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Global Bifurcation in Symmetric Systems of Nonlinear Wave Equations

Carlos Garcia-Azpeitia, Ziad Ghanem, Wieslaw Krawcewicz

Abstract

In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and non-trivial damping. As a motivating example, we consider an application of our result to a system of $N$ identical vibrating strings with dihedral coupling relations.

Global Bifurcation in Symmetric Systems of Nonlinear Wave Equations

Abstract

In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and non-trivial damping. As a motivating example, we consider an application of our result to a system of identical vibrating strings with dihedral coupling relations.

Paper Structure

This paper contains 15 sections, 27 theorems, 258 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Functional Space Reformulation of \ref{['eq:system']}
  3. Local and Global Bifurcation in \ref{['eq:system']}
  4. The Local Bifurcation Invariant and Krasnosel'skii's Theorem
  5. The $G$-Isotypic Decomposition of Our Functional Space
  6. The $s$-Folding Homomorphism
  7. Computation of the Local Bifurcation Invariant
  8. The Rabinowitz Alternative
  9. Resolution of the Rabinowitz Alternative and the $\mathbf{H}$-Fixed Point Setting
  10. A Motivating Example: Symmetric System of $N$ Vibrating Strings With Nonlinear Forces, Damping and Delay
  11. The Special Case of Dihedral Symmetries
  12. The $G$-Equivariant Degree
  13. The $S^1$-Equivariant Degree
  14. The Twisted Equivariant Degree
  15. A Computational Formula for the Twisted $G$-Equivariant Leray-Schauder Degree

Key Result

Theorem 1.1

The trivial solution to eq:system with the assignments $A(\alpha):= \zeta(\alpha)(L+\mathop{\mathrm{Id}}\nolimits)$, $\zeta:\mathbb R \rightarrow \mathbb R$ differentiable, strictly monotonic and under the conditions $\nu \in {\mathbb{Q}}$, $\delta > 0$, $\tau \neq \pi {\mathbb{Q}}$, undergoes a glo for every index quadruple $(m,n,j,k) \in 2 {\mathbb{N}} - 1 \times {\mathbb{N}} \times \{0,\ldots,r

Figures (3)

  • Figure 1: Cycle of $N$ Vibrating Strings with $\Gamma=D_{N}$ symmetries
  • Figure 2: Graph of the sigmoid function $\zeta(\alpha)$.
  • Figure 3: For the Dihedral symmetry group $\Gamma = D_7$, the functions $U_1(t,x) = \cos(x)(\cos(t)\operatorname{Re}(v_1) - \sin(t)\operatorname{Im}(v_1))$, $U_2(t,x) = \cos(x)\cos(t)\operatorname{Re}(v_1)$ and $U_3(t,x) = \cos(x)\cos(t)\operatorname{Im}(v_1)$ exhibit symmetries at least $(H_{1,1}^0)$, $(T_{1,1}^0)$ and $(S_{1,1}^0)$, respectively

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.2
  • ...and 53 more