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Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 313--327 | DOI:10.5890/DNC.2023.06.007

Ranses Alfonso Rodriguez, S. Roy Choudhury

Department of Mathematics, University of Central Florida, Orlando FL32816-8005, USA

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Abstract

The dynamics of a delayed multiparameter nonlinear Mathieu equation: $$ \ddot{x}+(\delta+\epsilon\alpha \cos{t})x+\epsilon\gamma x^3=\epsilon\beta\int_{-\infty}^{t}{x(\tau)\xi e^{-\xi(t-\tau)}}d\tau,$$ is investigated in the neighborhood of $\delta=1/4$. Three different features interact here: a distributed delay, cubic nonlinearity and 2:1 parametric resonance. The averaging method is used to obtain a slow flow that is analyzed for stability and bifurcations, and the resulting predictions are compared against actual system responses. In particular, we find regimes where: i. the slow flow has a zero stable fixed point (implying Amplitude Death), or ii. the slow flow goes to a stable non-zero fixed point (implying periodic solutions), or iii. the slow flow goes to a stable periodic solution at large times (corresponding to a quasiperiodic system response). All of these types of behavior would be very difficult to isolate otherwise, except by intensive numerical searching of the multiparameter space. However, there are also parameter regimes where the slow flow predictions may occasionally disagree with the actual system response $x(t)$ in cases where that has large amplitude or exhibits bounded aperiodicity. The reasons for these discrepancies are also carefully considered.}[\hfill Dynamics\par\hfill Averaging\par \hfill Nonlinear Mathieu \par\hfill Distributed delay][A.C.J. Luo][28 February 2021][19 November 2021][1 April 2023][2023 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle %\thispagestyle{fancy} \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} Mathieu's equation is one of the canonical equations in nonlinear vibrations theory, applied mathematics, and applications in a variety of engineering areas [1--7]. A recent comprehensive review\cite{Ver} covers the classical analytical techniques, as well as a variety of nonlinear effects and their use in various engineering models. A second review\cite{Kov-Ran} also covers the basic analytical techniques, and then more recent extensions including nonlinearities, discrete delays\cite{Mor-Ran}, as well as fractional\cite{Ran-Sah} and quasiperiodic\cite{Zou-Ran} extensions of the classical equation. More recent treatments include, but are not restricted to, studies which add effects such as fast harmonic forcing, and applications to systems with high-speed milling and large gains\cite{Ham-Bel13}. In this paper, the first of two, we investigate the effect of coupling a distributed delay to nonlinearity and the 2:1 parametric resonance. The results here will then serve as a foundation to add other nonlinearities and a second parametric forcing term in the following paper. More specifically, we investigate the dynamics of a parametrically excited, nonlinear differential delay equation (DDE) containing the so-called `weak generic kernel'\cite{Cus,Mac} \begin{equation}\label{EQ1} \ddot{x}(t)+( \delta+a \cos{t} )x(t)+cx(t)^3=b\int_{-\infty}^{t}x(\tau)\xi e^{-\xi (t-\tau)}d\tau, \end{equation} where $a$ is the amplitude of the parametric resonance, $b$ is the amplitude of the delay, $c$ is the amplitude of the cubic nonlinearity, $\delta$ is the frequency squared of the simple harmonic oscillator, and $\xi$ is the integral delay parameter. Note that the delay increases as the parameter $\xi$ decreases, since the exponential term inside