Frequency conditions for the global stability of nonlinear delay equations with several equilibria

TL;DR

The paper develops a frequency-domain framework for guaranteeing global stability of nonlinear delay equations with several equilibria by analyzing derivative cocycles on exterior powers. It introduces a robust, spectrally informed method based on a quadratic-constraint formulation and a nonautonomous perturbation of the $m$-fold exterior-derivative semigroup, together with an approximation scheme to verify the resulting frequency inequalities in practice. The theory is applied to scalar delay models, notably the Suarez–Schopf oscillator and Mackey–Glass equations, yielding sharper global stability regions and demonstrating convergence of the numerical scheme to the true transfer-operator bounds. The work integrates spectral theory, operator-perturbation analysis, and delay-equation numerics to offer a practical pathway toward establishing global stability via propagated spectral-dichotomy properties and adapted metrics, with implications for multistable and delayed-dynamics systems in infinite-dimensional settings.

Abstract

In our adjacent work, we developed a spectral comparison principle for compound cocycles generated by delay equations. It allows to derive frequency inequalities for the uniform exponential stability of such cocycles by means of their comparison with stationary problems. Such inequalities are hard to verify purely analytically, and in this work we develop approximation schemes to verify some of the arising frequency inequalities. Besides some general theoretical results, in applications we stick to the case of scalar equations. By means of the Suarez--Schopf delayed oscillator and the Mackey--Glass equations, we demonstrate applications of the theory to reveal regions in the space of parameters where the absence of closed invariant contours can be guaranteed. Since the frequency inequalities are robust, so close systems also satisfy them, we expect the method to actually imply the global stability, as in known finite-dimensional results utilizing variants of the closing lemma, which is still awaiting developments in infinite dimensions.

Figures (4)

• Figure 1: An illustration to the decomposition \ref{['EQ: TensorSpaceDelayCompoundDecompositionBoundarySubspaces']} with $m=2$ and $n=1$. Here $\mathcal{L}^{\otimes}_{2} = L_{2}([-\tau,0]^{2};\mu^{\otimes 2};\mathbb{R})$ is decomposed into the sum of the boundary subspaces $\partial_{0}\mathcal{L}^{\otimes}_{2}$, $\partial_{1}\mathcal{L}^{\otimes}_{2}$, $\partial_{2}\mathcal{L}^{\otimes}_{2}$, and $\partial_{12}\mathcal{L}^{\otimes}_{2}$ over the faces $\mathcal{B}_{0}$, $\mathcal{B}_{1}$, $\mathcal{B}_{2}$, and $\mathcal{B}_{12}$, respectively. These subspaces are naturally isomorphic to appropriate $L_{2}$-spaces via the restriction operators $R_{0}$, $R_{1}$, $R_{2}$, and $R_{12}$, respectively.
• Figure 2: Graphs of the largest singular values $\alpha_{N}(-\nu_{0} + i \omega)$ for $\nu_{0}=0.01$ versus $\omega$ computed via the explicit representation of the transfer operator in the case of \ref{['EQ: ExplicitResolventParamatersForTest']} (left) and \ref{['EQ: ExplicitResolventParamatersForTestSuarezSchopf']} (right). Truncation parameters are taken as $N=5$ (blue), $N=10$ (green), $N=20$ (purple), and $N=30$ (brown). The red graph is the $L_{2}$-norm of the kernel from \ref{['EQ: ExplicitEstimateForWViaL2NormK']}. The horizontal lines pass through the threshold value $\Lambda^{-1}$ (orange), the $L_{2}$-norms $|e^{p\theta}\bar{K}_{2}|$ (left) and $|\bar{K}_{1}|$ (right) of the asymptotic kernels (red), and the norm $\bar{W}$ of the asymptotic integral operator (olive) on the vertical axis. See the repository for implementation details.
• Figure 3: Graphs of the largest singular values $\alpha_{T,N}(-\nu_{0} + i \omega)$ versus $\omega$ computed via the numerical implementation of the approximation scheme \ref{['DESC: AS1DelayCompound']}--\ref{['DESC: AS4DelayCompound']} applied to \ref{['EQ: Suarez-SchopfLinearizedRewriten']} with $\alpha = 0.6$, $\tau = 0.83$, and $R=R_{0}(\alpha,\tau)$ given by Lemma \ref{['LEM: SSmodelRadiusEstimateWithRestr']}. Parameters of the scheme are taken as $m=2$, $\Lambda = \Lambda_{R}$, $\nu_{0} = 0.01$, $T=15.77$, $\Omega = 37.5$, and (Left): $N=2$ (blue), $N=5$ (orange), $N=10$ (green), and (Right): $N=10$ (blue), $N=20$ (orange), $N=30$ (green). The horizontal lines pass through the threshold value $\Lambda^{-1}$ (red) and the norm $\bar{W}$ of the asymptotic integral operator (olive) on the vertical axis. See Remark \ref{['REM: DelayCompoundNumerical']} and the repository for implementation details.
• Figure 4: Graphs of the largest singular values $\alpha_{T,N}(-\nu_{0} + i \omega)$ versus $\omega$ computed via the numerical implementation of the approximation scheme \ref{['DESC: AS1DelayCompound']}--\ref{['DESC: AS4DelayCompound']} applied to \ref{['EQ: MackeyGlassNumericalSchemeExample']} with $\gamma = 0.1$, $\beta = 0.2$, $\kappa = 10$, $\tau = 4.5$, and $\Lambda$ given by \ref{['EQ: MackeyGlassStabilityLambdaDef']}. Parameters of the scheme are taken as $m=2$, $\Lambda$ as above, $\nu_{0} = 0.01$, $T=15$, $\Omega = 37.5$, and (Left): $N=2$ (blue), $N=5$ (orange), $N=10$ (green), and (Right): $N=10$ (blue), $N=20$ (orange), $N=30$ (green). The horizontal lines pass through the threshold value $\Lambda^{-1}$ (red) and the norm $\bar{W}$ of the asymptotic integral operator (olive) on the vertical axis. See the repository for implementation details.

Theorems & Definitions (58)

• Proposition 2.1
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Proposition 3.1
• Corollary 3.1
• Remark 3.6