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New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System

Abstract

We provide sufficient conditions for the existence of a pair of symmetric periodic solutions in the Maxwell-Bloch differential equations modeling laser systems. These periodic solutions come from a zero-Hopf bifurcation studied using recent results in averaging theory.

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Acknowledgments

The first author is supported by FAPESP 2018/07344-0. The second author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

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Appendices

Appendix A: Averaging Theory

We consider differential systems of the form

$$ \dot{\textbf{x}}=\textbf{F}_{0}(t,\textbf{x})+ \epsilon \textbf{F}_{1}(t,{\textbf{x}})+\epsilon^{2} \textbf{F}_{2}(t,\textbf{x})+\epsilon^{3}\textbf{F}_{3}(t,{\textbf{x}})+\epsilon^{4}\textbf{F}_{4}(t,\textbf{x})+\epsilon^{5} \widetilde{{\textbf{F}}}(t,\textbf{x},\epsilon), $$
(1)

with x in some open subset Ω of \(\mathbb {R}^{n}\), t ∈ [0,), 𝜖 ∈ [−𝜖0,𝜖0]. We assume Fi and \(\widetilde {\textbf {F}}\) for all i = 1, 2, 3, 4 are T–periodic in the variable t. Let x(t,z, 0) be the solution of the unperturbed system

$$ \dot{\textbf{x}}=\textbf{F}_{0}(t,\textbf{x}), $$

such that x(0,z, 0) = z. We define M(t,z) the fundamental matrix of the linear differential system

$$ \dot{\textbf{y}}=\frac{\partial \textbf{F}_{0}(t,\textbf{x} (t,\textbf{z},0))}{\partial \textbf{x}}\textbf{y}, $$

such that M(0,z) is the n × n identity matrix. The displacement map of system (1) is defined as

$$ \textbf{d}(\textbf{z}, \varepsilon)=\textbf{x}(T,\textbf{z},\varepsilon)-\textbf{z}. $$
(2)

In order to have d(z,ε) well defined we assume that for |ε|≠ 0 sufficiently small the following hypothesis holds:

  • there exists an open set U ⊂Ω such that for all zU the solution x(t,z,ε) is defined on the interval [0,t(z,ε)) with t(z,ε) > T.

This hypothesis is always satisfied when the unperturbed system has a manifold of T-periodic solutions. The standard method of averaging for finding periodic solutions consists in write the displacement map (2) in power series of ε as follows

$$ \textbf{d}(\textbf{z},\varepsilon)= \textbf{g}_{0}(\textbf{z})+\epsilon \textbf{g}_{1}(\textbf{z})+\epsilon^{2} {\textbf{g}}_{2}(\textbf{z})+\epsilon^{3} \textbf{g}_{3}(\textbf{z})+\epsilon^{4} \widetilde{{\textbf{g}}}(\textbf{z},\epsilon), $$

Where for i = 0, 1, 2, 3, 4 we have

$$ \textbf{g}_{i}(\textbf{z})=M(T,\textbf{z})^{-1}\frac{\textbf{y}_{i}(T,\textbf{z})}{i!}, $$

