Recent zbMATH articles in MSC 37J40 https://www.zbmath.org/atom/cc/37J40 2022-04-28T18:15:44.554837Z Werkzeug On limit cycles near two centres and a double homoclinic loop in Liénard differential system https://www.zbmath.org/1482.34098 2022-04-28T18:15:44.554837Z "Wei, Lijun" https://www.zbmath.org/authors/?q=ai:wei.lijun "Zhang, Qingjing" https://www.zbmath.org/authors/?q=ai:zhang.qingjing "Zhang, Xiang" https://www.zbmath.org/authors/?q=ai:zhang.xiang The number of limit cycles in a Liénard differential system are bifurcate from the families of periodic orbits near two centres and a double homoclinic loop of a Hamiltonian system with an elliptic Hamiltonian function is studied. The lower bounds on the number of limit cycles in different cases of this system are reached as can be shown by means of Maple programs. Reviewer: Valery A. Gaiko (Minsk) Lyapunov functions and asymptotics at infinity of solutions of equations that are close to Hamiltonian equations https://www.zbmath.org/1482.37053 2022-04-28T18:15:44.554837Z "Sultanov, O. A." https://www.zbmath.org/authors/?q=ai:sultanov.oskar-a The author considers perturbed Hamiltonian systems of second order: $\frac{dx}{dt} = \partial_y H(x,y,t)\ ,\ \frac{dy}{dt} = - \partial_x H(x,y,t) + F(x,y,t),$ displaying the asymptotic properties for $$t\rightarrow\infty$$ $H(x,y) = H_0(x,y) + \sum_{k=1}^\infty t^{-k/q}H_k(x,y), \; F(x,y,t)=\sum_{k=1}^\infty t^{-k/q}F_k(x,y)\ ,\ q\in \mathbb{N},$ and having $$(0,0)$$ as center-type fixed point. The asymptotic properties are studied by considering a Lyapunov function of the form $V(x,y,t)=2H(x,y,t) + \sum_{k=1}^p t^{-k/q}v_k(x,y) ,$ where the $$v_k$$'s are subject to $(\partial_y H_0)\partial_x v_k - (\partial_x H_0)\partial_y v_k = G_k(x,y) .$ The right hand side of the above equation is defined in terms of the $$v_k$$'s as follows: $G_1(x,y) = - (2F_1)\partial_y H_0,$ $G_2(x,y) = - (2F_2)\partial_y H_0 - \partial_y v_1 F_1 - [(\partial_y H_1)\partial_x v_1 - (\partial_x H_1)\partial_y v_1],$ etc. If for some $$p\in \mathbb{N}$$ the system is solved with $$v_k= O(x^2+y^2)$$ and $G_p(x,y)=-\gamma(x^2+y^2) + O((x^2+y^2)^{3/2}),$ the Lyapunov function is constructed. Several examples are considered, e.g., the 4-th Painlevé equation, auto-resonance systems, etc. Reviewer: Vladimir Răsvan (Craiova) Theoretical and numerical results on Birkhoff normal forms and resonances in the Born-Oppenheimer approximation https://www.zbmath.org/1482.37062 2022-04-28T18:15:44.554837Z "Latigui, Nawel" https://www.zbmath.org/authors/?q=ai:latigui.nawel "Ghomari, Kaoutar" https://www.zbmath.org/authors/?q=ai:ghomari.kaoutar "Messirdi, Bekkai" https://www.zbmath.org/authors/?q=ai:messirdi.bekkai Summary: This paper mainly focuses on the Birkhoff normal form theorem for the Born-Oppenheimer Hamiltonians. Normal forms are accessible via those of the effective Hamiltonian obtained by the Grushin reduction method and the pseudodifferential calculus with operator-valued symbols. Resonance situations are discussed; the theoretical computations of Birkhoff normal form in the 1:1 resonance are written explicitly. Our approach gives compatible numerical results while using a computer program. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials https://www.zbmath.org/1482.37063 2022-04-28T18:15:44.554837Z "Sun, Yingte" https://www.zbmath.org/authors/?q=ai:sun.yingte The author considers the ordinary differential equation $-y''+u(\omega t)y=Ey,$ where $$y=y(t)$$, $$t\in\mathbb{R}$$, $$u$$ is a real-analytic quasi-periodic function such that $|\ell \, \omega|\geq M(\alpha/(|\ell|^\tau)),$ where $$\alpha>0$$, $$\tau>d-1$$, $$\ell\in\mathbb{Z}^{d}\backslash 0$$. It is proved that for any $$a>0$$, $$r>0$$ and $$\alpha>0$$ there exists $$M^*=M^*(d,r,a,\alpha)$$ such that if $$M\geq M^*$$ for the interval $$[a,\infty)$$ there exists a Cantor subset $$\Delta_\alpha\subset\Delta$$ such that for any $$\sqrt{E}\in\Delta_\alpha$$, the above equation has two linearly independent solutions. Reviewer: Ekin Uğurlu (Ankara)