On conditions of oscillations and multi-homogeneity

Abstract

The notion of homogeneity in the bi-limit from Andrieu et al. (SIAM J Control Optim 47(4):1814–1850, 2008) is extended to local homogeneity and then to homogeneity in the multi-limit. The converse Lyapunov/Chetaev theorems on (homogeneous) system instability are obtained. The problem of oscillation detection for nonlinear systems is addressed. The sufficient conditions of oscillation existence for systems homogeneous in the multi-limit are formulated. The proposed approach estimates the number of oscillating modes and the regions of their location. Efficiency of the technique is demonstrated on several examples.

Appendix

Proof of Lemma 1

The sufficient part is clear (appearance of the corresponding Lyapunov function implies instability [16]). The necessary existence of the locally Lipschitz continuous Lyapunov function V with the properties (a) and (b) has been proven in [25]. To prove the existence of a continuously differentiable and homogeneous Lyapunov function simply note that the time reversing in the system (1) implies that the system $\dot{\mathbf{x}}=-\mathbf{f}(\mathbf{x})$ with the trajectories $\mathbf{x}(-t,\mathbf{x}_{0})$ is locally asymptotically stable. This fact according to Theorem 1 implies the existence of a homogeneous and continuously differentiable Lyapunov function satisfying (a), (b) and (c). $\square$

Proof of Lemma 2

According to the lemma conditions, under constraints $0<|\mathbf{x}_{0}|<\delta$ and $\mathbf{x}_{0}\in K$ one can take initial conditions as close as possible to the origin such that the corresponding trajectory leaves the set $B_{\delta }$ in the finite time $T_{\mathbf{x}_{0}}$ through the cone K basement (the lateral borders ${\text {lb}}(K)$ are all points of entrance). Then, the sufficient part ((ii)$\Rightarrow$(i)) follows by the Chetaev theorem: if there exists the defined function V with $B_{\delta }$, then the system (1) is unstable [16].

To prove the necessary part ((i)$\Rightarrow$(ii)), note that by conditions for all $\mathbf{x}_{0}\in B_{\delta }$ there exists $T_{\mathbf{x}_{0}}\in R_{+}$ such that $\mathbf{x}(t,\mathbf{x}_{0})\notin B_{\delta }$ for $t\ge T_{\mathbf{x}_{0}}$. Define

$$\begin{aligned} v(\mathbf{x}_{0})=\inf _{0\le t\le T_{\mathbf{x}_{0}}}|\mathbf{x}(t,\mathbf{x}_{0})|, \end{aligned}$$

by construction $\eta (|\mathbf{x}_{0}|)\le v(\mathbf{x}_{0})\le |\mathbf{x}_{0}|$, where $\eta (s)=s(1+s)^{-1}\inf _{s\le |\mathbf{x}|\le \delta ,\mathbf{x}\in K}v(\mathbf{x})$, $\eta \in {\mathcal {K}}$ and $v(0)=0$. To analyze continuity property of the function v consider

$$\begin{aligned} |v(\mathbf{x}_{1})-v(\mathbf{x}_{2})|= & {} |\inf _{0\le t\le T_{\mathbf{x}_{1}}}|\mathbf{x}(t,\mathbf{x}_{1})|-\inf _{0\le t\le T_{\mathbf{x}_{2}}}|\mathbf{x}(t,\mathbf{x}_{2})||\le \sup _{0\le t\le T}||\mathbf{x}(t,\mathbf{x}_{1})|\\&-|\mathbf{x}(t,\mathbf{x}_{2})||, \end{aligned}$$

where $T=\max \{T_{\mathbf{x}_{1}},T_{\mathbf{x}_{2}}\}$. Due to Lipschitz continuity of the system (1) solutions on any compact set of initial conditions ${\mathcal {D}}\subset B_{\delta }$ and time $0\le T<+\infty$ there exists $L\in R_{+}$ such that $|\mathbf{x}(t,\mathbf{x}_{1})-\mathbf{x}(t,\mathbf{x}_{2})|\le L|\mathbf{x}_{1}-\mathbf{x}_{2}|$, for all $0\le t\le T$ and any $\mathbf{x}_{1},\mathbf{x}_{2}\in {\mathcal {D}}$. For all $0<\delta '<\delta$ there exists $T_{\delta '}=\sup _{\mathbf{x}_{0}\in B_{\delta }\backslash B_{\delta '}}T_{\mathbf{x}_{0}}$ with the properties $T_{\delta '}<+\infty$ and $T_{\delta '}\rightarrow +\infty$ for $\delta '\rightarrow 0$; then

$$\begin{aligned} |v(\mathbf{x}_{1})-v(\mathbf{x}_{2})|\le \sup _{0\le t\le T_{\delta '}}||\mathbf{x}(t,\mathbf{x}_{1})|-|\mathbf{x}(t,\mathbf{x}_{2})||\le L|\mathbf{x}_{1}-\mathbf{x}_{2}| \end{aligned}$$

for all $\mathbf{x}_{1},\mathbf{x}_{2}\in B_{\delta }{\mathcal {\backslash }}B_{\delta '}$ and the function v is locally Lipschitz continuous on the set $B_{\delta }{\mathcal {\backslash }}B_{\delta '}$ for any fixed $0<\delta '<\delta$. Therefore, it is locally Lipschitz continuous on $B_{\delta }$ and continuous on $B_{\delta }\cup \{0\}$. The function v is not decreasing on trajectories of the system (1) for any $\mathbf{x}_{0}\in B_{\delta }$:

$$\begin{aligned} v(\mathbf{x}(t,\mathbf{x}_{0}))= & {} \inf _{0\le \tau \le T_{\mathbf{x}(t,\mathbf{x}_{0})}}|\mathbf{x}(\tau ,\mathbf{x}(t,\mathbf{x}_{0}))|=\inf _{t\le \tau \le T_{\mathbf{x}_{0}}}|\mathbf{x}(\tau ,\mathbf{x}_{0})|\\\ge & {} \inf _{0\le \tau \le T_{\mathbf{x}_{0}}}|\mathbf{x}(\tau ,\mathbf{x}_{0})|=v(\mathbf{x}_{0}). \end{aligned}$$

