Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems
Sören Schwenker
备注：
19, pages, 1 figure; Minor revisions in the example: cosmetic changes in Figure 1 and its explanation on page 14, replaced "saddle-node bifurcation" with "transcritical bifurcation" on page 15
We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.
With the help of computer algebra system-\textsc{Mathematica}, this paper considers the weak center problem and local critical periods for bi-center of a $Z_2$-Equivariant quintic system with eight parameters. The order of weak bi-center is identified and the exact maximum number of bifurcation of critical periods generated from the bi-center is given via the combination of symbolic calculation and numerical analysis.
Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the assumption that the nonlinear terms satisfy a linear estimate in a ball. If the estimate is global, then the branch is unbounded. The results are formulated in an equivariant setting where the system can have multiple branches of periodic orbits characterized by different groups of symmetries. The non-local analysis is based on the equivariant degree method, which…
Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural nets. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on $n$ symbols, $S_n$; in particular, the standard representation of $S_n$. It is shown that spurious minima do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that ca…
We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This $G$-equivariant spectral flow shares all common properties of the integer valued classical spectral flow, and it can be non-trivial even if the classical spectral flow vanishes. Our main theorem uses the $G$-equivariant spectral flow to study bifurcation of periodic solutions for autonomous Hamiltonian systems with symmetries.
We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks [Hepworth, Theory Appl. Categ. 22 (2009), 542-587]. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and thei…
Equivariant Bifurcation and Absolute Irreducibility in $\R^8$ A contribution to the Ize conjecture
M. Field [5]引用了J. Ize的一个未发表的工作，其结果是通过紧李群的绝对不可约表示的稳定性损失导致稳定态的分支。证明的主要成分是假设，即对于紧李群的绝对不可约表示，存在一个闭子群，其不动点空间是奇数维的。然后，使用Brouwer度，得到结果。代数Ize猜想（AIC）是关于紧李群的绝对不可约表示存在至少一个子群具有奇维不动点空间的假设. Lauterbach和Matthews [10]已经证明了（AIC）一般来说是不正确的。事实上，他们已经构建了SO{4}的有限子群的三个无限族，它们在$\R^4$上绝对不可约，并且对于它们中的每一个，任何稳定子群都有一个偶数维的不动点空间。此外，在[10]中表明，尽管（AIC）失败，但原始猜想至少对这三个家族中的两个家族中的群是正确的。本文对文[10]中定义的第三族给出了一个类似的分支结果。我们继续下去，并建立一个家庭的群体作用绝对不可约的$\R^8$只有偶数维不动点空间。然后我们讨论了这种情况下的定态分岔。本文最后讨论了如何将文[10]中的结果推广到作用于$\R^{4}$的更大的群集合。
The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing $F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is $\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$ using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces, that play an important role in this model. We exploit them in order to generalise results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to p…
We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a clus…
The Bose--Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose--Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that des…
We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $Σ\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $τ\colon Σ\to B^G$, a subshift $Δ\subset Σ$ is of finite type, resp. sofic, if and only if so is the image $τ(Δ)$. Generaliza…
