Hopf-zero bifurcation in a generalized Gopalsamy neural network model

Abstract

In this paper, we study Hopf-zero bifurcation in a generalized Gopalsamy neural network model. By using multiple time scales and center manifold reduction methods, we obtain the normal forms near a Hopf-zero critical point. A comparison between these two methods shows that the two normal forms are equivalent. Moreover, bifurcations are classified in two-dimensional parameter space near the critical point, and numerical simulations are presented to demonstrate the applicability of the theoretical results.

Appendix

In this Appendix, we compute the normal form of (22) on center manifold near the Hopf-zero bifurcation critical point (b _c ,τ _c ) using the center manifold reduction (CMR) method. To achieve this, first re-scale time by $\tilde{t}\mapsto (t/\tau)$ to normalize the delay so that system (22) becomes

$$ \left \{ \begin{aligned} \dot{y_1}( \tilde{t})&=-\tau y_1(\tilde{t})+\tau a_1y_3( \tilde{t})-\tau a_1by_3(\tilde{t}-1) \\ &\quad {} -\frac{\tau a_1}{3}y_3^3(\tilde{t})+ \frac{\tau a_1b}{3}y_3^3(\tilde{t}-1), \\ \dot{y_2}(\tilde{t})&=-\tau y_2( \tilde{t})+\tau a_2y_1(\tilde{t})-\tau a_2by_1(\tilde{t}-1) \\ &\quad {}-\frac{\tau a_2}{3}y_1^3(\tilde{t}) + \frac{\tau a_2b}{3}y_1^3(\tilde{t}-1), \\ \dot{y_3}(\tilde{t})&=-\tau y_3( \tilde{t})+\tau a_3y_2(\tilde{t})-\tau a_3by_2(\tilde{t}-1) \\ &\quad {}-\frac{\tau a_3}{3}y_2^3(\tilde{t})+ \frac{\tau a_3b}{3}y_2^3(\tilde{t}-1). \end{aligned} \right . $$

(34)

The trivial equilibrium of (34) is y ₁=y ₂=y ₃=0. At the critical point (b,τ)=(b _c ,τ _c ), we choose

$$ \eta(\theta)= \begin{cases} \tau_cN_1, &\theta=0, \\ 0, &\theta\in(-1,0), \\ -\tau_cN_2, &\theta=-1, \end{cases} $$

with

Then the linearized equation of (34) at the trivial equilibrium can be written as

$$ \frac{\mathrm{d}X(\tilde{t})}{\mathrm{d}\tilde{t}}=L_0X_{\tilde{t}}, $$

where $L_{0}\phi=\int_{-1}^{0}\,\mathrm{d}\eta(\theta)\phi(\theta)$, ϕ∈C=C([−1,0],R³), and the bilinear form [51] on C^∗×C (∗ stands for adjoint) is

in which ϕ∈C, ψ∈C^∗. Then the phase space C is decomposed by Λ={±iωτ _c ,0} as C=P⊕Q, where , and the bases for P and its adjoint P ^∗ are given respectively by

$$ \varPhi(\theta) =\left (\begin{array}{c@{\quad}c@{\quad}c} e^{\mathrm{i}\omega\tau_c\theta}~& e^{-\mathrm{i}\omega\tau_c\theta} & \frac{1}{2} \\[1mm] \frac{a_2(1-b_ce^{-\mathrm{i}\omega\tau_c})e^{\mathrm{i}\omega\tau_c\theta}}{1+\mathrm{i}\omega} & \frac{a_2(1-b_ce^{\mathrm{i}\omega\tau_c})e^{-\mathrm{i}\omega\tau_c\theta}}{1-\mathrm{i}\omega} & \frac{a_2(1-b_c)}{2} \\[1.5mm] \frac{(1+\mathrm{i}\omega)e^{\mathrm{i}\omega\tau_c\theta}}{a_1(1-b_ce^{-\mathrm{i}\omega\tau_c})} & \frac{(1-\mathrm{i}\omega)e^{-\mathrm{i}\omega\tau_c\theta}}{a_1(1-b_ce^{\mathrm{i}\omega\tau_c})} &\frac{1}{2a_1(1-b_c)} \end{array} \right ) $$

