Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry

Vector fields tangent to foliations, Hamiltonian systems and first integrals of vector fields are correlated themes very exploit in the literature about Dynamical Systems. In fact the list of papers on these subjects is extremely large and we cite just the books [1, 7, 31, 38] for a brief notion on these issues.

Many authors have used the theoretical aspects about vector fields tangent to foliations, Hamiltonian systems and first integrals of vector fields in order to obtain dynamical properties of models describing some system in applied science. A far from exhaustive list of books in this sense is given by [3, 13, 24].

In recent years, scientists are realizing the importance and applicability of a new branch of dynamical systems that are powerful tools in phenomena where some “on-off” phenomena take place. For example, in control theory (see [34]), mechanics models (see [4, 17, 30]), electrical circuits (see [27]), relay systems (see [16, 25]), biological models (with refuge see [28], foraging predators see [33]), among others where an instantaneous change on the system is observed when any barrier is broken. These dynamical systems are modeled by “pieces” and are called piecewise smooth vector fields (PSVFs for short).

Many authors have contributed to provide a general and consistent theory about PSVFs. We cite here the works [15, 35] where a non familiar reader can found the main definitions, conventions and results on this theory. However, very little have been studied about PSVFs tangent to (piecewise) foliations, Hamiltonian PSVFs and first integrals of PSVFs. Addressing this topic we cite [8, 26, 32].

The present paper deals precisely with PSVFs tangent to piecewise foliations. We found first integrals for them and perform bifurcations on the unstable PSVFs obtained. In fact, a very rich behavior is observed and, which it is very important, an almost fully exploit study area is brought to the surface.

Consider the notation $X.f(p)=\left\langle\nabla f(p),X(p)\right\rangle$ and, for $i\geq 2$, $X^{i}.f(p)=\left\langle\nabla X^{i-1}.f(p),X(p)\right\rangle$, where $\langle.,.\rangle$ is the usual inner product in $\mathbb{R}^{3}$. We say that a point $p\in\Sigma$ is a $\Sigma$-fold point of $X$ if $X.f(p)=0$ but $X^{2}.f(p)\neq 0.$ Moreover, $p\in\Sigma$ is a visible (respectively invisible) $\Sigma$-fold point of $X$ if $X.f(p)=0$ and $X^{2}.f(p)>0$ (respectively $X^{2}.f(p)<0$). We say that $p\in\Sigma$ is a two-fold singularity of $Z$ if $p$ is a $\Sigma$-fold point for both $X$ and $Y$. In this work, we consider only two-fold singularities of type invisible-invisible, ie, the fold points are invisible for both, $X$ and $Y$.

Consider the case when the PSVF $Z=(X,Y)$ has $q$ as two-fold singularity. We can define the positive half-return map as $\varphi_{X}(\rho)=\rho^{+}$, and the negative half-return map as $\varphi_{Y}(\rho^{+})=\rho^{-}$ (see Figure 3). The complete return map associated to $Z$ is given by the composition of these two maps

Note that by Proposition 4, we get $\varphi_{Z_{0}}(x,y)=(\varphi_{Z_{0}}^{1}(x),\varphi_{Z_{0}}^{2}(y))$, where $\varphi_{Z_{0}}^{1}(x)=x$ and $\varphi_{Z_{0}}^{2}(y)=y$. In order to obtain isolated $Z_{0}$-invariant planes we perturb the map $\varphi_{Z_{0}}^{1}(x)$ (see Theorem A), and in order to obtain isolated $Z_{0}$-topological cylinders we perturb the map $\varphi_{Z_{0}}^{2}(y)$ (see Theorem B). When we perturb both we are able to obtain hyperbolic limit cycles (see Theorem C).

In this section we construct homeomorphism that sends orbits of $Z=(X,Y)$, that has a continuous of topological cylinders filled by periodic orbits, to orbits of $Z_{0}=(X_{0},Y_{0})$ given by (1).

Without loss of generality consider that orbits of $Z$ are oriented in an anti-clockwise sense. Let $L$ (respectively, $\overline{L}$) be a set of two-fold singularity of $Z_{0}$ (respectively, $Z$) with length $\mathcal{R}_{1}>0$. By arc length parametrization we identify $L$ with $\overline{L}$. By $p$ (respectively, $\overline{p}$), we mark the line segment $\mathcal{S}_{p}$ (respectively, $\mathcal{S}_{\overline{p}}$) of length $\mathcal{R}_{2}$ orthogonal to $\Sigma$ (see Figure 4). This segment reaches a topological cylinder $\mathcal{M}$ of $Z_{0}$ (respectively, $\overline{\mathcal{M}}$ of $Z$) at a point $p^{1}$ (respectively, $\overline{p}^{1}$).

In each point $\alpha\in L=[p,r]$ (respectively, $\overline{\alpha}\in\overline{L}$) mark the line segment $\mathcal{S}_{\alpha}$ orthogonal to $\Sigma$ (respectively, $\mathcal{S}_{\overline{\alpha}}$) with final point in $\mathcal{M}$ (respectively, $\overline{\mathcal{M}}$). Once $L$ and $\overline{L}$ are identified, identify each $S_{\alpha}$ with $\mathcal{S}_{\overline{\alpha}}$ by arc length parametrization.

