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...
Transmitted torque is an important performance index of coaxial magnetic gears. This study focuses on the effects of pole-pair numbers on the maximum torque capacity and torque ripple of coaxial magnetic gearsOwing to the magnetic resistance between the pole piece and the medium of air; a torque ripple is generated during magnetic gear operation. In magnetic gear design, maximum torque capacity is considered the higher, the better, and the torque ripple should be the lower, the better. A commercial Finite-element analysis software AN-SYS/Maxwell 2D is applied to simulate the transmitted torque and torque ripple of coaxial magnetic gears by solving the magnetic ield, and mechanical motion coupled models. Model parameters, including size and inertia, are designed via the Autodesk/Inventor. The results of initeelement analysis have shown that there are certain relationships between pole-pair numbers and torque performance. It reveals that the sum of the pole-pair numbers is inversely proportional to the maximum torque capacity; the differences of maximum torque between different polepair number designs could be up to 82%. However, the torque ripple has not been proven for the relationship. For future research, the torque capability per cost would be necessary for magnetic gear product, especially when the price of rare earth metal is seen rising in the past years. Also, the ef iciency of magnetic gear is still important for powertrain system, but it's not easy to reach both high torque and ef iciency, owing to the lux saturation problem. The balance of torque capacity, ef iciency, and cost would be the next challenge.
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without 1 : k resonance). Physical examples motivating considered settings are related to stationary solutions to PDEs with non-local interaction: reversible mixed delay differential equations (MDDEs) and integro-differential equations (IDEs). In the case of S 4 -symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. Algorithms and computational procedures used in this paper are also included.
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