Here, we will pay attention to determine the direction and stability of flip bifurcation and Neimark-Sacker bifurcation of system (3) around *E*_{2} via application of center manifold theory [19]. The integral step size *δ* is being taken as a real bifurcation parameter.

### Flip bifurcation: direction and stability

We take the parameters (*a*,*α*,*β*,*δ*) arbitrarily locate in \({\text {FB}}^{1}_{E_{2}}\). For other set \({\text {FB}}^{2}_{E_{2}}\), one can apply similar reasoning. Consider the system (3) at fixed point *E*_{2}(*x*^{∗},*y*^{∗}) with parameters lie in \({\text {FB}}^{1}_{E_{2}}\).

Let

$$\delta=\delta_{F}=\frac{- L - \sqrt{L^{2} - 4 M}}{M},$$

then the eigenvalues of *J*(*E*_{2}) are

$$\lambda_{1}(\delta_{F})=-1 \;\; \text{and} \;\; \lambda_{2}(\delta_{F})=3 + L \delta_{F}.$$

In order for |*λ*_{2}(*δ*_{F})|≠1, we have

$$ L \delta_{F} \ne -2, -4. $$

(7)

Assume that \(\tilde {x}=x-x^{*}, \;\;\tilde {y}=y-y^{*}\), and set *A*(*δ*)=*J*(*x*^{∗},*y*^{∗}). Then, we transform the fixed point (*x*^{∗},*y*^{∗}) of system (3) to the origin. By Taylor expansion, system (3) can be written as

$$ \left(\begin{array}{l} \tilde{x} \\ \tilde{y} \end{array}\right) \rightarrow A(\delta) \left(\begin{array}{l} \tilde{x} \\ \tilde{y} \end{array}\right) + \left(\begin{array}{l} F_{1}(\tilde{x}, \tilde{y}, \delta) \\ F_{2}(\tilde{x}, \tilde{y}, \delta) \end{array}\right) $$

(8)

where \(X=(\tilde {x}, \tilde {y})^{T}\) is the vector of the transformed system and

$$ {\begin{aligned} F_{1}(\tilde{x}, \tilde{y}, \delta) &= \frac{1}{6} \left[-\frac{6 a \delta {y^{*}}^{2}}{(x^{*}+ay^{*})^{4}} \tilde{x}^{3} - \frac{6 a^{2} \delta {x^{*}}^{2}}{(x^{*}+ay^{*})^{4}} \tilde{x}^{3} - \frac{6a \delta y^{*}(-2x^{*} +a y^{*})}{(x^{*}+ay^{*})^{4}} \tilde{x}^{2} \tilde{y} + \frac{6a \delta x^{*}(-x^{*} + 2a y^{*})}{(x^{*}+ay^{*})^{4}} \tilde{x} \tilde{y}^{2} \right] \\ & + \frac{1}{2} \left[\delta \left(-2 - \frac{2x^{*} y^{*}}{(x^{*}+ay^{*})^{3}} + \frac{2y^{*}}{(x^{*}+ay^{*})^{2}} \right) \tilde{x}^{2} + \frac{2a \delta {x^{*}}^{2}}{(x^{*}+ay^{*})^{3}} \tilde{y}^{2} - \frac{4 a \delta x^{*} y^{*}}{(x^{*}+ay^{*})^{3}} \tilde{x} \tilde{y} \right] \\ & \qquad+ O(\|X\|^{4}) \\ F_{2}(\tilde{x}, \tilde{y}, \delta) &= \frac{d \delta}{{x^{*}}^{4}} \tilde{x} \left({x^{*} \tilde{y} - y^{*} \tilde{x}} \right)^{2} - \frac{d \delta}{{x^{*}}^{3}} \tilde{x} \left({x^{*} \tilde{y} - y^{*} \tilde{x}} \right)^{2} + O(\|X\|^{4}) \end{aligned}} $$

(9)

The system (8) can be expressed as \(X_{n+1}=AX_{n} + \frac {1}{2}B(X_{n},X_{n}) + \frac {1}{6}C(X_{n},X_{n},X_{n}) + O({X_{n}}^{4})\) where \( B(x, y) = \left (\begin {array}{l} B_{1}(x, y) \\ B_{2}(x, y) \end {array}\right) \) and \( C(x, y, u) = \left (\begin {array}{l} C_{1}(x, y, u) \\ C_{2}(x, y, u) \end {array}\right) \)are symmetric multi-linear vector functions of \(x, y, u \in \mathbb {R}^{2}\) and defined as follows:

$${\begin{aligned} B_{1}(x, y) & = \sum_{j, k=1}^{2} \left. \frac{\delta^{2} F_{1}(\xi, \delta)}{\delta \xi_{j} \delta \xi_{k}} \right|_{\xi=0}\; {x_{j} y_{k}} = -\frac{2a \delta x^{*}y^{*}}{(x^{*}+ay^{*})^{3}} ({x_{1} y_{2}} + {x_{2} y_{1}}) + \frac{2a \delta {x^{*}}^{2}}{(x^{*}+ay^{*})^{3}}{x_{2} y_{2}}\\ & \qquad + \delta \left(-2 - \frac{2x^{*} y^{*}}{(x^{*}+ay^{*})^{3}} + \frac{2y^{*}}{(x^{*}+ay^{*})^{2}} \right) {x_{1} y_{1}}, \\ B_{2}(x, y) & = \sum_{j, k=1}^{2} \left. \frac{\delta^{2} F_{2}(\xi, \delta)}{\delta \xi_{j} \delta \xi_{k}} \right|_{\xi=0}\; {x_{j} y_{k}} = - \frac{2 d \delta}{x^{*}} {x_{2} y_{2}} + \frac{2 d \delta y^{*}}{{x^{*}}^{2}} ({x_{1} y_{2}} + {x_{2} y_{1}}) - \frac{2 d \delta {y^{*}}^{2}}{{x^{*}}^{3}} {x_{1} y_{1}}, \end{aligned}} $$

$${\begin{aligned} C_{1}(x, y, u) & = \sum_{j, k, l=1}^{2} \left. \frac{\delta^{2} F_{1}(\xi, \delta)}{\delta \xi_{j} \delta \xi_{k} \delta \xi_{l}} \right|_{\xi=0}\; {x_{j} y_{k} u_{l}} = - \frac{6a \delta {y^{*}}^{2}}{(x^{*}+ay^{*})^{4}} {x_{1} y_{1} u_{1}} - \frac{6a^{2} \delta {x^{*}}^{2}}{(x^{*}+ay^{*})^{4}} {x_{2} y_{2} u_{2}}\\ & \qquad - \frac{2ay^{*} (-2x^{*} + ay^{*}) \delta}{(x^{*}+ay^{*})^{4}} (x_{1} y_{2} u_{1} + x_{2} y_{1} u_{1} + x_{1} y_{1} u_{2}) \\ & \qquad + \frac{2ax^{*} (-x^{*} + 2ay^{*}) \delta}{(x^{*}+ay^{*})^{4}} (x_{1} y_{2} u_{2} + x_{2} y_{1} u_{2} + x_{2} y_{2} u_{1}),\\ C_{2}(x, y, u) & = \sum_{j, k, l=1}^{2} \left. \frac{\delta^{2} F_{2}(\xi, \delta)}{\delta \xi_{j} \delta \xi_{k} \delta \xi_{l}} \right|_{\xi=0}\; {x_{j} y_{k} u_{l}} = \frac{2 d \delta}{{x^{*}}^{2}} (x_{1} y_{2} u_{2} + x_{2} y_{1} u_{2} + x_{2} y_{2} u_{1}) \\ & \qquad - \frac{4 d \delta {y^{*}}}{{x^{*}}^{3}} (x_{1} y_{1} u_{2} + x_{1} y_{2} u_{1} + x_{2} y_{1} u_{1}) + \frac{6 d \delta {y^{*}}^{2}}{{x^{*}}^{4}} {x_{1} y_{1} u_{1}}. \end{aligned}} $$

Let \(p, q \in \mathbb {R}^{2}\) be eigenvectors of *A* and transposed matrix *A*^{T} respectively for *λ*_{1}(*δ*_{F})=−1 Then, we have

$$A(\delta_{F})q=-q \quad \text{and} \quad A^{T}(\delta_{F})p=-p.$$

Direct computation shows

$$\begin{array}{*{20}l} q & \sim \left(2 + bd \delta_{F} - \frac{2 d \delta_{F} {y^{*}}}{x^{*}}, - \frac{d \delta_{F} {y^{*}}^{2}}{{x^{*}}^{2}} \right)^{T}, \\ p & \sim \left(2 + bd \delta_{F} - \frac{2d \delta_{F} {y^{*}}}{x^{*}}, \frac{\delta_{F} {x^{*}}^{2}}{(x^{*}+ay^{*})^{2}} \right)^{T}. \end{array} $$

We use 〈*p*,*q*〉=*p*_{1}*q*_{1}+*p*_{2}*q*_{2}, standard scalar product in \(\mathbb {R}^{2}\) to normalize the vectors *p* and *q*. Setting the normalized vectors as

$$\begin{array}{*{20}l} q & = \left(2 + bd \delta_{F} - \frac{2 d \delta_{F} {y^{*}}}{x^{*}}, - \frac{d \delta_{F} {y^{*}}^{2}}{{x^{*}}^{2}} \right)^{T}, \\ p & = \gamma_{1} \left(2 + bd \delta_{F} - \frac{2d \delta_{F} {y^{*}}}{x^{*}}, \frac{\delta_{F} {x^{*}}^{2}}{(x^{*}+ay^{*})^{2}} \right)^{T}. \end{array} $$

where \(\gamma _{1} =\frac {1}{(2 + bd \delta _{F} - \frac {2d \delta _{F} {y^{*}}}{x^{*}})^{2} - \frac {d \delta ^{2}_{F} {y^{*}}^{2}}{(x^{*}+ay^{*})^{2}}}.\) We see that 〈*p*,*q*〉=1.

The sign of coefficient *l*_{1}(*δ*_{F}) determines the direction of flip bifurcation and is computed by

$$ l_{1}(\delta_{F}) = \frac{1}{6} \langle p, C(q, q, q) \rangle - \frac{1}{2} \langle p, B(q, (A-I)^{-1} B(q, q)) \rangle $$

(10)

We summarize above discussion into the following theorem for direction and stability of flip bifurcation.

###
**Theorem 1**

Assume that (7) holds. Then, if *l*_{1}(*δ*_{F})≠0 and the parameter *δ* varies its value in a small vicinity of \(FB^{1}_{E_{2}}\), the system (3) experiences a flip bifurcation at positive fixed point *E*_{2}(*x*^{∗},*y*^{∗}). Moreover, if *l*_{1}(*δ*_{F})>0 (resp., *l*_{1}(*δ*_{F})<0), then there exists stable (resp., unstable) period-2 orbits bifurcate from *E*_{2}(*x*^{∗},*y*^{∗}).

### Neimark-Sacker bifurcation: direction and stability

Next, we take the parameters (*a*,*α*,*β*,*δ*) arbitrarily located in \({\text {NSB}}_{E_{2}}\). Consider the system (3) at fixed point *E*_{2}(*x*^{∗},*y*^{∗}) with parameters vary in the vicinity of \({\text {NSB}}_{E_{2}}\). Then, the roots (eigenvalues) of Eq. 6 are pair of complex conjugate and given by

$$\lambda, \bar{\lambda}=\frac{-p(\delta) \pm i \sqrt{4q(\delta)-p(\delta)^{2}}}{2}=1 + \frac{L \delta}{2} \pm \frac{i \delta}{2} \sqrt{4M - L^{2}}.$$

Let

$$ \delta = \delta_{{\text{NS}}} =- \frac{L}{M} $$

(11)

Then, we have \(|\lambda |=\sqrt {q(\delta _{{\text {NS}}})}=1.\)

From the transversality condition, we get

$$ \frac{d|\lambda(\delta)|}{d\delta}|_{\delta = \delta_{{\text{NS}}}} =-\frac{L}{2} \ne 0 $$

(12)

Moreover, the nonresonance condition *p*(*δ*_{NS})≠0,1 obviously satisfies

$$ \frac{L^{2}}{M}\ne 2, 3 $$

(13)

and we have

$$ \lambda^{k}(\delta_{{\text{NS}}}) \ne 1 \;\; \text{for} \; k=1, 2, 3, 4 $$

(14)

Let \(q, p \in \mathbb {C}^{2}\) be eigenvectors of *A*(*δ*_{NS}) and *A*^{T}(*δ*_{NS}) for eigenvalues *λ*(*δ*_{NS}) and \(\bar {\lambda }(\delta _{{\text {NS}}})\) respectively such that

$$A(\delta_{{\text{NS}}}) q = \lambda(\delta_{{\text{NS}}}) q, \;\quad A(\delta_{{\text{NS}}}) \bar{q} = \bar{\lambda}(\delta_{{\text{NS}}}) \bar{q} $$

and

$$A^{T}(\delta_{{\text{NS}}}) p = \bar{\lambda}(\delta_{{\text{NS}}}) p, \;\quad A^{T}(\delta_{{\text{NS}}}) \bar{p} = \lambda(\delta_{{\text{NS}}}) \bar{p}. $$

By direct calculation, we obtain

$$\begin{array}{*{20}l} q & \sim \left(1 + bd \delta_{{\text{NS}}} - \frac{2d \delta_{{\text{NS}}} y^{*}}{x^{*}} - \lambda, - \frac{d \delta_{{\text{NS}}} {y^{*}}^{2}}{{x^{*}}^{2}} \right)^{T}, \\ p & \sim \left(1 + bd \delta_{{\text{NS}}} - \frac{2 d \delta_{{\text{NS}}} y^{*}}{x^{*}} - \bar{\lambda}, \frac{\delta_{{\text{NS}}} x^{*}}{(x^{*}+ay^{*})^{2}} \right)^{T}. \end{array} $$

For normalization of the vectors *p* and *q*, we set \(p = \gamma _{2} \left (1 + bd \delta _{{\text {NS}}} - \frac {2 d \delta _{{\text {NS}}} y^{*}}{x^{*}} - \bar {\lambda }, \frac {\delta _{{\text {NS}}} x^{*}}{(x^{*}+ay^{*})^{2}} \right)^{T}\) where

$$\gamma_{2} =\frac{1}{(1 + bd \delta_{{\text{NS}}} - \frac{2 d \delta_{{\text{NS}}} y^{*}}{x^{*}} - \bar{\lambda})^{2} - \frac{d \delta^{2}_{{\text{NS}}} {x^{*}}^{2} {y^{*}}^{2}}{x^{*}(x^{*}+ay^{*})^{2}}}.$$

Then we see that \(\langle p, q \rangle =\bar {p_{1}} q_{2} + \bar {p_{2}} q_{1}=1\).

When *δ* close to *δ*_{NS} and \(z \in \mathbb {C}\), the vector \(X \in \mathbb {R}^{2}\) can be decomposed uniquely as \(X = zq + \bar {z}\bar {q}\).

It is obvious that *z*=〈*p*,*X*〉. Thus, we obtain the following transformed form of system (8) for all sufficiently small |*δ*| near *δ*_{NS}:

$$z \mapsto \lambda(\delta)z + g(z, \bar{z}, \delta),$$

where *λ*(*δ*)=(1+*φ*(*δ*))*e*^{iθ(δ)} with *φ*(*δ*_{NS})=0 and \(g(z, \bar {z}, \delta)\) is a smooth complex-valued function. According to Taylor expression, the function *g* can be written as

$$g(z, \bar{z}, \delta) = \sum_{k+l \ge2} {\frac{1}{k! l!}} g_{kl}(\delta) z^{k} {\bar{z}}^{l}, \;\; \text{with} \;\; g_{kl} \in \mathbb{C}, \; k, l = 0, 1,\cdots.$$

By symmetric multi-linear vector functions, the Taylor coefficients *g*_{kl} are obtained as

$$\begin{array}{*{20}l} g_{20}(\delta_{{\text{NS}}}) & = \langle p, B(q, q) \rangle,\\ g_{11}(\delta_{{\text{NS}}}) & = \langle p, B(q, \bar{q}) \rangle\\ g_{02}(\delta_{{\text{NS}}}) & = \langle p, B(\bar{q}, \bar{q}) \rangle,\\ g_{21}(\delta_{{\text{NS}}}) & = \langle p, C(q, q, \bar{q}) \rangle, \end{array} $$

The coefficient *l*_{2}(*δ*_{NS}) which determines the direction of Neimark-Sacker bifurcation in a generic system exhibiting invariant closed curve can be calculated via

\(\phantom {\dot {i}\!}l_{2}(\delta _{{\text {NS}}}) = \text {Re} \left (\frac {e^{- i \theta (\delta _{{\text {NS}}})} g_{21}}{2} \right) - \text {Re} \left (\frac {(1-2e^{i \theta (\delta _{{\text {NS}}})}) e^{-2 i \theta (\delta _{{\text {NS}}})}}{2(1-e^{i \theta (\delta _{{\text {NS}}})})} g_{20} g_{11} \right) - \frac {1}{2} |g_{11}|^{2} - \frac {1}{4} |g_{02}|^{2},\) where \(\phantom {\dot {i}\!}e^{i \theta (\delta _{{\text {NS}}})} = \lambda (\delta _{{\text {NS}}})\).

Summarizing above analysis, we present the following theorem for direction and stability of Neimark-Sacker bifurcation.

###
**Theorem 2**

Suppose that (13) holds and *l*_{2}(*δ*_{NS})≠0. If the parameter *δ* varies its value in small neighborhood of \({\text {NSB}}_{E_{2}}\), then system (3) experiences a Neimark-Sacker bifurcation at positive fixed point *E*_{2}. Moreover, if the sign of *l*_{2}(*δ*_{NS}) is negative (resp., positive), then a unique invariant closed curve bifurcates from *E*_{2} which is attracting (resp., repelling) and the Neimark-Sacker bifurcation is supercritical (resp., subcritical).