By the Implicit Function Theorem (abbreviated by IFT), there exists a smallest time $t_{1}<0$ (respectively, $\overline{t}_{1}<0$), depending on $p^{1}$ (respectively, $\overline{p}^{1}$), such that $\phi_{X_{0}}(p^{1},t_{1}):=q\in\Sigma(+)$ (respectively, $\phi_{X}(\overline{p}^{1},\overline{t}_{1}):=\overline{q}\in\overline{\Sigma}(+)$), where $\Sigma(+)$ (respectively, $\overline{\Sigma}(+)$) is the set of all points of $\Sigma$ situated on the right of $L$ (respectively, $\overline{L}$) and $\phi_{W}$ denotes the flow of the vector field $W$. Identify the orbit arcs $\gamma_{q}^{p^{1}}(X_{0})$ and $\gamma_{\overline{q}}^{\overline{p}^{1}}(X)$ of $X_{0}$ and $X$ with initial points $q$ and $\overline{q}$ and final points $p^{1}$ and $\overline{p}^{1}$, respectively, by arc length parametrization. Again by IFT, there exists a smallest time $t_{2}>0$ (respectively, $\overline{t}_{2}>0$), depending on $p^{1}$ (respectively, $\overline{p}^{1}$), such that $\phi_{X_{0}}(p^{1},t_{2}):=q^{1}\in\Sigma(-)$ (respectively, $\phi_{X}(\overline{p}^{1},\overline{t}_{2}):=\overline{q}^{1}\in\overline{\Sigma}(-)$) where $\Sigma(-)$ (respectively, $\overline{\Sigma}(-)$) is the set of all points of $\Sigma$ situated on the left of $L$ (respectively, $\overline{L}$). Identify the orbit arcs $\gamma_{p^{1}}^{q^{1}}(X_{0})$ and $\gamma_{\overline{p}^{1}}^{\overline{q}^{1}}(X)$ of $X_{0}$ and $X$ with initial points $p^{1}$ and $\overline{p}^{1}$ and final points $q^{1}$ and $\overline{q}^{1}$, respectively, by arc length parametrization.

Now, since $Z_{0}$ (respectively, $Z$) presents a continuous of topological cylinders, and $L$ (respectively, $\overline{L}$) is an invisible $\Sigma$-fold set $Y_{0}$ (respectively, $Y$) by the IFT, there exists a smallest time $t_{3}>0$ (respectively, $\overline{t}_{3}>0$), depending on $q^{1}$ (respectively, $\overline{q}^{1}$), such that $\phi_{Y_{0}}(q^{1},t_{3}):=q\in\Sigma(+)$ (respectively, $\phi_{Y}(\overline{q}^{1},\overline{t}_{3}):=\overline{q}\in\overline{\Sigma}(+)$). Identify the orbit arcs $\gamma_{q^{1}}^{q}(Y_{0})$ and $\gamma_{\overline{q}^{1}}^{\overline{q}}(Y)$ of $Y_{0}$ and $Y$ with initial points $q^{1}$ and $\overline{q}^{1}$ and final points $q$ and $\overline{q}$, respectively, by arc length parametrization.

Do the same for all point $\beta\in S_{\alpha}$ (resp., $\overline{\beta}\in S_{\overline{\alpha}}$), and for all $\alpha\in L$ (resp., $\overline{\alpha}\in\overline{L}$).

Now we consider a perturbation on the normal form (1) that keeps invariant the nested cylinders and exactly $\mathcal{L}$ planes of the form $\pi_{i}=\{(x,y,z)\,|\,x=i\mu\}$, where $i\in\{0,1,2,\ldots,\mathcal{L}-1\}$ and $\mu>0$ is a small real number. In fact, consider

and

where $X_{\mathcal{L}}(x,y,z)=X_{0}(x,y,z)+\lambda\overline{X}_{\mathcal{L}}(x,y,z)$, with $\lambda$ a sufficiently smal real number and $X_{0}$ given in (1).

In order to prove Theorem B we need some lemmas. Observe that both vector fields $X_{0}$ and $Y_{0}$ in the normal form (1) are written as $W(x,y,z)=(0,\pm 1,g(y))$ (particularly, $g(y)=2y$ in such expression). Next lemma gives how are the trajectories of such systems.

Observe that an analogous result is obtained with $W(x,y,z)=(0,-1,g(y))$.

In what follows, $h:\mathbb{R}\rightarrow\mathbb{R}$ will denote the $C^{\infty}$-function given by

Since $h$ is a C^∞-function, the functions $\xi_{\varepsilon}^{f}(y)$ in Lemma 10 and $\xi_{\varepsilon}^{i}(y)$ in Lemma 11 are C^∞-functions. So $Z_{\varepsilon}^{\rho}\in\Omega$ given by

where either $\rho=f$ or $\rho=i$, is a small C^∞-perturbation of $Z_{0}$ given by (1) when $\varepsilon$ is sufficiently small. Moreover,

With the previous lemmas we can stated the following proposition.