and

$$ \varPsi(s)= \left (\begin{array}{c@{\quad}c@{\quad}c} he^{-\mathrm{i}\omega \tau_cs} & \frac{h(1+\mathrm{i}\omega)e^{-\mathrm{i}\omega\tau_cs}}{a_2(1-b_ce^{-\mathrm{i}\omega\tau_c})} & \frac{a_1h(1-b_ce^{-\mathrm{i}\omega\tau_c})e^{-\mathrm{i}\omega\tau_cs}}{1+\mathrm{i}\omega} \\[2mm] \bar{h}e^{\mathrm{i}\omega \tau_cs} & \frac{\bar{h}(1-\mathrm{i}\omega)e^{\mathrm{i}\omega\tau_cs}}{a_2(1-b_ce^{\mathrm{i}\omega\tau_c})} & \frac{a_1\bar{h}(1-b_ce^{\mathrm{i}\omega\tau_c})e^{\mathrm{i}\omega\tau_cs}}{1-\mathrm{i}\omega} \\[1.5mm] \frac{2}{3(1-\tau_cab_c)} & \frac{2a_1a_3(1-b_c)^2}{3(1-\tau_cab_c)} & \frac{2a_1(1-b_c)}{3(1-\tau_cab_c)} \end{array} \right ), $$

where $h=(3+\frac{3b_{c}\tau_{c}(1+\mathrm{i}\omega)e^{-\mathrm{i}\omega \tau_{c}}}{b_{c}e^{-\mathrm{i}\omega\tau_{c}}-1})^{-1}$.

We also use the same bifurcation parameters given by b=b _c +b _ε and τ=τ _c +τ _ε in (34), where b _ε and τ _ε are perturbation parameters, and denote ε=(b _ε ,τ _ε ). Then (34) can be written as

$$ \frac{\mathrm{d}X(\tilde{t})}{\mathrm{d}\tilde{t}} =L(\varepsilon)X_{\tilde{t}}+F(X_{\tilde{t}}, \varepsilon), $$

(35)

where

and

We now consider the enlarged phase space BC of functions from [−1,0] to R³, which are continuous on [−1,0) with a possible jump discontinuity at zero. This space can be identified as C×R³. Thus, its elements can be written in the form $\tilde{\varphi}=\varphi+X_{0}c$, where φ∈C, c∈R³ and X ₀ is a 3×3 matrix-valued function, defined by X ₀(θ)=0 for θ∈[−1,0) and X ₀(0)=I. In the BC, (35) becomes an abstract ODE,

$$ \frac{\mathrm{d}u}{\mathrm{d}\tilde{t}}=Au+X_0\tilde{F}(u,\varepsilon), $$

(36)

where u∈C, and A is defined by

$$ A:~C^1\rightarrow \mathrm{BC}, Au=\frac{\mathrm{d}u}{\mathrm{d}\tilde{t}} +X_0\biggl[L_0u-\frac{\mathrm{d}u(0)}{\mathrm{d}\tilde{t}}\biggr], $$

and

$$ \tilde{F}(u,\varepsilon)=\bigl[L(\varepsilon)-L_0\bigr]u+F(u, \varepsilon). $$

By using the continuous projection π: BC↦P, π(ϕ+X ₀ c)=Φ[(Ψ,ϕ)+Ψ(0)c], we can decompose the enlarged phase space by Λ={±iωτ _c ,0} as BC=P⊕Ker^π, where Ker^π={ϕ+X ₀ c: π(ϕ+X ₀ c)=0}, denoting the Kernel under the projection π. Let $\eta=(\eta_{1},\bar{\eta}_{1},\eta_{2})^{\mathrm{T}}$, $v_{\tilde{t}}\in Q^{1} :=Q\cap\mathrm{C}^{1}\subset\mathrm{Ker}^{\pi}$, and $A_{Q^{1}}$ the restriction of A as an operator from Q ¹ to the Banach space Ker^π. Further, denote $u_{\tilde{t}}=\varPhi \eta+v_{\tilde{t}}$. Then Eq. (36) is decomposed to the form:

$$ \left \{ \begin{aligned} &\frac{\mathrm{d}\eta}{\mathrm{d}\tilde{t}}=B\eta+\varPsi(0)\tilde{F}(\varPhi \eta+v_{\tilde{t}},\varepsilon), \\ &\frac{\mathrm{d}v_{\tilde{t}}}{\mathrm{d}\tilde{t}}=A_{Q^1}v_{\tilde{t}}+(\mathrm{I}- \pi)X_0\tilde{F}(\varPhi\eta+v_{\tilde{t}},\varepsilon), \end{aligned} \right . $$

(37)

where B=diag{iωτ _c ,−iωτ _c ,0}.

Next, let $M_{2}^{1}$ denote the operator defined in $V_{2}^{5}(\mathrm{C}^{3}\times \mathrm{Ker}^{\pi})$, with

where $V_{2}^{5}(\mathrm{C}^{3})$ represents the linear space of the second-order homogeneous polynomials in five variables $(\eta_{1},\bar{\eta}_{1},\eta_{2},b_{\varepsilon},\tau_{\varepsilon})$ with coefficients in C³. Then we may choose the decomposition $V_{2}^{5}(\mathrm{C}^{3})= \mathrm{Im}(M_{2}^{1})\oplus\mathrm{Im}(M_{2}^{1})^{c}$ with complementary space $\mathrm{Im}(M_{2}^{1})^{c}$ spanned by the elements b _ε η ₁ e ₁, τ _ε η ₁ e ₁, η ₁ η ₂ e ₁, $b_{\varepsilon}\bar{\eta}_{1}e_{2}$, $\tau_{\varepsilon}\bar{\eta}_{1}e_{2}$, $\bar{\eta}_{1}\eta_{2}e_{2}$, τ _ε η ₂ e ₃, b _ε η ₂ e ₃, $\eta_{2}^{2}e_{3}$, $\eta_{1}\bar{\eta}_{1}e_{3}$, where e _i (i=1,2,3) are unit vectors.

Consequently, the normal form of (35) on the center manifold near the equilibrium (0,0,0) associated with the critical point (b _ε ,τ _ε )=(0,0) has the form

$$\frac{\mathrm{d}\eta}{\mathrm{d}\tilde{t}}=B\eta+\frac{1}{2}g_2^1(\eta,0,\varepsilon)+\mathrm{h.o.t.}, $$

where $g_{2}^{1}$ is the function giving the quadratic terms in (η,ε) for $v_{\tilde{t}}=0$, and is determined by $g_{2}^{1}(\eta,0,\varepsilon)=\mathit{Proj}_{(\mathrm{Im}(M_{2}^{1}))^{c}}\times f_{2}^{1}(\eta,0,\varepsilon)$, where $f_{2}^{1}(\eta,0,\varepsilon)$ is the function giving the quadratic terms in (η,ε) for $v_{\tilde{t}}=0$ defined by the first equation of (37). Thus, the normal form, truncated at the quadratic order terms, is given by

$$ \left \{ \begin{aligned} \frac{\mathrm{d}\eta_1}{\mathrm{d}\tilde{t}}&=\mathrm{i}\omega \tau_c\eta_1+3\mathrm{i}\omega h\tau_\varepsilon \eta_1 \\ &\quad {}-\frac{3h\tau_ce^{-\mathrm{i}\omega\tau_c}(1+\mathrm{i}\omega)}{1-b_ce^{-\mathrm{i}\omega\tau_c}}b_\varepsilon\eta_1, \\ \frac{\mathrm{d}\bar{\eta}_1}{\mathrm{d}\tilde{t}}&=-\mathrm{i}\omega\tau_c\bar{\eta}_1-3 \mathrm{i}\omega \bar{h}\tau_\varepsilon\bar{\eta}_1 \\ &\quad {}- \frac{3\bar{h}\tau_ce^{\mathrm{i}\omega\tau_c}(1-\mathrm{i}\omega)}{ 1-b_ce^{\mathrm{i}\omega\tau_c}}b_\varepsilon\bar{\eta}_1, \\ \frac{\mathrm{d}\eta_2}{\mathrm{d}\tilde{t}}&=\frac{a\tau_c}{\tau_cab_c-1}b_\varepsilon\eta_2, \end{aligned} \right . $$

(38)

where $h=(3+\frac{3b_{c}\tau_{c}(1+\mathrm{i}\omega)e^{-\mathrm{i}\omega \tau_{c}}}{b_{c}e^{-\mathrm{i}\omega\tau_{c}}-1})^{-1}$.

To find the normal form up to third order, similarly, let $M_{3}^{1}$ denote the operator defined in $V_{3}^{3}(\mathrm{C}^{3}\times \mathrm{Ker}^{\pi})$, with

where $V_{3}^{3}(\mathrm{C}^{3})$ denotes the linear space of the third-order homogeneous polynomials in three variables: η ₁, $\bar{\eta}_{1}$, and η ₂ with coefficients in C³. Then one may choose the decomposition $V_{3}^{3}(\mathrm{C}^{3})=\mathrm{Im}(M_{3}^{1})\oplus\mathrm{Im}(M_{3}^{1})^{c}$ with complementary space $\mathrm{Im}(M_{3}^{1})^{c}$ spanned by the elements $\eta_{1}\eta_{2}^{2}e_{1}$, $\eta_{1}^{2}\bar{\eta}_{1}e_{2}$, $\bar{\eta}_{1}\eta_{2}^{2}e_{2}$, $\eta_{1}\bar{\eta}_{1}^{2}e_{3}$, $\eta_{2}^{3}e_{1}$, $\eta_{1}\bar{\eta}_{1}\eta_{2}e_{3}$, where e _i (i=1,2,3) are unit vectors.

Therefore, the normal form up to third-order terms is given by

$$ \frac{\mathrm{d}\eta}{\mathrm{d}\tilde{t}}=B\eta+\frac{1}{2!}g_2^1( \eta,0,\varepsilon) +\frac{1}{3!}g_3^1(\eta,0, \varepsilon)+\mathrm{h.o.t.}, $$

(39)

where

$$\frac{1}{3!}g_3^1(\eta,0,0)=\frac{1}{3!} \bigl(I-P^1_{I,3}\bigr)f_3^1( \eta,0,0), $$

and $f_{3}^{1}(\eta,0,0)$, is the function giving the cubic terms in $(\eta, \varepsilon, v_{\tilde{t}})$ for ε=0, and $v_{\tilde{t}}=0$, is defined by the first equation of (37). Finally, the normal form on the center manifold arising from (37) becomes

$$ \left \{ \begin{aligned} & \frac{\mathrm{d}\eta_1}{\mathrm{d}{t}}=\mathrm{i}\omega \eta_1+d_1\eta_1+P_1 \eta_1^2\bar{\eta}_1+ P_2 \eta_1\eta_2^2, \\ & \frac{\mathrm{d}\bar{\eta}_1}{\mathrm{d}{t}}=- \mathrm{i}\omega\bar{\eta}_1+\bar{d}_1\bar{ \eta}_1 +\bar{P}_1\eta_1\bar{ \eta}_1^2+\bar{P}_2\bar{\eta}_1 \eta_2^2, \\ & \frac{\mathrm{d}\eta_2}{\mathrm{d}{t}}=d_2 \eta_2+ P_3\eta_2^3+P_4 \eta_1\bar{\eta}_1\eta_2, \end{aligned} \right . $$

(40)

where d _i (i=1,2) and P _i (i=1,2,3,4) are given in (31).