of the proposed weak form of PD method include the following: (1) a higher-order approximation than the non-ordinary state-based peridynamic (NOSB-PD) in the strain construction; (2) the NOSB-PD is demonstrated as a special case of the weak form of the PD method; (3) as an extension of the NOSB-PD, the zero-energy mode oscillations in the weak form of the PD can be significantly reduced by introducing higher-order PD derivatives. In addition, a series of numerical tests are conducted. The results show the following: (1) the three proposed stabilization items containing higher-order PD derivatives have a better accuracy and stability than the traditional items of the NOSB-PD. In particular, the stress point stabilization item is preferred since it has the highest accuracy and efficiency and does not introduce any additional parameters; (2) the weak form of PD method is very suitable in dealing with the crack propagation and bifurcation problems.

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.