Nonuniqueness phenomena in discontinuous dynamical systems and their regularizations (Q6564692)

Let \(\alpha\), \(\beta\colon\mathbb{R}^n\to\mathbb{R}\) be sufficiently smooth functions and consider the sets\N\[\N\Sigma_\alpha=\{y\in\mathbb{R}^n\colon\alpha(y)=0\}, \quad \Sigma_\beta=\{y\in\mathbb{R}^n\colon\beta(y)=0\}.\N\]\NConsider the regions\N\[\N\mathcal{R}^{++}=\{y\in\mathbb{R}^n\colon\alpha(y)>0, \;\beta(y)>0\},\N\]\N\N\[\N\mathcal{R}^{+-}=\{y\in\mathbb{R}^n\colon\alpha(y)>0, \;\beta(y)<0\},\N\]\Nand similarly the regions \(\mathcal{R}^{--}\) and \(\mathcal{R}^{-+}\). Consider also the piecewise smooth system of differential equations\N\[\N\dot y=\left\{\begin{array}{ll} f^{++}(y), & \text{if } y\in\mathcal{R}^{++}, \\\Nf^{+-}(y), & \text{if } y\in\mathcal{R}^{+-}, \\\Nf^{-+}(y), & \text{if } y\in\mathcal{R}^{-+}, \\\Nf^{--}(y), & \text{if } y\in\mathcal{R}^{--}, \\\N\end{array}\right.\N\]\Nwhere all the right-hand sides are sufficiently smooth functions. Let \(\Sigma=\Sigma_\alpha\cap\Sigma_\beta\) and suppose that \(\Sigma_\alpha\) and \(\Sigma_\beta\) intersect transversely. Given \(y\in\Sigma\), the authors consider the regularized vector field\N\[\N\begin{array}{l} \displaystyle \dot y =\displaystyle \frac{1}{4}\Bigl[(1+\pi(u))(1+\pi(v))f^{++}(y)+(1+\pi(u))(1-\pi(v))f^{+-}(y) \\\N\qquad\quad\displaystyle+ (1-\pi(u))(1+\pi(v))f^{-+}(y)+(1-\pi(u))(1-\pi(v))f^{--}(y)\Bigr], \end{array}\N\]\Nwhere \(u=\alpha(y)/\varepsilon\), \(v=\beta(y)/\varepsilon\), \(\varepsilon>0\) is small and \(\pi\colon\mathbb{R}\to\mathbb{R}\) is a switching function. To study the hidden dynamics of the above system in a neighborhood of a point \(y\in\Sigma\), the authors consider the planar system\N\[\N\begin{array}{rl} \displaystyle u^\prime &=\displaystyle \frac{1}{4}\Bigl[(1+\pi(u))(1+\pi(v))f^{++}_\alpha+(1+\pi(u))(1-\pi(v))f^{+-}_\alpha \\\N&\qquad\displaystyle+ (1-\pi(u))(1+\pi(v))f^{-+}_\alpha+(1-\pi(u))(1-\pi(v))f^{--}_\alpha\Bigr], \\\N\displaystyle v^\prime &=\displaystyle \frac{1}{4}\Bigl[(1+\pi(u))(1+\pi(v))f^{++}_\beta+(1+\pi(u))(1-\pi(v))f^{+-}_\beta \\\N&\qquad\displaystyle+ (1-\pi(u))(1+\pi(v))f^{-+}_\beta+(1-\pi(u))(1-\pi(v))f^{--}_\beta\Bigr], \end{array}\N\]\Nwhere the prime denotes the derivative in relation to the new time variable \(\tau=t/\varepsilon\) and \(f^{++}_\alpha\), \(f^{++}_\beta\), etc, are real numbers whose expressions depend on \(\alpha\), \(\beta\), \(f^{++}\) and etc.\N\NUnder this framework, the authors provide several examples showing that the dynamics of the above planar system depends highly on the choosen switching function \(\pi\).