being

$$ \begin{array}{@{}rcl@{}} \textbf{y}_{0}(t,\textbf{z})&=&\textbf{x}(t,\textbf{z},0)-\textbf{z}, \vspace{0.3cm}\\ \textbf{y}_{1}(t,\textbf{z})&=&M(t,\textbf{z}){{\int}_{0}^{t}} M(\tau,\textbf{z})^{-1}\textbf{F}_{1}(\tau,\textbf{x}(\tau,\textbf{z},0))\mathrm{d}\tau,\vspace{0.3cm}\\ \textbf{y}_{2}(t,\textbf{z})&=&M(t,\textbf{z}){{\int}_{0}^{t}} M(\tau,\textbf{z})^{-1}\left[2\textbf{F}_{2}(\tau,\textbf{x}(\tau,\textbf{z},0))+2\frac{\partial \textbf{F}_{1}}{\partial \textbf{x}}(\tau,\textbf{x}(\tau,\textbf{x},0))\textbf{y}_{1}(\tau,\textbf{z}) \vspace{0.3cm}\right.\\ &&\left.+\frac{\partial^{2} \textbf{F}_{0}}{\partial \textbf{x}^{2}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{2} \right]\mathrm{d}\tau,\vspace{0.3cm}\\ \textbf{y}_{3}(t,\textbf{z})&=&M(t,\textbf{z}){{\int}_{0}^{t}} M(\tau,\textbf{z})^{-1}\left[6\textbf{F}_{3}(\tau,\textbf{x}(\tau,\textbf{z},0))+6\frac{\partial \textbf{F}_{2}}{\partial \textbf{x}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z}) \vspace{0.3cm}\right.\\ &&\left.+3\frac{\partial^{2} \textbf{F}_{1}}{\partial \textbf{x}^{2}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{2}+3\frac{\partial \textbf{F}_{1}}{\partial \textbf{x}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{2}(\tau,\textbf{z})+3\frac{\partial^{2} \textbf{F}_{0}}{\partial \textbf{x}^{2}}\vspace{0.3cm}\right.\\ &&\times\left.(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})\odot \textbf{y}_{2}(\tau,\textbf{z})+\frac{\partial^{3} \textbf{F}_{0}}{\partial \textbf{x}^{3}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{3}\right]\mathrm{d}\tau,\vspace{0.3cm}\\ \textbf{y}_{4}(t,\textbf{z})&=& M(t,\textbf{z}){{\int}_{0}^{t}} M(\tau,\textbf{z})^{-1}\left[24\textbf{F}_{4}(\tau,\textbf{x}(\tau,\textbf{z},0)){\kern1.7pt}+{\kern1.7pt}24\frac{\partial \textbf{F}_{3}}{\partial x}(\tau,\textbf{x}(\tau,x,0))\textbf{y}_{1}(\tau,\textbf{z}) \vspace{0.3cm}\right.\\ &&\left.+12\frac{\partial^{2} \textbf{F}_{2}}{\partial x^{2}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{2}+12\frac{\partial \textbf{F}_{2}}{\partial x}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{2}(\tau,\textbf{z})\vspace{0.3cm}\right.\\ &&\left.+12\frac{\partial^{2} \textbf{F}_{1}}{\partial x^{2}}(\tau,\textbf{x}(\tau,\textbf{z},\!0))\textbf{y}_{1}(\tau,\textbf{z})\!\odot\! \textbf{y}_{2}(\tau,\textbf{z}){\kern1.7pt}+{\kern1.7pt}4\frac{\partial^{3} \textbf{F}_{1}}{\partial x^{3}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{3}\vspace{0.3cm}\right.\\ &&\left.+4\frac{\partial \textbf{F}_{1}}{\partial x}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{3}(\tau,\textbf{z})+3\frac{\partial^{2} \textbf{F}_{0}}{\partial x^{2}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{2}(\tau,\textbf{z})^{2} \vspace{0.3cm}\right.\\ &&\left.+4\frac{\partial^{2} \textbf{F}_{0}}{\partial x^{2}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})\odot \textbf{y}_{3}(\tau,\textbf{z}){\kern1.7pt}+{\kern1.7pt}6\frac{\partial^{3} \textbf{F}_{0}}{\partial x^{3}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{2}\vspace{0.3cm}\right.\\ &&\left.\odot \textbf{y}_{2}(\tau,\textbf{z})+\frac{\partial^{4} \textbf{F}_{0}}{\partial x^{4}}(\tau,\textbf{x}(\tau,\textbf{z},0))\textbf{y}_{1}(\tau,\textbf{z})^{4}\right]\mathrm{d}\tau.\vspace{0.3cm} \end{array} $$

The functions g1, g2, g3 and g4 will be called here the averaged functions of order 1, 2, 3 and 4 respectively of system (1).

We say that system (1) has a periodic solution bifurcating from the point z0 if there exists a branch of solutions x(t,z(ε),ε) such that the displacement function satisfies d(z(ε),ε) = 0 and z(0) = z0.

