Dynamic analysis of nonlinear variable frequency water supply system with time delay

Abstract

In this paper, we study the mathematical model about variable frequency water supply system in the process of applying glue to particleboard. Based on the original linear ordinary differential model, the effects of time delay and the nonlinear factor are considered. Then, we obtain the delayed nonlinear differential equation associated with variable frequency water supply system. Further, we consider the existence and stability of the equilibria and the existence of several types of bifurcations in this functional differential equation. Next, we derive the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation by using the multiple time scales method and the center manifold reduction method, respectively, and analyze the classifications of local dynamics. Finally, by using Matlab software, we obtain numerical simulations with estimated values of parameters and show the existence of stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and a complex chaotic attractor from a sequence of period-doubling bifurcations.

Appendices

Appendix A

When $\tau =\tau _k^{(j)},k=1,2;~j=0,1,2,\ldots$, where $\tau _k^{(j)}$ is given by (9), the characteristic equation (6) has one pair of pure imaginary roots $\lambda ={\pm }i\beta _k$. We treat the time delay $\tau$ as the bifurcation parameter. Suppose system (5) undergoes a Hopf bifurcation from the trivial equilibrium at the critical point: $\tau =\tau _c$. Next, we use the multiple time scales method to derive the normal form of system (5) associated with Hopf bifurcation. Further, by the MTS, the solution of (5) is assumed to take the form:

$$\begin{aligned} \omega (t)=\epsilon \omega _1+\epsilon ^2\omega _2+\epsilon ^3\omega _3+\cdots , \end{aligned}$$

(17)

where $\omega _j=\omega _j(T_0,T_1,T_2,\ldots ),~j=1,2,3,\ldots$ with $T_k=\epsilon ^kt,~k=0,1,2,\ldots$. The derivative with respect to t is now transformed into

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}= & {} \frac{\partial }{\partial T_0}+\epsilon \frac{\partial }{\partial T_1}+\epsilon ^2\frac{\partial }{\partial T_2}+\cdots \\= & {} \mathrm {D}_0+\epsilon \mathrm {D}_1+\epsilon ^2 \mathrm {D}_2+\cdots , \end{aligned}$$

where the differential operator $\mathrm {D}_i=\frac{\partial }{\partial T_i},~i=0,1,2,\ldots .$

We take perturbation as $\tau =\tau _c+\epsilon \tau _\epsilon$ in (5), where $\tau _\epsilon$ is called a detuning parameter [7]. To deal with the delayed terms, we expand $\omega _j(T_0-\tau _c-\epsilon \tau _\epsilon ,T_1-\epsilon (\tau _c+\epsilon \tau _\epsilon ),T_2-\epsilon ^2(\tau _c+\epsilon \tau _\epsilon ),\cdots )$ at $\omega _j(T_0-\tau _c,T_1,T_2,\ldots )$ for $j=1,2,3,\ldots$, then

$$\begin{aligned}&\omega (t-\tau _c-\epsilon \tau _\epsilon ,\epsilon (t-\tau _c-\epsilon \tau _\epsilon ),\epsilon ^2(t-\tau _c-\epsilon \tau _\epsilon ),\cdots )\nonumber \\&\quad =\epsilon \omega _{1\tau _c} +\epsilon ^2(\omega _{2\tau _c}-\tau _\epsilon \mathrm {D}_0\omega _{1\tau _c}-\tau _c\mathrm {D}_1\omega _{1\tau _c})\nonumber \\&\qquad +\, \epsilon ^3(\omega _{3\tau _c}-\tau _\epsilon \mathrm {D}_0\omega _{2\tau _c}-\tau _\epsilon \mathrm {D}_1\omega _{1\tau _c}- \tau _c\mathrm {D}_1\omega _{2\tau _c}\nonumber \\&\qquad -\, \tau _c\mathrm {D}_2\omega _{1\tau _c})+\cdots , \end{aligned}$$

(18)

where $\omega _{j\tau _c}=\omega _j(T_0-\tau _c,T_1,T_2,\ldots ),~j=1,2,3,\ldots$.

Then, substituting the solutions with the multiple scales (17) and (18) into (5) and balancing the coefficients of $\epsilon ^n~(n=1,2,3,\ldots )$ yields a set of ordered linear differential equations.

First, for the $\epsilon ^1-$order terms, we have

$$\begin{aligned} \mathrm {D}_0\omega _1-A_k\omega _1-B_k\omega _{1\tau _c}=0,~k=1,2, \end{aligned}$$

(19)

where $A_k=-\frac{D+pK_0+2pK_mE_k}{J}$ and $B_k=\frac{a+2bE_k}{J}$. Since ${\pm }i\beta _k$ is the eigenvalues of the characteristic equation (6), the solution of (19) can be expressed in the form of

$$\begin{aligned} \omega _1=&G\mathrm {e}^{i\beta _k T_0}+\bar{G}\mathrm {e}^{-i\beta _k T_0},~k=1,2, \end{aligned}$$

(20)

where $G=G(T_1,T_2,\ldots )$ and $\bar{G}=\bar{G}(T_1,T_2,\ldots )$.

Next, for the $\epsilon ^2-$order terms, we obtain

$$\begin{aligned}&\mathrm {D}_0\omega _2-A_k\omega _2-B_k\omega _{2\tau _c}\nonumber \\&\quad =-\frac{pK_m}{J}\omega _1^2+\frac{b}{J}\omega _{1\tau _c}^2 -B_k(\tau _\epsilon \mathrm {D}_0\omega _{1\tau _c}+\tau _c\mathrm {D}_1\omega _{1\tau _c})\nonumber \\&\qquad -\, \mathrm {D}_1\omega _{1},~k=1,2. \end{aligned}$$

(21)

Substituting solution (20) into (21) and simplifying, we obtain the following equation:

$$\begin{aligned} \begin{aligned}&\mathrm {D}_0\omega _2-A_k\omega _2-B_k\omega _{2\tau _c}\\&\quad =\frac{b\mathrm {e}^{-2i\beta _k \tau _c}-pK_m}{J}G^2\mathrm {e}^{2i\beta _k T_0}\\&\quad -B_k(i\beta _k\tau _\epsilon \mathrm {e}^{-i\beta _k \tau _c} G +\tau _c\mathrm {e}^{-i\beta _k \tau _c}\frac{\partial G}{\partial T_1})\mathrm {e}^{i\beta _k T_0}\\&\quad -\frac{\partial G}{\partial T_1}\mathrm {e}^{i\beta _k T_0}+c.c.+\frac{2(b-pK_m)}{J}G\bar{G},~k=1,2, \end{aligned} \end{aligned}$$

(22)

where c.c. stands for the complex conjugate of the preceding terms.

Non-homogeneous equation (22) has a solution if and only if the so-called solvability condition is satisfied [1]. That is, the right-hand side of non-homogeneous equation (22) is orthogonal to every solution of the adjoint homogeneous problem. As a matter of fact, finding the solvability conditions is equivalent to finding the conditions resulted from eliminating the secular terms. To avoid occurrence of secular terms in the solution of (22), the coefficients of $\mathrm {e}^{i\beta _k T_0}$ in (22) must be set zero. Then, $\frac{\partial G}{\partial T_1}$ is solved to yield

$$\begin{aligned} \frac{\partial G}{\partial T_1}=f_k\tau _\epsilon G,~k=1,2, \end{aligned}$$

(23)

where $f_k=-\frac{B_ki\beta _k\mathrm {e}^{-i\beta _k \tau _c}}{1+B_k\tau _c\mathrm {e}^{-i\beta _k \tau _c}}$.

Then, Eq. (22) is reduced to

$$\begin{aligned} \begin{aligned}&\mathrm {D}_0\omega _2-A_k\omega _2-B_k\omega _{2\tau _c}\\&\quad =\frac{b\mathrm {e}^{-2i\beta _k \tau _c}-pK_m}{J}G^2\mathrm {e}^{2i\beta _k T_0}+c.c.\\&\qquad +\frac{2(b-pK_m)}{J}G\bar{G},~k=1,2, \end{aligned} \end{aligned}$$

(24)

thus, the particular solution of $\omega _2(t)$ is obtained from the resulting equation of (24) as

$$\begin{aligned} \omega _2=H_0+H_1\mathrm {e}^{2i\beta _k T_0}+\bar{H}_1\mathrm {e}^{-2i\beta _k T_0},~k=1,2, \end{aligned}$$

(25)

where

$$\begin{aligned} \begin{aligned}&H_0=H_0(T_1,T_2,\ldots )=h_0^{(k)}G\bar{G},\\&H_1=H_1(T_1,T_2,\ldots )=h_1^{(k)}G^2. \end{aligned} \end{aligned}$$

with $h_0^{(k)}=-\frac{2(b-pK_m)}{J(A_k+B_k)}$, $h_1^{(k)}=\frac{b\mathrm {e}^{-2i\beta _k \tau _c}-pK_m}{J(2i\beta _k -A_k-B_k\mathrm {e}^{-2i\beta _k \tau _c})}$, $k=1,2$.

Further, for the $\epsilon ^3-$order terms, we similarly obtain

$$\begin{aligned}&\mathrm {D}_0\omega _3-A_k\omega _3-B_k\omega _{3\tau _c}\nonumber \\&\quad =\,-\mathrm {D}_1\omega _2-\mathrm {D}_2\omega _1-\frac{2pK_m\omega _1\omega _2}{J}\nonumber \\&\qquad -B_k(\tau _\epsilon \mathrm {D}_0\omega _{2\tau _c}+\tau _\epsilon \mathrm {D}_1\omega _{1\tau _c} \,+\tau _c\mathrm {D}_1\omega _{2\tau _c}+\tau _c\mathrm {D}_2\omega _{1\tau _c})\nonumber \\&\qquad +\,\frac{2b\omega _{1\tau _c}}{J}(\omega _{2\tau _c}-\tau _\epsilon \mathrm {D}_0\omega _{1\tau _c}\nonumber \\&\qquad -\,\tau _c\mathrm {D}_1\omega _{1\tau _c}). \end{aligned}$$

(26)

Substituting the solutions (23) and (25) into (26) and letting the coefficients of the terms which may generate secular terms in the solution equal zero, yields the derivatives $\frac{\partial G}{\partial T_2}$ expressed in terms of G,

$$\begin{aligned} \begin{aligned} \frac{\partial G}{\partial T_2}=&g_0^{(k)}\tau ^2_\epsilon G+g_kG^2\bar{G},~k=1,2. \end{aligned} \end{aligned}$$

(27)

where

$$\begin{aligned} \begin{aligned}&g_0^{(k)}=-\frac{B_k\mathrm {e}^{-i\beta _k \tau _c}f_k}{1+B_k\tau _c\mathrm {e}^{-i\beta _k \tau _c}},\\&g_k=-\frac{2(pK_m-b\mathrm {e}^{-i\beta _k \tau _c})(h_0^{(k)}+h_1^{(k)})}{J(1+B_k\tau _c\mathrm {e}^{-i\beta _k \tau _c})}. \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&f_k=-\frac{B_ki\beta _k\mathrm {e}^{-i\beta _k \tau _c}}{1+B_k\tau _c\mathrm {e}^{-i\beta _k \tau _c}},\\&\quad h_0^{(k)}=-\frac{2(b-pK_m)}{J(A_k+B_k)},\\&\quad h_1^{(k)}=\frac{b\mathrm {e}^{-2i\beta _k \tau _c}-pK_m}{J(2i\beta _k -A_k-B_k\mathrm {e}^{-2i\beta _k \tau _c})}. \end{aligned} \end{aligned}$$

The above procedure can in principle continue indefinitely (to any high order). Finally, the equation for $\frac{\partial G}{\partial T_0}$ is given by

$$\begin{aligned} \frac{\partial G}{\partial T_0}=\epsilon \frac{\partial G}{\partial T_1}+\epsilon ^2 \frac{\partial G}{\partial T_2}+\cdots , \end{aligned}$$

Let $\epsilon \rightarrow 1/\epsilon$, and by (23) and (27), we derive the following normal form

$$\begin{aligned} \frac{\partial G}{\partial T_0}=f_k\tau _\epsilon G+g_0^{(k)}\tau ^2_\epsilon G+g_kG^2\bar{G},~k=1,2. \end{aligned}$$

Note that $\tau ^2_\epsilon G$ is small enough for small unfolding parameter $\tau _\epsilon$, and we can omit term $g_0^{(k)}\tau ^2_\epsilon G$ for local normal form. Moreover, the normal forms derived by using the MTS and CMR methods are identical, and we also denote $\frac{\partial G}{\partial T_0}$ as $\dot{G}$ for consistency [24] and thus yield the following normal form up to the third order:

$$\begin{aligned} \dot{G}=f_k\tau _\epsilon G+g_kG^2\bar{G},~(k=1,2), \end{aligned}$$

(28)

where $f_k$ and $g_k$ are given by (23) and (27), respectively.

Appendix B

In this section, we consider the system (13) and derive the normal form of Bogdanov–Takens bifurcation. We can check that the Bogdanov–Takens bifurcation occurs when $A+B=0$ and $B\tau +1=0$ in system (13). Thus, we introduce two bifurcation parameters as $\tau =\tau _c+\mu _1$ and $a=a_c+\mu _2$ and choose

$$\begin{aligned} \eta (\theta )={\left\{ \begin{array}{ll}\tau _cA, &{}\theta =0,\\ 0, &{}\theta \in (-1,0),\\ -\tau _cB, &{}\theta =-1, \end{array}\right. } \end{aligned}$$

where $A=-\frac{1}{J}(D+pK_0+2pK_mE_0)$ and $B=\frac{1}{J}(a_c+2bE_0)$ with $E_0=\frac{-a+D+pK_0}{2(b-pK_m)}$.

Then, the linearized equation of (13) at the trivial equilibrium is

$$\begin{aligned} \frac{\mathrm {d}X(t)}{\mathrm {d}{t}}=L_0X_{t}, \end{aligned}$$

where $L_0\phi =\int _{-1}^{0}d\eta (\theta )\phi (\theta ),~\phi \in C = \mathrm {C}([-1,0],~R^{1})$, and the bilinear form on $C^*\times \mathrm {C}$ ($*$ stands for adjoint) is

$$\begin{aligned} \langle \psi (s)~,~\phi (\theta )\rangle= & {} \psi (0)\phi (0)\\&-\int _{-1}^{0}\int _{\xi =0}^{\theta }\psi (\xi -\theta )d\eta (\theta ) \phi (\xi )d\xi , \end{aligned}$$

in which $\phi \in C$, $\psi \in C^*$. Then, the phase space C is decomposed by $\varLambda =\{0\}$ as $C = P\oplus Q$, where $Q=\{\varphi \in C:~(\psi ,\varphi )=0, \mathrm {for~ all}~\psi \in P^*$}, and the bases for P and its adjoint $P^*$ are given respectively by

$$\begin{aligned} \varPhi (\theta )= \left( \begin{array}{cc} 1,~&\theta \end{array}\right) , \end{aligned}$$

and

$$\begin{aligned} \varPsi (s)= \left( \begin{array}{c} -2s+\frac{2}{3}\\ 2 \end{array}\right) , \end{aligned}$$

We use the bifurcation parameters given by $\tau =\tau _c+\mu _1$ and $a=a_c+\mu _2$ in (13), where $\mu _1$ and $\mu _2$ are perturbation parameters and denote $\mu =(\mu _1,\mu _2)$. Then, (13) can be written as

$$\begin{aligned} \frac{\mathrm {d}X({t})}{\mathrm {d}{t}}=L(\mu )X_{{t}}+F(X_{{t}},\mu ), \end{aligned}$$

