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 Dec 6 comment Periodic solution of nonlinear differential equation The typo is in the last $z(t)$. the first $z(t)$ is ok. No, we restrict $u(0)\neq 0$. The main point is: Is it true if the linearization system of differential equation have a periodic solution then the original nonlinear have a periodic solution too Dec 6 revised Periodic solution of nonlinear differential equation edir z' (t) Dec 6 revised Periodic solution of nonlinear differential equation added 2 characters in body Dec 6 asked Periodic solution of nonlinear differential equation Dec 5 answered Periodic parametric curve on cylinder Dec 5 comment Differential equation, Stability , Lyapunov function How about this argument: We can reparameterize $S=\{(\sqrt{C}\cos u,\frac{\sqrt{C}}{\sqrt{2}}\sin u, v): u,v\in \mathbb{R}\}$. Let $r(t)= (x(t),y(t),z(t))$ and $r(0)=(x_0,y_0,z_0)$, then $x(t)=\sqrt{C}\cos (t-t_0)$ and $y(t)=\frac{\sqrt{C}}{\sqrt{2}}\sin (t-t_0)$ with $t_0$ satisfying $x_0=\sqrt{C}\cos t_0$ and $y_0=-\frac{\sqrt{C}}{\sqrt{2}}\sin t_0$. Since $z'=xy$ then $z'(t)=\frac{C}{2\sqrt{2}}\sin(2t-t_0)$, hence $z(t)=-\frac{C}{4\sqrt{2}}\cos(2t-t_0)$. Since $r(2\pi)=(x_0,y_0,z_0)=r(0)$ then $r(t)$ is periodic. Dec 5 revised Parametric curve on cylinder surface added 411 characters in body Dec 5 answered Parametric curve on cylinder surface Dec 5 comment Differential equation, Stability , Lyapunov function How about the possiblity if the solution only trace the first octan? In this octan, z'=xy>0 so z always increasing. What make you sure that the solution trace along all quadrant of domain Dec 5 comment Differential equation, Stability , Lyapunov function How about the possiblity if the solution only trace the first octan? In this octan, z'=xy>0 so z always increasing. What make you sure that the solution trace along all quadrant of domain Dec 4 accepted Space of bounded functions is reflexive if the domain is finite Dec 4 accepted Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space Dec 3 awarded Citizen Patrol Dec 3 asked Space of bounded functions is reflexive if the domain is finite Dec 3 comment Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space No. I find the question from exercise of conway book Dec 3 asked Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space Nov 30 comment Strictly convex Inequality in $l^p$ If the vectors is multiple of the other by scalar implies the equality is trivial. How about the converse? Nov 30 asked Strictly convex Inequality in $l^p$ Nov 24 comment Differential equation, Stability , Lyapunov function I just took the definition from Hirsch and Smale's Book. And it say like that. $dV/dt<0$ for asymptotically stable and $dV/dt\leq 0$ for stable one. You can check the book now. Nov 24 comment Differential equation, Stability , Lyapunov function In Lyapunov theorem, if $dV/dt<0$ then the equilibrium is asymptotically stable and $dV/dt\leq 0$ then the equilibrium is stable (An equilibrium point $x_0$ is stable iff for every neighboorhood $U$ of $x_0$ there is a neighboorhood $U_1$ of $x_0$ in $U$ such that every solution $r(t)$ with $r(0)$ in $U_1$ is defined and in $U$ for all $t>0$)