论文标题
关于三阶差分方程家族的封闭轨迹和伪 - 突发设备的存在
On the existence of closed trajectories and pseudo-trajectories for a family of third order differential equations
论文作者
论文摘要
本文的目的是研究微分方程的封闭轨迹$ \ dddot {z}+a \ ddot {z}+b \ dot {z}+abz = \ varepsilon f（z，z，\ dot {z}，\ dot {z}，\ ddot {z} {z}）$。在第一种情况下，我们考虑$ f（z，\ dot {z}，\ ddot {z}）= 1 $ and $ b = {\ rm sgn}（h（z，\ dot {z}}，\ ddot {z}），其中$ h（z，\ dot {z}，\ ddot {z}）= z^2+（\ dot {z}）^2+（\ ddot {z}）^2-1 $。我们表明，微分方程等效于分段平滑的微分系统，该系统将单位球作为不连续性歧管。在这种情况下，我们获得了存在封闭的伪 - 区域的条件。在第二种情况下，我们考虑$ \ varepsilon \ neq 0 $足够小，$ b> 0 $和$ f（z，\ dot {z}，\ ddot {z}）$我们表明，不受干扰的微分方程具有一个填充不变平面的等效周期性解决方案。然后，我们研究了使用平均理论从该二维等方面分叉的最大极限循环数。因此，在同一家族中，我们具有定期解决方案（在参数创建平滑方程式的情况下）和伪周期解决方案（在Filippov Systems的情况下）。
The goal of this article is to study the existence of closed trajectories for the differential equation $\dddot{z}+a\ddot{z}+b\dot{z}+abz=\varepsilon F(z,\dot{z},\ddot{z})$ in two situations. In the first situation, we consider $F(z,\dot{z},\ddot{z})=1$ and $b={\rm sgn}(h(z,\dot{z},\ddot{z}))$, where $h(z,\dot{z},\ddot{z})=z^2+(\dot{z})^2+(\ddot{z})^2-1$. We show that the differential equation is equivalent to a piecewise smooth differential system that admits the unit sphere as the discontinuity manifold. We obtain conditions for the existence of a closed pseudo-trajectory in this case. In the second situation, we consider $\varepsilon \neq 0$ sufficiently small, $b>0$, and $F(z,\dot{z},\ddot{z})$ a $n$-degree polynomial. We show that the unperturbed differential equation has a family of isochronous periodic solutions filling an invariant plane. Then, we study the maximum number of limit cycles which bifurcate from this 2-dimensional isochronous using the averaging theory. Thus, within the same family, we have periodic solutions (in the case where the parameters create a smooth equation) and also pseudo-periodic solutions (in the case of Filippov systems).