Let \(\pi :{\mathbb {R}}^{m}\times {\mathbb {R}}^{n-m} \rightarrow {\mathbb {R}}^{m}\) and \(\pi ^{\perp }:{\mathbb {R}}^{m}\times {\mathbb {R}}^{n-m} \rightarrow {\mathbb {R}}^{n-m}\) denote the projections onto the first m coordinates and onto the last nm coordinates, respectively. For a point zU we also consider \(\textbf {z}=(a,b)\in {\mathbb {R}}^{m}\times {\mathbb {R}}^{n-m}\). Consider the graph

$$ {\mathcal{Z}}=\lbrace \textbf{z}_{\alpha}=(\alpha,\beta(\alpha)):\alpha\in \overline V \rbrace\subset U $$

such that m < n, V is an open set of \({\mathbb {R}}^{m}\) and \(\beta :\overline V\rightarrow {\mathbb {R}}^{n-m}\) is a \(\mathcal {C}^{4}\) function.

The next theorem provides sufficient conditions for the existence of periodic solutions of the differential system (1) when the set \({\mathcal {Z}}\) is a continuum of zeros to the first non vanishing averaged equation.

Theorem 1

Let r ∈{0, 1, 2, 3}such that r is the first subindex such that gr≢0. In addition to hypothesis (H) assume that

  • (i) the averaged function gr vanishes on \({\mathcal {Z}}\).That is gr(zα) = 0 for all \(\alpha \in \overline V\),and

  • (ii) the Jacobian matrix

    $$D \textbf{g}_{r}(\textbf{z}_{\alpha})=\left( \begin{array}{cc} {\Lambda}_{\alpha} & {\Gamma}_{\alpha} \\ B_{\alpha} & {\Delta}_{\alpha} \end{array}\right), $$

    where Λα = Daπgr(zα), Γα = Dbπgr(zα), Bα = Daπgr(zα) and Δα = Dbπgr(zα), satisfies that det(Δα)≠ 0 for all \(\alpha \in \overline V\).

We define the function

$$ f(\alpha)=-{\Gamma}_{\alpha}{\Delta}^{-1}_{\alpha}\pi^{\perp}\textbf{g}_{r+1}({\textbf{z}}_{\alpha})+\pi \textbf{g}_{r+1}(\textbf{z}_{\alpha}), $$

Then the following statements hold.

  • (a) If there exists αV such that f(α) = 0 and det (Df(α)) ≠ 0, for |ε|≠ 0 sufficiently small, then there is an initial condition z(ε) ∈ U such that \(\textbf {z}(0)=\textbf {z}_{\alpha ^{*}}\) and the solution x(t,z(ε),ε) of system (1) is T-periodic.

For a proof of Theorem 3 see [3]. The ideas of the proof were first presented in [8].

Appendix B: The Functions F i(r,z,𝜃) for i = 1, 2, 3, 4

In the following functions we take S = sin 𝜃, C = cos 𝜃, S2 = sin(2𝜃), C2 = cos(2𝜃) and C3 = cos(3𝜃).