the integral becomes larger. This paper is organized as follows. Section \ref{sec:2} derives the averaged equations, also referred to as slow flow, and Section \ref{sec:3} analyzes the stability and bifurcations of the slow flow fixed points. Detailed numerical examples are considered in Section \ref{sec:4} including comparisons of slow flow predictions with actual system responses, while Section \ref{sec:conclusions} summarizes the conclusions. In particular, analysis of the slow flow allows the direct identification of parameter regimes with very different types of dynamics. In multiparameter systems like ours, this can otherwise only be done by very time-consuming numerical searching, almost like finding a needle in a haystack. However, the slow flow approach does have limitations, and that is also considered carefully in Section \ref{sec:4}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% NEW SECTION (SECTION 2) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Averaging} \label{sec:2} Defining $z(t)=\int_{-\infty}^{t}x(\tau)\xi e^{-\xi (t-\tau)}d\tau$ and $y(t)=\dot{x}(t)$, \eqref{EQ1} may be reformulated as the extended system of ordinary differential equations \begin{subequations}\label{eqreform} \begin{align} \dot{x}(t) & =y(t), \\ \dot{y}(t) & =bz(t)-( \delta+a \cos{t} )x(t)-cx(t)^3, \\ \dot{z}(t) & =\xi( x(t)-z(t) ). \end{align} \end{subequations} We now introduce a small parameter $\varepsilon$ by scaling $a=\varepsilon\alpha$, $b=\varepsilon\beta$, and $c=\varepsilon\gamma$; in addition, we detune near the 2:1 resonance\cite{Nay-Moo} by setting $\delta=1/4+\varepsilon\delta_1$. Hence \begin{subequations} \label{EQSys} \begin{align} \dot{x}(t) & =y(t), \\ \dot{y}(t)& =-\frac{1}{4}x(t) +\varepsilon ( \beta z(t)-( \delta_1+\alpha \cos{t} )x(t)-\gamma x(t)^3), \label{EQ2.1a} \\ \dot{z}(t) & =\xi( x(t)-z(t) ). \label{EQ2.2b} \end{align} \end{subequations} When $\varepsilon=0$, Eq. \eqref{EQ2.1a} reduces to $\ddot{x}(t) =-\frac{1}{4}x(t)$, which implies the System \eqref{EQSys} has solution \begin{subequations}\label{EqSol} \begin{align} x(t) & =A\cos{( \frac{t}{2}+\phi )}, \\ y(t) & =\dot{x}(t) =-\frac{A}{2}\sin{( \frac{t}{2}+\phi )}, \\ z(t) & =\int_{-\infty}^{t}x(\tau)\xi e^{-\xi (t-\tau)}d\tau. \end{align} \end{subequations} Then, for $\varepsilon >0$, we look for a solution in the form of \eqref{EqSol}, but treating $A$ and $\phi$ as time dependant. Variation of parameters now gives the following equations for the slow $O(\epsilon)$ variation of $A(t)$ and $\phi(t)$ \cite{Nay-Moo} \begin{subequations}\label{EQVP} \begin{align} \dot{A}(t) & =-2\varepsilon\sin{( \frac{t}{2}+\phi )}F(x(t),y(t),z(t),t), \\ \dot{\phi}(t) & = -2\frac{\varepsilon}{A}\cos{( \frac{t}{2}+\phi )}F(x(t),y(t),z(t),t), \end{align} \end{subequations} where $F(x(t),y(t),z(t),t)=( \beta z(t)-( \delta_1+\alpha \cos{t} )x(t)-\gamma x(t)^3)$ and $x(t)$, $y(t)$, $z(t)$ are given by Eqs. \eqref{EqSol}. Since $\varepsilon$ is small,the variation of $A(t)$ and $\phi(t)$ is slow, and hence these equations \eqref{EQVP} may be averaged as usual, replacing the right hand sides by their averages over one $2\pi$ period of the forcing function $\cos{t}$ to give: \begin{subequations}\label{EQVP1} \begin{align} \dot{A}(t) & \approx -\frac{\varepsilon}{\pi} \int_0^{2\pi}\sin{( \frac{t}{2}+\phi )}F(x(t),y(t),z(t),t) dt, \\ \dot{\phi}(t) & \approx -\frac{\varepsilon}{A\pi} \int_0^{2\pi}\cos{( \frac{t}{2}+\phi )}F(x(t),y(t),z(t),t) dt. \end{align} \end{subequations} Evaluating the integrals in Eqs. \eqref{EQVP1}, one gets \begin{subequations}\label{EQVP2} \begin{align} \dot{A} & =\varepsilon A( \frac{\alpha}{2}\sin{2\phi} -\beta\frac{2\xi}{1+4\xi^2}), \\ \dot{\phi} & = \varepsilon ( \frac{\alpha}{2}\cos{2\phi} -\beta\frac{4\xi}{1+4\xi^2} +\frac{3\gamma}{4}A^2+\delta_1 ). \end{align} \end{subequations} From the last equation, we have \begin{subequations}\label{SloF1} \begin{align} A' & =A( \frac{\alpha}{2}\sin{2\phi} -\beta\hat{\xi}), \\ \phi' & = \frac{\alpha}{2}\cos{2\phi} -2\beta\hat{\xi} +\frac{3\gamma}{4}A^2+\delta_1, \end{align} \end{subequations} where primes now represent differentiation with respect to the \textit{slow time} $\eta=\varepsilon t$, and $\hat{\xi}=\frac{2\xi}{1+4\xi^2}$. For future use in deriving a second set of slow flow equations, we also recast these equations into an equivalent representation. Transforming from polar coordinates $A$ and $\phi$ to rectangular ones $u$ and $v$ through the substitutions $u=A\cos{\phi}$, $v=-A\sin{\phi}$, yields an alternate form of the slow flow equations \begin{subequations}\label{SloF2} \begin{align} u' & =-\beta\hat{\xi}u + ( \delta_1-\frac{\alpha}{2} -2\beta\hat{\xi} )v +\frac{3\gamma}{4}v(u^2+v^2), \label{SloF2a} \\ v' & = ( -\delta_1-\frac{\alpha}{2} +2\beta\hat{\xi} )u -\beta\hat{\xi}v -\frac{3\gamma}{4}u(u^2+v^2) . \label{SloF2b} \end{align} \end{subequations} Also, using this transformation in \eqref{EqSol} yields the lowest order system response \begin{equation}\label{SloF-OrS} x(t)=A\cos{( \frac{t}{2}+\phi )}=u\cos{( \frac{t}{2})}+v\sin{( \frac{t}{2} )}. \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% NEW SECTION (SECTION 3) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Slow flow equilibria: stability and bifurcation} \label{sec:3} As usual, from Eq. \eqref{SloF-OrS}, any non-trivial equilibrium point of the slow flow \eqref{SloF1} or \eqref{SloF2} corresponds to a periodic motion in the original system \eqref{EQSys}, whose stability is also given by the stability of the fixed point. The origin $u=v=0$ is an exception, since it is an equilibrium point in both the slow flow and the original system. In order to find slow flow equilibrium points, we set $A'=\phi'=0$ in Eqs. \eqref{SloF1}. Then, by some algebraic work and using the trigonometric identity $\sin^2{2\phi}+\cos^2{2\phi}=1$, we get the following condition on $R=A^2$ \begin{equation}\label{Quad} 9\gamma^2R^2+24\gamma( \delta_1-2\beta\hat{\xi} )R +4( 20\beta^2\hat{\xi}^2-\alpha^2+4\delta_1^2-16\beta\delta_1 \hat{\xi} )=0. \end{equation} Since Eq. \eqref{Quad} is quadratic in $R$, we can find two solutions, which are \begin{equation}\label{RSol} R=\frac{4}{3\gamma}[ 2\beta\hat{\xi}-\delta_1\pm\frac{1}{2}\sqrt{\alpha^2-4\beta^2\hat{\xi}^2} ]. \end{equation} Eqs. \eqref{SloF2} are invariant under the transformation $(u,v)\mapsto (-u,-v)$, which means that each value of $R$ corresponds to two nontrivial slow flow equilibria located $180^\circ$ apart. For nontrivial equilibrium points we must require $R$ to be nonegative, yielding the conditions \begin{equation} \delta_1\le 2\beta\hat{\xi}\pm\frac{1}{2}\sqrt{\alpha^2-4\beta^2\hat{\xi}^2}. \end{equation} For real roots, we need the term inside the square root in Eq. \eqref{RSol} to be positive, leading to the condition \begin{equation}\label{In21} | \hat{\xi} |\le \frac{1}{2}| \frac{\alpha}{\beta} |. \end{equation} For a given $\alpha$, the inequality \eqref{In21} gives a condition on the delay parameters $\beta$ and $\xi$ (through $\hat{\xi}$) for the existence of nontrivial slow flow equilibrium points. In the case that $|\beta/\alpha|\le 1$, nontrivial fixed points exist for all $\hat{\xi}$. When inequality \eqref{In21} is satisfied, there are at least two nontrivial slow flow equilibrium points if \begin{equation}\label{In22} \delta_1 < 2\beta\hat{\xi}+\frac{1}{2}\sqrt{\alpha^2-4\beta^2\hat{\xi}^2}, \end{equation} with