To design a strictly increasing function, let us define a new function for all $\mathbf{x}_{0}\in B_{\delta }$:

$$\begin{aligned} V(\mathbf{x}_{0})=\inf _{0\le t\le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}(t,\mathbf{x}_{0}))k(t)\}, \end{aligned}$$

where $k:R_{+}\rightarrow R_{+}$ is a continuously differentiable function with the properties $0<\kappa _{1}\le k(t)\le \kappa _{2}<+\infty$ and $\dot{k}(t)\le -\kappa _{3}(t)<0$ for all $t\ge 0$ [25, 27]. An example of such a function is as follows:

$$\begin{aligned} k(t)=\kappa _{1}+(\kappa _{2}-\kappa _{1})\mathrm{e}^{-\omega t},\quad \;\dot{k}(t)=\omega (\kappa _{1}-\kappa _{2})\mathrm{e}^{-\omega t},\quad \;\omega >0. \end{aligned}$$

The function V has bounds $\kappa _{1}\eta (|\mathbf{x}_{0}|)\le V(\mathbf{x}_{0})\le \kappa _{2}|\mathbf{x}_{0}|$ and $V(0)=0$. Next, for all $\mathbf{x}_{1},\mathbf{x}_{2}\in B_{\delta }$

$$\begin{aligned} |V(\mathbf{x}_{1})-V(\mathbf{x}_{2})|= & {} |\inf _{0\le t\le T_{\mathbf{x}_{1}}}\{v(\mathbf{x}(t,\mathbf{x}_{1}))k(t)\}-\inf _{0\le t\le T_{\mathbf{x}_{2}}}\{v(\mathbf{x}(t,\mathbf{x}_{2}))k(t)\}|\\= & {} |\inf _{0\le t\le T}\{v(\mathbf{x}(t,\mathbf{x}_{1}))k(t)\}-\inf _{0\le t\le T}\{v(\mathbf{x}(t,\mathbf{x}_{2}))k(t)\}|\\\le & {} \sup _{0\le t\le T}|k(t)[v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))]|\\\le & {} \kappa _{2}\sup _{0\le t\le T}|v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))|, \end{aligned}$$

where $T=\max \{T_{\mathbf{x}_{1}},T_{\mathbf{x}_{2}}\}$. For all $0<\delta '<\delta$ there exists $T_{\delta '}=\sup _{\mathbf{x}_{0}\in B_{\delta }\backslash B_{\delta '}}T_{\mathbf{x}_{0}}<+\infty$ and

$$\begin{aligned} |V(\mathbf{x}_{1})-V(\mathbf{x}_{2})|\le & {} \kappa _{2}\sup _{0\le t\le T_{\delta '}}|v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))|\\\le & {} \kappa _{2}L|\mathbf{x}(t,\mathbf{x}_{1})-\mathbf{x}(t,\mathbf{x}_{2})|\le \kappa _{2}L^{2}|\mathbf{x}_{1}-\mathbf{x}_{2}| \end{aligned}$$

for all $\mathbf{x}_{1},\mathbf{x}_{2}\in B_{\delta }{\mathcal {\backslash }}B_{\delta '}$. Then the function V is locally Lipschitz continuous on the set $B_{\delta }$ and continuous onto $B_{\delta }\cup \{0\}$. It is strictly increasing for any $\mathbf{x}_{0}\in B_{\delta }$:

$$\begin{aligned} V(\mathbf{x}(t,\mathbf{x}_{0}))= & {} \inf _{0\le \tau \le T_{\mathbf{x}(t,\mathbf{x}_{0})}}\{v(\mathbf{x}[\tau ,\mathbf{x}(t,\mathbf{x}_{0})])k(\tau )\}\\\ge & {} \inf _{t\le \tau \le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}[\tau ,\mathbf{x}_{0}])k(\tau -t)\}\\> & {} \inf _{0\le \tau \le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}[\tau ,\mathbf{x}_{0}])k(\tau )\}=V(\mathbf{x}_{0}). \end{aligned}$$

Since V(t) is identically zero at the origin,

$$\begin{aligned} L_{\mathbf{f}(\mathbf{x}_{0})}V(\mathbf{x}_{0})=\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}[V(\mathbf{x}(h,\mathbf{x}_{0}))-V(\mathbf{x}_{0})]>0 \end{aligned}$$

for a.e. $\mathbf{x}_{0}\in B_{\delta }$. By definition

$$\begin{aligned} V(\mathbf{x}(h,\mathbf{x}_{0}))= & {} \inf _{0\le t\le T_{\mathbf{x}(h,\mathbf{x}_{0})}}\{v(\mathbf{x}[t,\mathbf{x}(h,\mathbf{x}_{0})])k(t)\}\\= & {} \inf _{h\le t\le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}[t,\mathbf{x}_{0}])k(t-h)\}=\inf _{h\le t\le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}[t,\mathbf{x}_{0}])k(t)k(t)^{-1}k(t{-}h)\}\\\ge & {} \inf _{h\le t\le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}[t,\mathbf{x}_{0}])k(t)\}\inf _{h\le t\le T_{\mathbf{x}_{0}}}\{k(t)^{-1}k(t-h)\}\\\ge & {} V(\mathbf{x}_{0})\inf _{h\le t\le T_{\mathbf{x}_{0}}}\{k(t)^{-1}k(t-h)\}. \end{aligned}$$

Finally,

$$\begin{aligned} \mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\left[ V(\mathbf{x}(h,\mathbf{x}_{0})){-}V(\mathbf{x}_{0})\right]\ge & {} \mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\left[ V(\mathbf{x}_{0})\inf _{h\le t\le T_{\mathbf{x}_{0}}}\{k(t)^{-1}k(t-h)\}{-}V(\mathbf{x}_{0})\right] \\= & {} V(\mathbf{x}_{0})\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\left[ \inf _{h\le t\le T_{\mathbf{x}_{0}}}\{k(t)^{-1}k(t-h)\}-1\right] \\= & {} V(\mathbf{x}_{0})\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\inf _{h\le t\le T_{\mathbf{x}_{0}}}k(t)^{-1}\{k(t-h)-k(t)\}\\\ge & {} V(\mathbf{x}_{0})\inf _{0\le t\le T_{\mathbf{x}_{0}}}k(t)^{-1}\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\{k(t-h)-k(t)\}\\= & {} V(\mathbf{x}_{0})\inf _{0\le t\le T_{\mathbf{x}_{0}}}\{-k(t)^{-1}\dot{k}(t)\}\ge \kappa _{2}^{-1}\kappa _{3}(T_{\mathbf{x}_{0}})V(\mathbf{x}_{0}). \end{aligned}$$

We substantiate the inequality

$$\begin{aligned} L_{\mathbf{f}(\mathbf{x})}V(\mathbf{x})\ge \alpha _{3}(V(\mathbf{x})),\alpha _{3}(s)=\kappa _{2}^{-1}s\inf _{V(\mathbf{x})\ge s\wedge \mathbf{x}\in B_{\delta }}\kappa _{3}(T_{\mathbf{x}}) \end{aligned}$$

for a.e. $\mathbf{x}\in B_{\delta }$ (the inequality is additionally valid at $\mathbf{x}=0$), where $\alpha _{1}(s)=\kappa _{1}\eta (s)$, $\alpha _{2}(s)=\kappa _{2}s$. Existence of a continuously differentiable function V follows next by standard smoothing arguments [27].

Let $\mathbf{f}:R^{n}\rightarrow R^{n}$ be $\mathbf{r}$-homogeneous with degree d. In this case it is worth reformulating all conditions of the theorem using the homogeneous norm $|\cdot |_{\mathbf{r}}$ instead of the Euclidean norm $|\cdot |$; in this case the value of $\delta >0$ can be chosen arbitrary, i.e. $B_{\delta }=K$. In addition, in this case K should be invariant with respect to dilation transformation $\Lambda _{\mathbf{r}}$. Let $V:R^{n}\rightarrow R_{+}$ be the continuously differentiable Chetaev function obtained on the previous step with this new norm. As in [7] define

$$\begin{aligned} W(\mathbf{x})=\left\{ \begin{array}{*{20}l} \int _{0}^{+\infty }{a[V(\Lambda _{\mathbf{r}}{} \mathbf{x})]/\lambda ^{k+1}\mathrm{d}\lambda } &{}\quad \mathrm{{if}}\,\,\mathbf{x}\in K\backslash \{0\};\\ 0 &{}\quad \mathrm{{if}}\,\,\,\mathbf{x}=0,\end{array}\right. \end{aligned}$$

where $a:R_{+}\rightarrow R_{+}$ is a smooth function with the following properties:

• (i) $a(s)=0$ for all $s\le 1$;

• (ii) $a'(s)>0$ for all $1<s<2$, $a'(s)=0$ for all $s\ge 2$.

By this construction the function W is positive definite and continuously differentiable. A direct computation shows that this function is $\mathbf{r}$-homogeneous, i.e. for $\mathbf{x}\ne 0$

$$\begin{aligned} W(\Lambda _{\mathbf{r}}{} \mathbf{x})= & {} \int _{0}^{+\infty }{a[V(\Lambda _{\mathbf{r}}\Lambda '_{\mathbf{r}}\mathbf{x})]/\lambda '^{k+1}\mathrm{d}\lambda '}\\= & {} \lambda ^{k} \int _{0}^{+\infty }{a[V(\Lambda ''_{\mathbf{r}}\mathbf{x})]/\lambda ''^{k+1}\mathrm{d}\lambda ''}=\lambda ^{k}W(\mathbf{x}), \end{aligned}$$

where the change of variables $\lambda ''=\lambda \lambda '$ has been used on the last step. Then

$$\begin{aligned} \alpha _{1}(|\mathbf{x}|_{\mathbf{r}})=\min _{\mathbf{y}\in S_{\mathbf{r}}}\{W(\mathbf{y})\}|\mathbf{x}|_{\mathbf{r}}^{k},\;\alpha _{2}(|\mathbf{x}|_{\mathbf{r}})=\max _{\mathbf{y}\in S_{\mathbf{r}}}\{W(\mathbf{y})\}|\mathbf{x}|_{\mathbf{r}}^{k}. \end{aligned}$$

Its time derivative for the system (1) is a continuous function and it has the following form for all $\mathbf{x}\in K\backslash \{0\}$:

$$\begin{aligned} \dot{W}(\mathbf{x})= & {} \int _{0}^{+\infty }{a'[V(\Lambda _{\mathbf{r}}\mathbf{x})]DV(\Lambda _{\mathbf{r}}{} \mathbf{x})\Lambda _{\mathbf{r}}{} \mathbf{f}(\mathbf{x})/\lambda ^{k+1}\mathrm{d}\lambda }\\= & {} \int _{0}^{+\infty }{a'[V(\Lambda _{\mathbf{r}}\mathbf{x})]DV(\Lambda _{\mathbf{r}}{} \mathbf{x})\mathbf{f}(\Lambda _{\mathbf{r}}\mathbf{x})/\lambda ^{k+d+1}\mathrm{d}\lambda }\\= & {} \int _{\underline{\lambda }({\mathbf {x}})}^{\overline{\lambda }({\mathbf {x}})}{a'[V(\Lambda _{\mathbf{r}}{} \mathbf{x})]DV(\Lambda _{\mathbf{r}}{} \mathbf{x})\mathbf{f}(\Lambda _{\mathbf{r}}{} \mathbf{x})/\lambda ^{k+d+1}\mathrm{d}\lambda }, \end{aligned}$$

where $\underline{\lambda }({\mathbf {x}})={\text {arg min}}_{\lambda \in R_{+}}\{V(\Lambda _{\mathbf{r}}{{\mathbf {x}}})=1\}$ and $\overline{\lambda }({\mathbf {x}})={\text {arg max}}_{\lambda \in R_{+}}\{V(\Lambda _{\mathbf{r}}{\mathbf {x}})=2\}$. Owing the functions a and V properties we have $a'[V(\Lambda _{\mathbf{r}}\mathbf{x})]DV(\Lambda _{\mathbf{r}}{} \mathbf{x})\mathbf{f}(\Lambda _{\mathbf{r}}\mathbf{x})/\lambda ^{k+d+1}>0$ for all $\mathbf{x}\in K\backslash \{0\}$ and $\lambda \in [\underline{\lambda }({\mathbf {x}}),\overline{\lambda }({\mathbf {x}})]$ ($a'[V(\Lambda _{\mathbf{r}}{} \mathbf{x})]=0$ for other $\lambda$). Thus $\dot{W}(\mathbf{x})>0$ for all $\mathbf{x}\in K\backslash \{0\}$ and $\dot{W}(0)=0$; then $\alpha _{3}(s)=\widetilde{\alpha }_{3}\circ \alpha _{2}^{-1}(s)$ where

$$\begin{aligned} \widetilde{\alpha }_{3}(s)=\inf _{s<|\mathbf{x}|_{\mathbf{r}},\mathbf{x}\in K}\int _{\underline{\lambda }({\mathbf {x}})}^{\overline{\lambda }({\mathbf {x}})}{a'[V(\Lambda _{\mathbf{r}}{} \mathbf{x})]DV(\Lambda _{\mathbf{r}}{} \mathbf{x})\mathbf{f}(\Lambda _{\mathbf{r}}{} \mathbf{x})/\lambda ^{k+d+1}\mathrm{d}\lambda }. \end{aligned}$$

$\square$

Proof of Lemma 3

The sufficient part ((ii)$\Rightarrow$(i)) again follows by the Chetaev theorem: if there exists the defined function V with such a $B_{\delta }$, then the system (1) is unstable [16].

To prove that (i) $\Rightarrow$ (ii) note that if the set $B_{\delta }\cup \{0\}$ is backward invariant, it means that any not invariant trajectory of the system (1) leaves the set in a finite time and there is no trajectory entering $B_{\delta }$. Considering system (1) in the backward time, i.e. $\dot{\mathbf{x}}=-\mathbf{f}(\mathbf{x})$, we make the set $B_{\delta }\cup \{0\}$ forward invariant. By assumptions the only invariant solution is at the origin; thus all trajectories into the forward invariant set $B_{\delta }\cup \{0\}$ converge asymptotically to the equilibrium $\mathbf{x}=0$ in the backward time. In [38] (Theorem 1.5.40) it is shown that under these conditions the system is asymptotically stable at the origin into the set $B_{\delta }\cup \{0\}$. Let us show that in this case there exists a locally Lipschitz continuous function $V:B_{\delta }\cup \{0\}\rightarrow R_{+}$ (continuous at the origin) such that

$$\begin{aligned} \alpha _{1}(|\mathbf{x}|)\le V(\mathbf{x})\le \alpha _{2}(|\mathbf{x}|),\alpha _{1},\alpha _{2}\in {\mathcal {K}}_{\infty },\; L_{-\mathbf{f}(\mathbf{x})}V(\mathbf{x})\le -\alpha _{3}(V(\mathbf{x})),\alpha _{3}\in {\mathcal {K}}, \end{aligned}$$

for a.e. $\mathbf{x}\in B_{\delta }\cup \{0\}$. For this purpose for any $\mathbf{x}_{0}\in B_{\delta }\cup \{0\}$ define

$$\begin{aligned} v(\mathbf{x}_{0})=\sup _{t\ge 0}|\mathbf{x}(t,\mathbf{x}_{0})|, \end{aligned}$$

by construction $|\mathbf{x}_{0}|\le v(\mathbf{x}_{0})\le \phi (|\mathbf{x}_{0}|)\le \delta$, $\phi (s)=(1+s)\sup _{|\mathbf{x}|\le s,\mathbf{x}\in B_{\delta }}v(\mathbf{x})$ and $v(0)=0$. From attractivity for any $\mathbf{x}_{0}\in B_{\delta }$ there exists $T_{\mathbf{x}_{0}}\in R_{+}$ such that $v(\mathbf{x}_{0})=\sup _{0\le t\le T_{\mathbf{x}_{0}}}|\mathbf{x}(t,\mathbf{x}_{0})|$. To analyze continuity property of the function v, consider

$$\begin{aligned} |v(\mathbf{x}_{1})-v(\mathbf{x}_{2})|= & {} |\sup _{t\ge 0}|\mathbf{x}(t,\mathbf{x}_{1})|-\sup _{t\ge 0}|\mathbf{x}(t,\mathbf{x}_{2})||\\= & {} |\sup _{0\le t\le T_{\mathbf{x}_{1}}}|\mathbf{x}(t,\mathbf{x}_{1})|-\sup _{0\le t\le T_{\mathbf{x}2}}|\mathbf{x}(t,\mathbf{x}_{2})||\\\le & {} \sup _{0\le t\le T}||\mathbf{x}(t,\mathbf{x}_{1})|-|\mathbf{x}(t,\mathbf{x}_{2})||, \end{aligned}$$

where $T=\max \{T_{\mathbf{x}_{1}},T_{\mathbf{x}_{2}}\}$, $\mathbf{x}_{1},\mathbf{x}_{2}\in B_{\delta }$. Due to Lipschitz continuity of the system (1) solutions on any compact set of initial conditions ${\mathcal {D}}\subset B_{\delta }$ and time $0\le T<+\infty$, there exists $L\in R_{+}$ such that

$$\begin{aligned} |\mathbf{x}(t,\mathbf{x}_{1})-\mathbf{x}(t,\mathbf{x}_{2})|\le L|\mathbf{x}_{1}-\mathbf{x}_{2}|, \end{aligned}$$

for all $0\le t\le T$ and any $\mathbf{x}_{1},\mathbf{x}_{2}\in {\mathcal {D}}$. For all $0<\delta '\le \delta$ for any compact ${\mathcal {D}}\subset B_{\delta }$, there exists $T_{\delta '}=\sup _{\mathbf{x}_{0}\in {\mathcal {D}}\backslash B_{\delta '}}T_{\mathbf{x}_{0}}$ with the property $T_{\delta '}<+\infty$; then

$$\begin{aligned} \begin{array}{l} |v(\mathbf{x}_{1})-v(\mathbf{x}_{2})|\le \sup _{0\le t\le T_{\delta '}}||\mathbf{x}(t,\mathbf{x}_{1})|-|\mathbf{x}(t,\mathbf{x}_{2})||\le L|\mathbf{x}_{1}-\mathbf{x}_{2}|\end{array} \end{aligned}$$

for all $\mathbf{x}_{1},\mathbf{x}_{2}\in {\mathcal {D\backslash }}B_{\delta '}$ and the function v is locally Lipschitz continuous on the set ${\mathcal {D\backslash }}B_{\delta '}$ for any fixed $0<\delta '\le \delta$. The function v is not increasing on any trajectory of the system (1); indeed, for any $\mathbf{x}_{0}\in B_{\delta }\cup \{0\}$:

$$\begin{aligned} v(\mathbf{x}(t,\mathbf{x}_{0}))= & {} \sup _{\tau \ge 0}|\mathbf{x}(\tau ,\mathbf{x}(t,\mathbf{x}_{0}))|=\sup _{\tau \ge t}|\mathbf{x}(\tau ,\mathbf{x}_{0})|\le \sup _{\tau \ge 0}|\mathbf{x}(\tau ,\mathbf{x}_{0})|=v(\mathbf{x}_{0}). \end{aligned}$$

To construct a strictly decreasing function, define a new function for all $\mathbf{x}_{0}\in B_{\delta }\cup \{0\}$:

$$\begin{aligned} V(\mathbf{x}_{0})=\sup _{t\ge 0}\{v(\mathbf{x}(t,\mathbf{x}_{0}))k(t)\}, \end{aligned}$$

where $k:R_{+}\rightarrow R_{+}$ is a continuously differentiable function with the properties $0<\kappa _{1}\le k(t)\le \kappa _{2}<+\infty$ and $\dot{k}(t)\ge \kappa _{3}(t)>0$ for all $t\ge 0$ [27]. An example of such function is as follows:

$$\begin{aligned} k(t)=(\kappa _{1}+\kappa _{2}t^{1+\omega })(1+t^{1+\omega })^{-1},\quad \dot{k}(t)=t^{\omega }(\kappa _{2}-\kappa _{1})(1+\omega )(1+t^{1+\omega })^{-2},\quad \omega \ge 0. \end{aligned}$$

The function V has bounds $\kappa _{1}|\mathbf{x}_{0}|\le V(\mathbf{x}_{0})\le \kappa _{2}\phi (|\mathbf{x}_{0}|)$ and $V(0)=0$. Again, for any $\mathbf{x}_{0}\in B_{\delta }\cup \{0\}$ there exists $T_{\mathbf{x}_{0}}\in R_{+}$ such that $V(\mathbf{x}_{0})=\sup _{0\le t\le T_{\mathbf{x}_{0}}}\{v(\mathbf{x}(t,\mathbf{x}_{0}))k(t)\}$. This claim follows from non strict decreasing of the function v. Next, for all $\mathbf{x}_{1},\mathbf{x}_{2}\in B_{\delta }$

$$\begin{aligned} |V(\mathbf{x}_{1})-V(\mathbf{x}_{2})|= & {} \left| \sup _{t\ge 0}\{v(\mathbf{x}(t,\mathbf{x}_{1}))k(t)\}-\sup _{t\ge 0}\{v(\mathbf{x}(t,\mathbf{x}_{2}))k(t)\}\right| \\= & {} \left| \sup _{0\le t\le T}\{v(\mathbf{x}(t,\mathbf{x}_{1}))k(t)\}-\sup _{0\le t\le T}\{v(\mathbf{x}(t,\mathbf{x}_{2}))k(t)\}\right| \\\le & {} \sup _{0\le t\le T}\left| k(t)[v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))]\right| \\\le & {} \kappa _{2}\sup _{0\le t\le T}\left| v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))\right| , \end{aligned}$$

where $T=\max \{T_{\mathbf{x}_{1}},T_{\mathbf{x}_{2}}\}$. For all $0<\delta '\le \delta$ for any compact ${\mathcal {D}}\subset B_{\delta }$ there exists $T_{\delta '}=\sup _{\mathbf{x}_{0}\in {\mathcal {D}}\backslash B_{\delta '}}T_{\mathbf{x}_{0}}$ with the property $T_{\delta '}<+\infty$ and

$$\begin{aligned} |V(\mathbf{x}_{1})-V(\mathbf{x}_{2})|\le & {} \kappa _{2}\sup _{0\le t\le T_{\delta '}}|v(\mathbf{x}(t,\mathbf{x}_{1}))-v(\mathbf{x}(t,\mathbf{x}_{2}))|\\\le & {} \kappa _{2}L|\mathbf{x}(t,\mathbf{x}_{1})-\mathbf{x}(t,\mathbf{x}_{2})|\le \kappa _{2}L^{2}|\mathbf{x}_{1}-\mathbf{x}_{2}| \end{aligned}$$

for all $\mathbf{x}_{1},\mathbf{x}_{2}\in {\mathcal {D\backslash }}B_{\delta '}$. Then the function V is locally Lipschitz continuous on the set ${\mathcal {D}}\backslash B_{\delta '}$ for any $0<\delta '\le \delta$ and it is strictly decreasing for any $\mathbf{x}_{0}\in B_{\delta }$:

$$\begin{aligned} V(\mathbf{x}(t,\mathbf{x}_{0}))= & {} \sup _{\tau \ge 0}\{v(\mathbf{x}[\tau ,\mathbf{x}(t,\mathbf{x}_{0})])k(\tau )\}=\sup _{\tau \ge t}\{v(\mathbf{x}[\tau ,\mathbf{x}_{0}])k(\tau -t)\}\\< & {} \sup _{\tau \ge 0}\{v(\mathbf{x}[\tau ,\mathbf{x}_{0}])k(\tau )\}=V(\mathbf{x}_{0}); \end{aligned}$$

V(t) is zero on the trajectories at the origin. Denote $L_{-\mathbf{f}(\mathbf{x}_{0})}V(\mathbf{x}_{0})=\mathop {\lim }\limits \nolimits _{h\rightarrow 0}h^{-1}[V(\mathbf{x}(h,\mathbf{x}_{0}))-V(\mathbf{x}_{0})]$; then

$$\begin{aligned} L_{-\mathbf{f}(\mathbf{x}_{0})}V(\mathbf{x}_{0})<0 \end{aligned}$$

for a.e. $\mathbf{x}_{0}\in B_{\delta }$. Denote $T_{\delta }=\sup _{\mathbf{x}_{0}\in B_{\delta }}T_{\mathbf{x}_{0}}$ and by the definition

$$\begin{aligned} V(\mathbf{x}(h,\mathbf{x}_{0}))= & {} \sup _{t\ge 0}\{v(\mathbf{x}[t,\mathbf{x}(h,\mathbf{x}_{0})])k(t)\}\\= & {} \sup _{h\le t\le T_{\delta }}\{v(\mathbf{x}[t,\mathbf{x}_{0}])k(t-h)\}=\sup _{0\le t\le T_{\delta }}\{v(\mathbf{x}[t,\mathbf{x}_{0}])K(t,h)\}, \end{aligned}$$

for a.e. $\mathbf{x}_{0}\in B_{\delta }$ and $K(t,h)=if[t<h,0,k(t-h)]$; further

$$\begin{aligned} \sup _{0\le t\le T_{\delta }}\{v(\mathbf{x}[t,\mathbf{x}_{0}])K(t,h)\}\le & {} V(\mathbf{x}_{0})\sup _{0\le t\le T_{\delta }}\{k(t)^{-1}K(t,h)\}\\= & {} V(\mathbf{x}_{0})\max \{0,\sup _{h\le t\le T_{\delta }}\{k(t)^{-1}k(t-h)\}\}\\= & {} V(\mathbf{x}_{0})\sup _{h\le t\le T_{\delta }}\{k(t)^{-1}k(t-h)\} \end{aligned}$$

and

$$\begin{aligned} \mathop {\lim }\limits _{h\rightarrow 0}h^{-1}[V(\mathbf{x}(h,\mathbf{x}_{0}))-V(\mathbf{x}_{0})]= & {} \mathop {\lim }\limits _{h\rightarrow 0}h^{-1}[V(\mathbf{x}_{0})\sup _{h\le t\le T_{\delta }}\{k(t)^{-1}k(t-h)\}-V(\mathbf{x}_{0})]\\= & {} V(\mathbf{x}_{0})\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\left[ \sup _{h\le t\le T_{\delta }}\{k(t)^{-1}k(t-h)\}-1\right] \\= & {} V(\mathbf{x}_{0})\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\sup _{h\le t\le T_{\delta }}k(t)^{-1}\{k(t-h)-k(t)\}\\\le & {} V(\mathbf{x}_{0})\sup _{0\le t\le T_{\delta }}k(t)^{-1}\mathop {\lim }\limits _{h\rightarrow 0}h^{-1}\{k(t-h)-k(t)\}\\= & {} V(\mathbf{x}_{0})\sup _{t\ge 0}\{-k(t)^{-1}\dot{k}(t)\}\le -\kappa _{2}^{-1}\kappa _{3}(T_{\delta })V(\mathbf{x}_{0}). \end{aligned}$$

We substantiate the inequality

$$\begin{aligned} L_{-\mathbf{f}(\mathbf{x})}V(\mathbf{x})\le -\kappa _{2}^{-1}\kappa _{3}(T_{\delta })V(\mathbf{x}) \end{aligned}$$

for a.e. $\mathbf{x}\in B_{\delta }\cup \{0\}$ (the inequality is additionally valid at the origin), $\alpha _{3}(s)=\kappa _{2}^{-1}\kappa _{3}(T_{\delta })s$ and $\alpha _{1}(s)=\kappa _{1}s$, $\alpha _{2}(s)=\kappa _{2}\phi (s)$. The existence of a continuously differentiable function V can be substantiated applying standard smoothing arguments [27]. Returning to the initial forward time we obtain the existence of the required continuously differentiable Chetaev function for the system (1) in $B_{\delta }\cup \{0\}$. To prove existence of a homogeneous Chetaev function for $\mathbf{r}$-homogeneous function $\mathbf{f}$ one can apply the same arguments as in the proof of Lemma 2. $\square$

Proof of Proposition 1

$1\Rightarrow 2$. Application of the dilation transformation for the ($\mathbf{r}$,$\lambda _{0}$,$\mathbf{f}_{0}$)-homogeneous system (1) gives

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})= & {} DV_{0}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})\\= & {} DV_{0}({\varvec{\Lambda }}_{\mathbf{r}}\mathbf{y})\lambda ^{d}{\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{f}_{0}(\mathbf{y})+DV_{0}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})[\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}\mathbf{y})-\lambda ^{d}{\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{f}_{0}(\mathbf{y})]\\= & {} \lambda ^{d+k}DV_{0}(\mathbf{y})\mathbf{f}_{0}(\mathbf{y})+\lambda ^{d+k}DV_{0}(\mathbf{y})[\lambda ^{-d}{\varvec{\Lambda }}_{\mathbf{r}}^{-1}\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})-\mathbf{f}_{0}(\mathbf{y})], \end{aligned}$$

where k and d are the homogeneity degree of the Lyapunov function $V_{0}$ and the function $\mathbf{f}_{0}$, respectively, $\mathbf{x}\in R^{n}$, $\mathbf{y}\in S_{\mathbf{r}}$. Due to continuity of the functions $\mathbf{f}$, $\mathbf{f}_{0}$ and the local homogeneity property definition, for any $\varepsilon >0$ there exist $\underline{\lambda }_{\varepsilon }(\mathbf{y})\le \lambda _{0}(\mathbf{y})\le \bar{\lambda }_{\varepsilon }(\mathbf{y})$ such that

$$\begin{aligned} \sup _{\lambda \in (\lambda _{\varepsilon }(\mathbf{y}),\bar{\lambda }_{\varepsilon }(\mathbf{y})),\mathbf{y}\in S_{\mathbf{r}}}|DV_{0}(\mathbf{y})[\lambda ^{-d}{\varvec{\Lambda }}_{\mathbf{r}}^{-1}{} \mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})-\mathbf{f}_{0}(\mathbf{y})]|\le \varepsilon . \end{aligned}$$

(10)

Therefore, for the properly chosen $\underline{\lambda }_{\varepsilon }$, $\bar{\lambda }_{\varepsilon }$ the sign of $DV_{0}(\mathbf{y})\mathbf{f}_{0}(\mathbf{y})$ can override the sign of $DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})$. Depending on this sign the set ${\mathcal {X}}$ can be (locally) asymptotically stable or unstable.

