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Consider the following oscillatory integral $$ I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac {(1 - \cos(2x)) (1 - \cos(2y))} {2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y. $$ where $ …

Consider the following system of coupled differential equations \begin{eqnarray*} \dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) & = & …

Consider the following matrix function $$ f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0, $$ where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers. N …

Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following differential e …

Let $a$, $b$ be two real positive parameters with $a>b$, and consider the following nonlinear differential equation: \begin{align} \dot{x}_{\varepsilon}(t) = a - b\sin(x_{\varepsilon}(t))+\varepsilon, …

Consider the following system of coupled differential equations \begin{align} \dot{x}_{1}&= -b_1\sin(x_{1})+c(\sin(x_{2})-\sin(x_{3})) \\ \dot{x}_{2}&= a-2c\sin(x_{2})+b_1\sin(x_{1})-b_4\sin(x_{4}) \\ …

Let us consider the following differential equation $$ \dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}. $$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the abov …

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(t) …

Consider a dynamical system of the form $$ \dot{x}=f(x), \quad x\in X, $$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ …