Before starting our work, we need two definitions and two lemmas. Consider the autonomous impulsive differential equation
\{\begin{array}{c}\frac{dx}{dt}=f(x,\epsilon ),\phantom{\rule{1em}{0ex}}x\notin \sigma (\epsilon ),\hfill \\ \mathrm{\u25b3}x=I(x,\epsilon ),\phantom{\rule{1em}{0ex}}x\in \sigma (\epsilon ),\hfill \end{array}
(3.1)
where \epsilon \in J=(\overline{\epsilon},\overline{\epsilon}) is a small parameter. For each \epsilon \in J, the \sigma (\epsilon ) is a hypersurface in {R}^{n}. Suppose \sigma (\epsilon ) consists of q nonintersecting smooth hypersurfaces {\sigma}_{k}(\epsilon ) which are given by the equations {\phi}_{k}(x,\epsilon )=0 (k=1,2,\dots ,q).
Let x=\varphi (t), t\in {R}_{+} be a solution of Equation (3.1) with moments of impulsive effect {\tau}_{k}:0<{\tau}_{1}<{\tau}_{2}<\cdots, {lim}_{k\to \mathrm{\infty}}{\tau}_{k}=+\mathrm{\infty}, and {L}_{+}=\{x\in {R}^{n}:x=\varphi (t),t\in {R}_{+}\}. Let x(t,{t}_{0},{x}_{0}) denote the solution of Equation (3.1) for which x({t}^{+},{t}_{0},{x}_{0})={x}_{0}, and let {J}^{+}({t}_{0},{x}_{0}) denote the right maximal interval of the existence of this solution.
Lemma 3.1 The solution x=\varphi (t) of Equation (3.1) is said to be

(i)
orbitally stable, if
\begin{array}{c}(\mathrm{\forall}\rho >0)\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}\eta >0)\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}{t}_{0}\in {R}_{+},t{\tau}_{k}>\eta )\phantom{\rule{0.25em}{0ex}}(\mathrm{\exists}\delta >0)\hfill \\ (\mathrm{\forall}{x}_{0}\in {R}^{n},d({x}_{0},{L}_{+})<\delta ,{x}_{0}\notin {B}_{\eta}\left(\varphi \left({\tau}^{+}\right)\right)\cup {B}_{\eta}(\varphi (\tau )))\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}t\in {J}^{+}({t}_{0},{x}_{0}))\hfill \\ d(x(t,{t}_{0},{x}_{0}),{L}_{+})<\rho ;\hfill \end{array}

(ii)
orbitally attractive, if
\begin{array}{c}(\mathrm{\forall}\eta >0)\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}{t}_{0}\in {R}_{+},t{\tau}_{k}>\eta )\phantom{\rule{0.25em}{0ex}}(\mathrm{\exists}\lambda >0)\hfill \\ (\mathrm{\forall}{x}_{0}\in {R}^{n},d({x}_{0},{L}_{+})<\lambda ,{x}_{0}\notin {B}_{\eta}\left(\varphi \left({\tau}^{+}\right)\right)\cup {B}_{\eta}(\varphi (\tau )))\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}t\in {J}^{+}({t}_{0},{x}_{0}))\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}\rho >0)\hfill \\ (\mathrm{\exists}\sigma >0,{t}_{0}+\sigma \in {J}^{+}({t}_{0},{x}_{0}))\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}t\ge {t}_{0}+\sigma ,t\in {J}^{+}({t}_{0},{x}_{0}))\hfill \\ d(x(t,{t}_{0},{x}_{0}),{L}_{+})<\rho ;\hfill \end{array}