$2\Rightarrow 1$. According to definition of ($\mathbf{r}$,$\lambda _{0}$,$\mathbf{f}_{0}$)-homogeneity, the functions $\mathbf{f}_{0}$ and $\mathbf{f}$ coincide on the set ${\mathcal {S}}$ and $\mathbf{f}_{0}$ is the unique local approximation of $\mathbf{f}$ for the given d, $d_{0}$, $\mathbf{r}$ and $\lambda _{0}$. Since the function $\mathbf{f}_{0}$ is homogeneous it may be globally stable or unstable at the origin in this case, actually applying the same dilation transformation, but in the inverse direction we can show that for ${\mathbf {x}}\in {\mathcal {S}}$ the sign of $DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})$ overrides the sign of $DV_{0}(\mathbf{x})\mathbf{f}_{0}(\mathbf{x})$, since both $V_{0}$ and ${\mathbf {f}}_{0}$ are homogeneous the result follows. $\square$

Proof of Proposition 2

As in the proof of Proposition 1, since ${\mathcal {S}}=\{{\varvec{\Lambda }}_{\mathbf{r},0}(\mathbf{y})\mathbf{y},\;\mathbf{y}\in S_{\mathbf{r}}\}=\{\mathbf{x}\in R^{n}:\; V_{0}(\mathbf{x})=b\}$, it is possible to select $\underline{\lambda }_{\varepsilon }$, $\bar{\lambda }_{\varepsilon }$ (the relations (10) are satisfied for $\underline{\lambda }_{\varepsilon }(\mathbf{y})\le \lambda _{0}(\mathbf{y})\le \bar{\lambda }_{\varepsilon }(\mathbf{y})$ for all ${\mathbf {y}}\in S_{{\mathbf {r}}}$) in such a way that the sign of $DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})$ for ${\mathbf {x}}\in X$ is predefined by $a_{z}$, $z\in \{s,u\}$.

Consider the case (i); then $a_{s}>0$ and for all $\mathbf{x}\in X$

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})\le -\lambda ^{d+k}(a_{s}-\varepsilon )=-|\mathbf{x}|_{\mathbf{r}}^{d+k}(a_{s}-\varepsilon ). \end{aligned}$$

If $\lambda _{0}=0$, then clearly $b=0$ and the system is locally asymptotically stable at the origin with the domain of asymptotic stability containing the set $X_{0}$ for some $0<\epsilon <+\infty$ by the standard arguments [26]. If $\lambda _{0}=+\infty$, then the function $V_{0}$ has strictly negative time derivative for the system (1) into the set $R^{n}\backslash X_{0}$ for some $0<\epsilon <+\infty$. Thus the set $X_{0}$ is forward invariant for (1) and according to [27] these facts imply the global asymptotic stability of the system (1) with respect to the set $X_{0}$. Finally, let $0<\lambda _{0}({\mathbf {x}})<+\infty$; then the function $V_{0}$ is strictly decreasing into the set X and all trajectories $\mathbf{x}(t,\mathbf{x}_{0})$ with initial conditions $\mathbf{x}_{0}\in X$ reach for the set $X_{1}$ in a finite time, which implies the desired conclusion.

Consider the case (ii), then $a_{u}>0$ and

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})\ge \lambda ^{d+k}(a_{u}-\varepsilon )=|\mathbf{x}|_{\mathbf{r}}^{d+k}(a_{u}-\varepsilon ) \end{aligned}$$

for all $\mathbf{x}\in X$. If $\lambda _{0}=0$, then the system is unstable into the set $X_{0}$ by the standard arguments [26]. Therefore, the set $R^{n}\backslash X_{0}$ is forward invariant and has the region of attraction $X_{0}\backslash \{0\}$. If $\lambda _{0}=+\infty$, then the function $V_{0}$ has a strictly positive time derivative for the system (1) into the set $R^{n}\backslash X_{0}$ for some $0<\epsilon <+\infty$. Thus the set $R^{n}\backslash X_{0}$ is forward invariant for (1). Finally, let $0<\lambda _{0}({\mathbf {x}})<+\infty$; then the function $V_{0}$ is strictly increasing into the set X and all trajectories $\mathbf{x}(t,\mathbf{x}_{0})$ with initial conditions $\mathbf{x}_{0}\in X$ reaching for the set $X_{2}$ in a finite time, which implies forward invariance of this set and finite time stability with the region of attraction X. $\square$

Proof of Corollary 1

Under the corollary conditions the time derivative of the function $V_{0}$ for the system (1) has the same form:

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})=\lambda ^{d+k}DV_{0}(\mathbf{y})\mathbf{f}_{0}(\mathbf{y})+\lambda ^{d+k}DV_{0}(\mathbf{y})[\lambda ^{-d}{\varvec{\Lambda }}_{\mathbf{r}}^{-1}\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})-\mathbf{f}_{0}(\mathbf{y})]. \end{aligned}$$

And for the cases (i) and (ii) the inequalities

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})\le -|\mathbf{x}|_{\mathbf{r}}^{d+k}(a-\varepsilon ),\quad \; DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})\ge |\mathbf{x}|_{\mathbf{r}}^{d+k}(a-\varepsilon ). \end{aligned}$$

Further, the conclusion follows by the same arguments as in Proposition 2. $\square$

Proof of Proposition 3

As in Proposition 2, transformation to the sphere $S_{\mathbf{r}}$ for the ($\mathbf{r}$,$\lambda _{0}$,$\mathbf{f}_{0}$)-homogeneous system (1) gives

