Twisted $$\Gamma \times \mathbb T^n$$ Γ × T n -equivariant degree with n-parameters: computational formulae and applications

Da̧bkowski, Mieczysław K.; Krawcewicz, Wiesław; Lv, Yanli

doi:10.1007/s11784-016-0309-9

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“…One should point out, that other algebraic structures were also used for the ranges of various versions of equivariant degrees. For instant the twisted Γ × S 1 -equivariant degree with one parameter is taking its values in the A(Γ)-module A 1 (Γ × S 1 ) (see [5]) and the twisted Γ × T n -equivariant degree with n-parameters is taking its values in A(Γ)-module A 1 (Γ × T n ) (see [11]). These additional structures, for which there is a well established computational base, can be used in order to simplify the computations of the Euler ring U (G).…”

Section: Introductionmentioning

confidence: 99%

Multiple periodic solutions for Γ-symmetric Newtonian systems

Da̧bkowski

Krawcewicz

et al. 2017

Journal of Differential Equations

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The existence of periodic solutions in Γ-symmetric Newtonian systemsẍ = −∇f (x) can be effectively studied by means of the Γ × O(2)-equivariant gradient degree with values in the Euler ring U (Γ × O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U (Γ × O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ × O(2)), and the reduced Γ × O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.1 for example a difference of the equivariant gradient degrees on large and small ball). Such cases are for example when the system (1) is asymptotically linear or satisfy a Nagumo-type growth condition. Then clearly the existence of non-trivial solutions (i.e. outside the small ball) can be concluded by the fact that ω = 0. However, one can be also interested to predict the existence of multiple 2π-periodic solutions with different types of symmetry. In such a case, the coefficients of ω corresponding to the so-called maximal orbit type can provide the crucial information in order to formulate such results. But the maximality of such obit types implies that it is a generator of the Burnside ring A(G), therefore it can actually be detected by the Γ × O(2)-equivariant degree with no free parameter, which can be much easier computed than the equivariant gradient degree. Similar arguments apply to the system (2), which we can consider as a bifurcation problem with a parameter λ. More precisely, in this case we are looking for critical values λ o of the parameter λ > 0, to which we can associate the Γ × O(2)-equivariant gradient bifurcation invariants ω(λ o ) ∈ U (Γ×O(2)) classifying the bifurcation of 2π-periodic solutions from the zero solution. The existence and multiplicity of such bifurcating branches of 2π-periodic solutions can be described from the information contained in the invariants ω(λ o ). Consequently, all the essential information needed to establish the existence and multiplicity results for the systems (1) and (2) can be extracted from the Γ × O(2)-equivariant degree (with no free parameter) of J which takes values in the Burnside ring A(G). It is clear that the Γ × O(2)-equivariant degree without free parameter can be easily computed (without getting entangled in complicated technical details), has similar properties and provides enough information for analyzing these problems.Nevertheless, let us emphasize that only the equivariant invariants ω ∈ U (G) (without truncation of its coefficients) provide a complete equivariant topological classification for the related solution sets to (1) or (2).To illustrate the usage and the computations of the associated with the systems (1) and (2) equivariant invariants, in section 7 we present several examples of symmetric Newtonian systems, for which the exact values of the a...

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Section: Introductionmentioning

confidence: 99%