$$ \begin{array}{@{}rcl@{}} \textbf{F}_{1}(r,z,\theta)&=&\left( \frac{2 r}{2 \omega^{2} \sqrt{{b_{0}^{2}}+\omega^{2}}} \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) (b_{0} S_{2}+\omega C_{2}) \right.\\ &&\left.-a_{3} \omega^{2} \left( {b_{0}^{2}}+\omega^{2}\right) C, 0\vphantom{\frac{2 r}{2 \omega^{2} \sqrt{{b_{0}^{2}}+\omega^{2}}}}\right)\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} \textbf{F}_{2}(r,z,\theta)&=&\left( \frac{1}{4 \omega^{4}}\left( \frac{1}{r \left( {b_{0}^{2}}+\omega^{2}\right)}\left( \omega S \left( a_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\right.\right.\\ &&\left.\left.\left.+4 r C \left( c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}-2 b_{0} z\right)\right) +2 b_{0} c_{1} r \sqrt{{b_{0}^{2}}+\omega^{2}}\right.\right.\\ &&+\left.\left.2 b_{0} r C_{2} \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right. -4 r z \left( {b_{0}^{2}}+\omega^{2}\right)\right) \\ &&\left.\left.\left( a_{3} \omega^{2} \left( {b_{0}^{2}}+\omega^{2}\right) C-2 r \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) (b_{0} S_{2}+\omega C_{2})\right)\right.\right.\\ &&\left.\left.+\frac{2 \omega} {b_{0} \sqrt{{b_{0}^{2}}+\omega^{2}}} \left( C \left( {b_{0}^{2}} \left( a_{3} c_{1} \omega^{3}-2 r^{2} \omega \right)+a_{3} c_{1} \omega^{5}-4 {b_{0}^{3}} r^{2} S_{2}\right)\right.\right.\right.\\ &&\left.\left.\left.-2 b_{0} r \omega \left( c_{2} \sqrt{{b_{0}^{2}}+\omega^{2}} (b_{0} S_{2}+\omega C_{2})+b_{0} r C_{3}\right)\right)\right),\frac{r} {\omega^{3} \sqrt{{b_{0}^{2}}+\omega^{2}}}\right.\\ && \left.\left( 2 S \left( -b_{0} c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}+2 {b_{0}^{2}} z+\omega^{2} z\right)\right.\right.\\ &&\left.\left.+\omega C \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\right),\\ \textbf{F}_{3}(r,z,\theta)&=&\left( \frac{1}{4 \omega^{5}}\left( \frac{1}{2 r^{2} \omega \sqrt{{b_{0}^{2}}+\omega^{2}}}\left( 2 r \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) \left( b_{0} S_{2}\right.\right.\right.\right.\\ &&\left.\left.+\omega C_{2}\right)-a_{3} \omega^{2} \left( {b_{0}^{2}}+\omega^{2}\right) C\right) \left( \frac{1}{{b_{0}^{2}}+\omega^{2}}\left( \omega S \left( a_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\right.\\ &&\left.+4 r C \left( c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}-2 b_{0} z\right)\right)+2 b_{0} c_{1} r\sqrt{{b_{0}^{2}}+\omega^{2}}+2 b_{0} r C_{2} \\ &&\left.\left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)-4 r z \left( {b_{0}^{2}}+\omega^{2}\right)\right)^{2}-\frac{2 r \omega}{b_{0} \sqrt{{b_{0}^{2}}+\omega^{2}}} \\ &&\left. \left( -\omega S \left( a_{3} c_{1} \omega^{2} \left( {b_{0}^{2}}+\omega^{2}\right)-4 b_{0} c_{2} r \sqrt{{b_{0}^{2}}+\omega^{2}} \right.\right.\right.\\ &&\left.\left.\left.(b_{0} S{\kern1.7pt}+{\kern1.7pt}\omega C)\right){\kern1.7pt}+{\kern1.7pt}4 b_{0} r^{2} \left( 2 {b_{0}^{2}}{\kern1.7pt}+{\kern1.7pt}\omega^{2}\right) S^{2} C{\kern1.7pt}+{\kern1.7pt}8 {b_{0}^{2}} r^{2} \omega S C^{2}{\kern1.7pt}+{\kern1.7pt}4 b_{0} r^{2} \omega^{2} C^{3}\right)\right)\\ &&+\frac{1}{b_{0} r \left( {b_{0}^{2}}+\omega^{2}\right)}\left( \omega C \left( 4 {b_{0}^{2}} r^{2} C_{2}-a_{3} c_{1} \omega^{2} \left( {b_{0}^{2}}+\omega^{2}\right)\right)+8 {b_{0}^{3}} r^{2} S C^{2}\right.\\ &&\left.+2 b_{0} c_{2} r \omega \sqrt{{b_{0}^{2}}+\omega^{2}} (b_{0} S_{2}+\omega C_{2})\right) \left( \omega S \left( a_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\\ &&\left.\left.+4 r C \left( c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}-2 b_{0} z\right)\right)+2 b_{0} c_{1} r \sqrt{{b_{0}^{2}}+\omega^{2}}+2 b_{0} r C_{2}\right.\\ &&\left. \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)-4 r z \left( {b_{0}^{2}}+\omega^{2}\right)\right)+\frac{2 \omega}{b_{0}} \\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} && \left( -4 a_{3} {b_{0}^{3}} r \omega C^{2}+a_{3} C \left( c_{2} \omega^{4} \sqrt{{b_{0}^{2}}+\omega^{2}}-4 {b_{0}^{4}} r S\right)\right.\\ &&\left.\left.-b_{0} r \omega^{2} ((a_{3}-2 b_{3}) (b_{0} S_{2}+\omega C_{2})+a_{3} \omega )\right)\right),\\ &&\frac{1}{2 \omega^{5}}\left( \frac{\omega}{b_{0} \sqrt{{b_{0}^{2}}+\omega^{2}}} \left( a_{3} \omega^{3} \left( {b_{0}^{2}} c_{1}-2 b_{0} z \sqrt{{b_{0}^{2}}+\omega^{2}}+c_{1} \omega^{2}\right)\right.\right.\\ &&-2 b_{0} c_{2} r \omega^{2} \sqrt{{b_{0}^{2}}+\omega^{2}} C-4 b_{0} r \left( b_{0} c_{2} \omega \sqrt{{b_{0}^{2}}+\omega^{2}} S\right.\\ &&\left.\left.+r \left( 2 {b_{0}^{2}}+\omega^{2}\right) S C+b_{0} r \omega C^{2}\right)\right)-\frac{1}{{b_{0}^{2}}+\omega^{2}}\left( 2 S\right.\\ &&\left. \left( -b_{0} c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}+2 {b_{0}^{2}} z+\omega^{2} z\right)+\omega C \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\\ && \left( \omega S \left( a_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)+4 r C \left( c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}-2 b_{0} z\right)\right)\right.\\ &&\left.\left.\left.+2 b_{0} c_{1} r \sqrt{{b_{0}^{2}}+\omega^{2}}+2 b_{0} r C_{2} \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right.\right.\right.\\ &&\left.\left.\left.-4 r z \left( {b_{0}^{2}}+\omega^{2}\right)\right)\right)\right),\\ \textbf{F}_{4}(r,z,\theta)&=&\left( \frac{1}{4 \omega^{4} \sqrt{{b_{0}^{2}}+\omega^{2}}}\left( -2 c_{3} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right) C \omega^{2}+\frac{1}{r}\left( 2 c_{3} r \omega C^{2}\right.\right.\right.\\ &&\left.\left.+\left( c_{2} \delta_{3} \sqrt{{b_{0}^{2}}+\omega^{2}}-2 b_{0} (b_{3}-c_{3}) r S\right) C+2b_{3} r \omega S^{2}\right)\right.\\ &&\left( -4 r z \omega^{2} C^{2}-S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\\ &&+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\\ &&\left.\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)\\ &&+\frac{1}{2 r^{2} \omega^{3}}\left( -4 b_{0} r^{2} \omega C^{3}-2 r \left( 4 r S {b_{0}^{2}}+c_{2} \omega^{2} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) C^{2}\right.\\ &&-\omega \left( -4 b_{0} r^{2} S^{2}-2 b_{0} (b_{2}-c_{2}) r \sqrt{{b_{0}^{2}}+\omega^{2}} S+c_{1} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)\right) C\\ &&\left.-2 b_{2} r \omega^{2} \sqrt{{b_{0}^{2}}+\omega^{2}} S^{2}\right) \left( \frac{1}{{b_{0}^{2}}+\omega^{2}}\left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\right.\\ &&+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\\ &&\left.\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)^{2}\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+\frac{2 r \omega} {\sqrt{{b_{0}^{2}}+\omega^{2}}} \left( -4 r^{2} \omega^{2} C^{3}-8 b_{0} r^{2} \omega S C^{2}-S \left( 2 \left( 2 {b_{0}^{2}}+\omega^{2}\right) S_{2} r^{2}\right.\right.\\ &&\left.\left.\left.\left.\left.