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})=\lambda ^{d+k}DV_{0}(\mathbf{y})\mathbf{f}_{0}(\mathbf{y})+\lambda ^{d+k}DV_{0}(\mathbf{y})[\lambda ^{-d}{\varvec{\Lambda }}_{\mathbf{r}}^{-1}\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})-\mathbf{f}_{0}(\mathbf{y})], \end{aligned}$$

where d and k are the homogeneity degree of the Lyapunov function $V_{0}$ and the function $\mathbf{f}$, respectively. Due to continuity of the functions $\mathbf{f}$, $\mathbf{f}_{0}$ and the local homogeneity property definition for any $\varepsilon >0$, there exists $0<\bar{\lambda }_{\varepsilon }$ such that

$$\begin{aligned} \sup _{\mathbf{y}\in S_{\mathbf{r}}}|DV_{0}(\mathbf{y})[\lambda ^{-d}{\varvec{\Lambda }}_{\mathbf{r}}^{-1}\mathbf{f}({\varvec{\Lambda }}_{\mathbf{r}}{} \mathbf{y})-\mathbf{f}_{0}(\mathbf{y})]|\le \varepsilon \end{aligned}$$

for all $\lambda \in (0,\bar{\lambda }_{\varepsilon })$. Therefore, for the properly chosen $\bar{\lambda }_{\varepsilon }$ the sign of $DV_{0}(\mathbf{y})\mathbf{f}_{0}(\mathbf{y})$ can override the sign of $DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})$ into the set $S_{\mathbf{r}}\cap K$ provided that $\varepsilon <a$:

$$\begin{aligned} DV_{0}(\mathbf{x})\mathbf{f}(\mathbf{x})\ge \lambda ^{d+k}(a-\varepsilon )=|\mathbf{x}|_{\mathbf{r}}^{d+k}(a-\varepsilon ) \end{aligned}$$

for all $\mathbf{x}\in X_{\mathbf{r}}=\{\mathbf{x}\in K:0<|\mathbf{x}|_{\mathbf{r}}<\bar{\lambda }_{\varepsilon }\}$. Therefore, the system is unstable at the origin [16]. $\square$

Proof of Theorem 6

For the part (i) the existence of such an index $j^{*}$ means that $\mathbf{f}_{j^{*}}$ is unstable and the vector field $\mathbf{f}_{j^{*}+1}$ is asymptotically stable. Thus, from part (ii)-3 and part (i)-3 of Proposition 2, there exist $\underline{\lambda }_{k}\le \lambda _{k}\le \bar{\lambda }_{k}$, for $k=j^{*},j^{*}+1$ such that the Lyapunov functions $V_{k}$, $k=j^{*},j^{*}+1$ have sign definite time derivatives for the system (1) into the sets

$$\begin{aligned} X_{\mathbf{r}_{k}}=\{\mathbf{x}\in R^{n}:\;\underline{\lambda }_{k}<|\mathbf{x}|_{\mathbf{r}_{k}}<\bar{\lambda }_{k}\},\quad \; k=j^{*},j^{*}+1, \end{aligned}$$

respectively. Since the sets $X_{k}$, $k=j*,j*+1$ are connected and nonempty, all trajectories of the system (1) with initial conditions into the set $\Omega _{j^{*}}$ stay there for all further instants of time. By the same arguments all trajectories of the system (1) with initial conditions into the set $\Omega _{j^{*}+1}$ stay there. Therefore, the set $\Omega =\Omega _{j*}\cap \Omega _{j*+1}$ being nonempty is forward invariant and compact. By the theorem conditions this set does not contain the system (1) equilibria ($\Omega \cap \Xi =\emptyset$).

Consider any trajectory $\mathbf{x}(t,\mathbf{x}_{0})$ of the system (1) with initial conditions $\mathbf{x}_{0}\in \Omega$, by consideration above $\mathbf{x}(t,\mathbf{x}_{0})\in \Omega$ for all $t\ge 0$ and the trajectories $\mathbf{x}(t,\mathbf{x}_{0})$ are bounded due to compactness of $\Omega$. Then for each $\mathbf{x}_{0}\in \Omega$ there should exist an index $1\le i^{*}\le n$ such that the solution $\mathbf{x}(t,\mathbf{x}_{0})$ is $[\pi ^{-},\pi ^{+}]$-oscillation with respect to the output $\psi =x_{i*}$ for some $-\infty <\pi ^{-}<\pi ^{+}<+\infty$. Indeed, the definition

$$\begin{aligned} \pi ^{-}=\lim \inf \limits _{t\rightarrow +\infty }\psi (t),\quad \;\pi ^{+}=\lim \sup \limits _{t\rightarrow +\infty }\psi (t), \end{aligned}$$

and the boundedness of the trajectory $\mathbf{x}(t,\mathbf{x}_{0})$ imply that the constants $\pi ^{-}$, $\pi ^{+}$ are finite. If for all $1\le i^{*}\le n$ for the corresponding constants the quantity $\pi ^{-}=\pi ^{+}$ holds, then it means that there is $\mathbf{x}_{\infty }\in \Omega$ such that $\mathop {\lim }\limits \nolimits _{t\rightarrow +\infty }{} \mathbf{x}(t,\mathbf{x}_{0})=\mathbf{x}_{\infty }$, which is a contradiction since $\mathbf{x}_{\infty }$ is an equilibrium and all equilibria are excluded from $\Omega$.

For the part (ii), $\mathbf{f}_{j^{*}}$ is stable and the vector field $\mathbf{f}_{j*+1}$ is unstable. Again, according to Proposition 2 there exist $\underline{\lambda }_{k}\le \lambda _{k}\le \bar{\lambda }_{k}$, for $k=j^{*},j^{*}+1$ such that the Lyapunov functions $V_{k}$, $k=j^{*},j^{*}+1$ have sign definite time derivatives for the system (1) into the sets $X_{\mathbf{r}_{k}}$, $k=j^{*},j^{*}+1$. Since the sets $X_{k}$, $k=j*,j*+1$ are connected and nonempty, then all trajectories of the system (1) with initial conditions into the sets $R^{n}\backslash \Omega _{j^{*}}$ and $R^{n}\backslash \Omega _{j^{*}+1}$ stay there for all positive times. Therefore, the complement set $\Omega$ being nonempty is backward invariant and compact. By the theorem conditions this set does not contain the system (1) equilibria. Since the system (1) is time-invariant, reversing the time we transform the set $\Omega$ to forward invariant and compact set for the system $\dot{\mathbf{x}}=-\mathbf{f}(\mathbf{x})$. Applying to this system the same arguments as in part (i) we can prove the existence of oscillating trajectories into the set $\Omega$, which implies the same conclusion for the original system (1). $\square$