(iii)
orbitally asymptotically stable, if it is orbitally stable and orbitally attractive.
Definition 3.2 The solution x=\varphi (t) of Equation (3.1) is said to enjoy the property of asymptotic phase if
\begin{array}{c}(\mathrm{\forall}\eta >0)\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}{t}_{0}\in {R}_{+},t{\tau}_{k}>\eta )\phantom{\rule{0.25em}{0ex}}(\mathrm{\exists}\lambda >0)\hfill \\ (\mathrm{\forall}{x}_{0}\in {R}^{n},{x}_{0}\varphi ({t}_{0})<\lambda )\phantom{\rule{0.25em}{0ex}}(\mathrm{\exists}c\in R)\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}\rho >0)\hfill \\ (\mathrm{\exists}\sigma >c,{t}_{0}+\sigma \in {J}^{+}({t}_{0},{x}_{0}))\phantom{\rule{0.25em}{0ex}}(\mathrm{\forall}t>{t}_{0}+\sigma ,t\in {J}^{+}({t}_{0},{x}_{0}),{t}_{0}{\tau}_{k}>\eta )\hfill \\ x(t,{t}_{0},{x}_{0})\varphi (t)<\rho .\hfill \end{array}
Lemma 3.3 [15]
For Equation (3.1) with \epsilon >0, the following conditions hold:
({c}_{1}) For \epsilon =0, Equation (3.1) has a {\tau}_{0}periodic solution x=\varphi (t) with moments of impulsive effect {\tau}_{k}:{\tau}_{k+q}={\tau}_{k}+{\tau}_{0} (k\in Z) and {\varphi}^{\prime}(t)\not\equiv 0 (t\in R).
({c}_{2}) For each k=1,2,\dots ,q, the function \phi (x,\epsilon ) is differentiable in some neighborhood of the point (\varphi ({\tau}_{k}),0) and
{\phi}_{k}(\varphi ({\tau}_{k}),0)=0,\phantom{\rule{2em}{0ex}}\frac{\partial {\phi}_{k}}{\partial x}(\varphi ({\tau}_{k}),0)f(\varphi ({\tau}_{k}),0)\ne 0.
({c}_{3}) There exists a \delta >0 such that for each \epsilon \in (\delta ,\delta ) and {x}_{0}\in {R}^{n}, {x}_{0}\varphi (0)<\delta, the solution x(t,{x}_{0},\epsilon ) of Equation (3.1) is defined for t\in [0,{\tau}_{0}+\delta ]. Let the multipliers {\mu}_{j} (j=1,2,\dots ,n) of the variational equation
\{\begin{array}{c}\frac{dz}{dt}=\frac{\partial f(\varphi (t),0)}{\partial x}z,\phantom{\rule{1em}{0ex}}t\ne {\tau}_{k},\hfill \\ \mathrm{\u25b3}z={N}_{k}z,\phantom{\rule{1em}{0ex}}t={\tau}_{k},\hfill \end{array}
(3.2)
where
\begin{array}{rcl}{N}_{k}& =& \frac{\partial I}{\partial x}(\varphi ({\tau}_{k}),0)\\ +[f(\varphi \left({\tau}_{k}^{+}\right),0)f(\varphi ({\tau}_{k}),0)(1+\frac{\partial I}{\partial x}(\varphi ({\tau}_{k}),0))]\frac{\frac{\partial \phi}{\partial x}}{\frac{\partial \phi}{\partial x}f},\end{array}
satisfy the condition
{\mu}_{1}=1,\phantom{\rule{1em}{0ex}}{\mu}_{j}<1\phantom{\rule{1em}{0ex}}(j=2,3,\dots ,n),
then the {\tau}_{0}periodic solution x=\varphi (t) of Equation (3.1) with \epsilon =0 is orbitally asymptotically stable and enjoys the property of asymptotic phase.
If n=2, Equation (3.1) has the form
\{\begin{array}{l}\begin{array}{l}\frac{dx}{dt}=P(x,y),\\ \frac{dy}{dt}=Q(x,y),\end{array}\}\phi (x,y)\ne 0,\\ \begin{array}{l}\mathrm{\u25b3}x=\alpha (x,y),\\ \mathrm{\u25b3}y=\beta (x,y),\end{array}\}\phi (x,y)=0.\end{array}
(3.3)
If Equation (3.3) has a {\tau}_{0}periodic solution x=\zeta (t), y=\eta (t) and the condition of Lemma 3.1 are satisfied, then it can be (check [13]) that the corresponding variational system has multipliers {\mu}_{1} and
{\mu}_{2}=\prod _{k=1}^{q}{\mathrm{\u25b3}}_{k1}exp\left\{{\int}_{0}^{{\tau}_{0}}(\frac{\partial P}{\partial x}(\zeta (t),\eta (t))+\frac{\partial Q}{\partial y}(\zeta (t),\eta (t)))\phantom{\rule{0.2em}{0ex}}dt\right\},
(3.4)
where
{\mathrm{\u25b3}}_{k}=\frac{{P}_{+}(\frac{\partial \beta}{\partial y}\frac{\partial \varphi}{\partial x}\frac{\partial \beta}{\partial x}\frac{\partial \varphi}{\partial y}+\frac{\partial \varphi}{\partial x})+{Q}_{+}(\frac{\partial \alpha}{\partial x}\frac{\partial \varphi}{\partial y}\frac{\partial \alpha}{\partial y}\frac{\partial \varphi}{\partial x}+\frac{\partial \varphi}{\partial y})}{P\frac{\partial \varphi}{\partial y}+Q\frac{\partial \varphi}{\partial y}},
and P, Q, \frac{\partial \alpha}{\partial x}, \frac{\partial \alpha}{\partial y}, \frac{\partial \beta}{\partial x}, \frac{\partial \beta}{\partial y}, \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y} are calculated at point (\zeta ({\tau}_{k}),\eta ({\tau}_{k})) and {P}_{+}=P(\zeta ({\tau}_{k}^{+}),\eta ({\tau}_{k}^{+})), {Q}_{+}=Q(\zeta ({\tau}_{k}^{+}),\eta ({\tau}_{k}^{+})).
Lemma 3.4 [13]
The Tperiodic solution (x(t),y(t))=(\zeta (t),\eta (t)) of system (3.3) is orbitally asymptotically stable and enjoys the property of asymptotic phase if the multiplier {\mu}_{2} calculated by (3.4) satisfies {\mu}_{2}<1.