at least two more if \begin{equation}\label{In23} \delta_1 < 2\beta\hat{\xi}-\frac{1}{2}\sqrt{\alpha^2-4\beta^2\hat{\xi}^2}, \end{equation} is also satisfied. Hence, it is possible to have up to five slow flow equilibria points, the origin plus four nontrivial ones, the latter corresponding to periodic solutions of the unaveraged system. Next, we investigate the parameter combinations of $\alpha$, $\beta$, $\gamma$, $\delta_1$ and $\hat{\xi}$ that can cause the slow flow equilibrium points to change stability, and the associated bifurcations. To do this, we use the trace and determinant of the Jacobian matrix evaluated at an equilibrium point in the standard way. From \eqref{SloF2}, the Jacobian matrix is \begin{equation}\label{JacMat} J=\left[ \begin{array}{cc} \frac{3\gamma}{2}uv-\beta\hat{\xi} & \delta_1-\frac{\alpha}{2}-2\beta\hat{\xi}+\frac{3\gamma}{4}u^2+\frac{9\gamma}{4}v^2 \\ -\delta_1-\frac{\alpha}{2}+2\beta\hat{\xi}-\frac{3\gamma}{4}u^2-\frac{9\gamma}{4}v^2 & -\frac{3\gamma}{2}uv-\beta\hat{\xi} \end{array} \right], \end{equation} where $u$ and $v$ are to be evaluated at the slow flow equilibria. The trace of the Jacobian matrix is \begin{equation}\label{Tr} \text{Tr}(J)=-2\beta\hat{\xi}. \end{equation} Note that $\text{Tr}(J)$ is a function of the delay parameters only, and in particular does not depend on $R$. Therefore, $\text{Tr}(J)=0$ at \textit{all} of the slow flow equilibrium points when $\beta=0$ or $\hat{\xi}=0$. Thus, a change of stability and a possible Hopf bifurcation (birth of a limit cycle) will occur at $\beta=0$ if $\text{Det}(J)>0$. The determinant of the Jacobian matrix is \begin{equation}\label{Det1} \text{Det}(J) = \frac{27}{16}\gamma^2R^2-\frac{\alpha^2}{4}+\delta_1^2 +5\beta^2\hat{\xi}^2-3\gamma R(2\beta\hat{\xi}-\delta_1) +\frac{3}{4}\alpha\gamma(v^2-u^2), \end{equation} where the $(u^2+v^2)$ terms were replaced with $R$. The $(v^2-u^2)$ term is simplified by multiplying the right hand side of Eq. \eqref{SloF2a} by $v$ and subtracting from it the right hand side of Eq. \eqref{SloF2b} multiplied by $u$, which gives \begin{equation} (v^2-u^2) = \frac{1}{2\alpha}[ 3\gamma R^2+4R( \delta_1-2\beta\hat{\xi} ) ]. \end{equation} Using this expression in Eq. \eqref{Det1} yields \begin{equation}\label{Det2} \text{Det}(J) = \frac{45}{16}\gamma^2R^2-\frac{\alpha^2}{4}+\delta_1^2 +5\beta^2\hat{\xi}^2 -4\beta\delta_1\hat{\xi}-\frac{9}{2}\gamma R(2\beta\hat{\xi}-\delta_1). \end{equation} Substituting the values of $R$ at the nontrivial slow flow equilibria, or Eq. \eqref{RSol}, into Eq. \eqref{Det2} yields \begin{equation}\label{Det3} \text{Det}(J)=\alpha^2-4\beta^2\hat{\xi}^2 \pm2 ( 2\beta\hat{\xi}-\delta_1 )\sqrt{\alpha^2-4\beta^2\hat{\xi}^2}. \end{equation} \begin{figure}[!t] \centering \includegraphics[width=.75\linewidth]{Figures/Fig1.eps} \caption{Bifurcation surfaces for the slow flow equilibria.} \label{Fig:1a} \end{figure} \begin{figure}[!t] \centering \includegraphics[width=.75\linewidth]{Figures/Fig2.eps} \caption{Bifurcation surfaces for fixed points of the slow flow equilibria.} \label{Fig:1b} \end{figure} Changes in stability occur when the determinant is zero. Enforcing this in Eq. \eqref{Det3}, we can solve for a critical value of the detuning parameter $\delta_1$, in terms of $\hat{\xi}$, $\alpha$ and $\beta$ \begin{equation}\label{Eq30} \delta_1=2\beta\hat{\xi} \pm\frac{1}{2}\sqrt{\alpha^2-4\beta^2\hat{\xi}^2}. \end{equation} \begin{table}[t!] \doublerulesep 0.1pt \tabcolsep 7.8mm \centering \caption{Values of the parameters at the points presented in Figs.\ref{Fig:1a} and \ref{Fig:1b}.} \label{tab:1}

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