-2 b_{0} (b_{2}{\kern1.7pt}-{\kern1.7pt}c_{2}) \omega \sqrt{{b_{0}^{2}}+\omega^{2}} S r{\kern1.7pt}+{\kern1.7pt}c_{1} \delta_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right){\kern1.7pt}+{\kern1.7pt}(b_{2}{\kern1.7pt}-{\kern1.7pt}c_{2}) r \omega^{2} S_{2} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\right.\right.\\ &&-\frac{1}{4 r^{3} \omega^{4}}\left( 2 r \omega \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right.\\ && C^{2}+b_{0} \left( \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r S \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right) C-2 r \omega \left( 2 b_{0} z\right.\\ &&\left.\left.+b_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) S^{2}\right) \left( -4 r^{2} S \left( 2 (b_{3}-c_{3}) r \omega C+2 b_{0} (b_{3}-c_{3}) r S\right.\right.\\ &&\left.-c_{2} \delta_{3} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) \omega^{4}+\frac{2 r \omega} {{b_{0}^{2}}+\omega^{2}} \left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\\ &&\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\\ &&\left.\left.+r \omega S_{2} \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right) \left( 4 r^{2} \omega^{2} C^{3}+8 b_{0} r^{2} \omega S C^{2}+\omega S\right.\right.\\ &&\left. \left( c_{1} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)-2 b_{0} (b_{2}-c_{2}) r \sqrt{{b_{0}^{2}}+\omega^{2}} S\right)+r \left( (c_{2}-b_{2}) \sqrt{{b_{0}^{2}}+\omega^{2}} \omega^{2}\right.\right.\\ &&\left.\left.+2 r \left( 2 {b_{0}^{2}}+\omega^{2}\right)S\right) S_{2}\right)+\frac{1}{\sqrt{{b_{0}^{2}}+\omega^{2}}}\left( -4 r z \omega^{2} C^{2}-S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)\right.\right.\\ &&+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\\ &&\left.\left.\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right) \left( \frac{1}{{b_{0}^{2}}+\omega^{2}}\right.\right.\\ &&\left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right.\right.\\ &&\left.\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)^{2}+\frac{2 r \omega}{\sqrt{{b_{0}^{2}}+\omega^{2}}} \left( -4 r^{2} \omega^{2} C^{3}\right.\\ &&-8 b_{0} r^{2} \omega S C^{2}-S \left( 2 \left( 2 {b_{0}^{2}}+\omega^{2}\right) S_{2} r^{2}-2 b_{0} (b_{2}-c_{2}) \omega \sqrt{{b_{0}^{2}}+\omega^{2}} S r\right.\\ &&\left.\left.\left.\left.\left.+c_{1} \delta_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right)+(b_{2}-c_{2}) r \omega^{2} S_{2} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\right)\right),\\ && -\frac{1}{2 \omega^{3}}\left( -2 a_{3} r \omega C+2 c_{3} r \omega C-2 b_{0}b_{3} r S+2 b_{0} c_{3} r S+\frac{1}{2 r \omega^{3} \left( {b_{0}^{2}}+\omega^{2}\right)}\right.\\ &&\left( 4 b_{0} r^{2} \omega C^{2}+2 r \left( (c_{2}-a_{2}) \sqrt{{b_{0}^{2}}+\omega^{2}} \omega^{2}+2 r \left( 2 {b_{0}^{2}}+\omega^{2}\right) S\right) C\right.\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&\left.+\omega \left( c_{1} \delta_{3} {b_{0}^{2}}-2 (b_{2}-c_{2}) r \sqrt{{b_{0}^{2}}+\omega^{2}} S b_{0}+\omega^{2} \left( c_{1} \delta_{3}+2 a_{3} z \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\right)\\ && \left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right.\right.\\ &&\left.\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)-\frac{1}{4 r^{2} \omega^{4} \sqrt{{b_{0}^{2}}+\omega^{2}}}\left( a_{3} \omega^{4}\right.\\ &&+a_{3} {b_{0}^{2}} \omega^{2}+b_{0} \delta_{3} \omega^{2}-2 a_{2} z \sqrt{{b_{0}^{2}}+\omega^{2}} \omega^{2}+2 r C \left( 2 b_{0} z-c_{1} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) \omega +{b_{0}^{3}} \delta_{3}\\ &&\left.+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right) \left( \frac{1}{{b_{0}^{2}}+\omega^{2}}\left( 4 r z \omega^{2} C^{2}+S\right.\right.\\ &&\left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r \omega C \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)+2 r S \left( 4 z {b_{0}^{2}}\right.\right.\\ &&\left.\left.\left.+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)^{2}+\frac{2 r \omega} {\sqrt{{b_{0}^{2}}+\omega^{2}}} \left( -4 r^{2} \omega^{2} C^{3}-8 b_{0} r^{2} \omega S\right.\\ && C^{2}-S \left( 2 \left( 2 {b_{0}^{2}}+\omega^{2}\right) S_{2} r^{2}-2 b_{0} (b_{2}-c_{2}) \omega \sqrt{{b_{0}^{2}}+\omega^{2}} S r\right.\\ &&\left.\left.\left.+c_{1} \delta_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right)+(b_{2}-c_{2}) r \omega^{2} S_{2} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)+\frac{a_{2} \sqrt{{b_{0}^{2}}+\omega^{2}}}{8 r^{3} \omega^{4}} \\ &&\left( -4 r^{2} S \left( 2 (b_{3}-c_{3}) r \omega C+2 b_{0} (b_{3}-c_{3}) r S-c_{2} \delta_{3} \sqrt{{b_{0}^{2}}+\omega^{2}}\right) \omega^{4}\right.\\ &&+\frac{2 r \omega} {{b_{0}^{2}}+\omega^{2}}\left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r S \left( 4 z {b_{0}^{2}}\right.\right.\right.\\ &&\left.\left.\left.+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)+r \omega S_{2} \left( 4 b_{0} z+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right) \\ &&\left( 4 r^{2} \omega^{2} C^{3}+8 b_{0} r^{2} \omega S C^{2}+\omega S \left( c_{1} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)-2 b_{0} (b_{2}-c_{2})\right.\right.\\ &&\left.\left. r \sqrt{{b_{0}^{2}}+\omega^{2}} S\right)+r \left( (c_{2}-b_{2}) \sqrt{{b_{0}^{2}}+\omega^{2}} \omega^{2}+2 r \left( 2 {b_{0}^{2}}+\omega^{2}\right) S\right) S_{2}\right)\\ &&+\frac{1}{\sqrt{{b_{0}^{2}}+\omega^{2}}}\left( -4 r z \omega^{2} C^{2}-S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r \omega C \left( 4 b_{0} z\right.\right.\right.\\ &&\left.\left.\left.+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)\\ && \left( \frac{1}{{b_{0}^{2}}+\omega^{2}}\left( 4 r z \omega^{2} C^{2}+S \left( b_{0} \delta_{3} \left( {b_{0}^{2}}+\omega^{2}\right)+2 r \omega C \left( 4 b_{0} z\right.\right.\right.\right.\\ &&\left.\left.\left.+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}}\right)+2 r S \left( 4 z {b_{0}^{2}}+(b_{1}-c_{1}) \sqrt{{b_{0}^{2}}+\omega^{2}} b_{0}+2 z \omega^{2}\right)\right)\right)^{2}\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+\frac{2 r \omega} {\sqrt{{b_{0}^{2}}+\omega^{2}}}\left( -4 r^{2} \omega^{2} C^{3}-8 b_{0} r^{2} \omega S C^{2}-S \left( 2 \left( 2 {b_{0}^{2}}+\omega^{2}\right) S_{2} r^{2}\right.\right.\\ &&\left.-2 b_{0} (b_{2}-c_{2}) \omega \sqrt{{b_{0}^{2}}+\omega^{2}} S r+c_{1} \delta_{3} \omega \left( {b_{0}^{2}}+\omega^{2}\right)\right)\\ &&\left.\left.\left.+(b_{2}-c_{2}) r \omega^{2} S_{2} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right)\right)\\ &&\left.\left.+c_{2} \delta_{3} \sqrt{{b_{0}^{2}}+\omega^{2}}\right)\right). \end{array} $$

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Cândido, M.R., Llibre, J. & Valls, C. New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System. Math Phys Anal Geom 22, 16 (2019). https://doi.org/10.1007/s11040-019-9313-9

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Keywords

  • Maxwell-Bloch
  • Averaging theory
  • Periodic solutions
  • Zero-Hopf bifurcations

Mathematics Subject Classification (2010)

  • 34C29
  • 37C27