(29)

where

$$\begin{aligned} \begin{aligned}&L(\mu )X_{t}=\frac{\tau _c+\mu _1}{J}[-(D+pK_0+2pK_mE_0)\phi (0)\\&\quad +(a_c+2bE_0)\phi (-1)]+\frac{\tau _c\mu _2}{J}\phi (-1), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} F(X_t,\mu )&=-\frac{\tau _c+\mu _1}{J}pK_m\phi ^2(0)\\&\quad +\frac{\tau _c+\mu _1}{J}b\phi ^2(-1)\\&\quad +\frac{\mu _1\mu _2}{J}\phi (-1). \end{aligned}$$

We now consider the enlarged phase space $\mathrm {BC}$ of functions from $[-1,0]$ to $R^2$, which are continuous on $[-1,0)$ with a possible jump discontinuity at zero. This space can be identified as $C\times R^2$. Thus, its elements can be written in the form $\psi =\varphi +X_0c$, where $\varphi \in C$, $c\in R^2$ and $X_0$ is a $2\times 2$ matrix-valued function, defined by $X_0(\theta )=0$ for $\theta \in [-1,0)$ and $X_0(0)=i$. In the $\mathrm {BC}$, (29) becomes an abstract ODE,

$$\begin{aligned} \frac{\mathrm {d}u}{\mathrm {d}{t}}=\mathcal {A}u+X_0\tilde{F}(u,\mu ), \end{aligned}$$

(30)

where $u\in C$, and $\mathcal {A}$ is defined by

$$\begin{aligned} \mathcal {A}:~C^1\rightarrow \mathrm {BC},~~\mathcal {A}u=\frac{\mathrm {d}u}{\mathrm {d}t}+X_0[L_0u-\frac{\mathrm {d}u(0)}{\mathrm {d}t}], \end{aligned}$$

and

$$\begin{aligned} ~\tilde{F}(u,\mu )=[L(\mu )-L_0]u+F(u,\mu ). \end{aligned}$$

By the continuous projection $\pi :~\mathrm {BC}\mapsto P$, $\pi (\varphi +X_0c)=\varPhi [(\varPsi ,\varphi )+\varPsi (0)c]$, we can decompose the enlarged phase space by $\varLambda =\{0\}$ as $\mathrm {BC}= P\oplus \mathrm {Ker} \pi$, where $\mathrm {Ker} \pi =\{\varphi +X_0c:~\pi (\varphi +X_0c)=0\}$, denoting the Kernel under the projection $\pi$. Let $x=(x_1,x_2)^{\mathrm {T}}$, $v_{t}\in Q^1 :=Q\cap \mathrm {C}^{1}\subset \mathrm {Ker} \pi$, and $\mathcal {A}_{Q^1}$ the restriction of $\mathcal {A}$ as an operator from $Q^1$ to the Banach space $\mathrm {Ker} \pi$. Further, denote $u_{t}=\varPhi x+v_{t}$. Then, Eq. (29) is decomposed as

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\mathrm {d}x}{\mathrm {d}t}=Nx+\varPsi (0)\tilde{F}(\varPhi x+v_{t},\mu ),\\&\frac{\mathrm {d}v_{t}}{\mathrm {d}t}=\mathcal {A}_{Q^1}v_{t}+(i-\pi )X_0\tilde{F}(\varPhi x+v_{t},\mu ), \end{aligned} \right. \end{aligned}$$

(31)

where

$$\begin{aligned} N= \left( \begin{array}{c@{\quad }c} 0~&{} 1\\ 0~&{}0 \end{array}\right) , \end{aligned}$$

Next, let $M_2^{1}$ denote the operator defined in $V_2^4(R^2 \times \mathrm {Ker} \pi )$, with

$$\begin{aligned} \begin{aligned} M_2^1:&V_2^4(R^2)\mapsto V_2^4(R^2),\\&(M_2^1p)(x,\mu )=D_x p(x,\mu )Nx-Np(x,\mu ), \end{aligned} \end{aligned}$$

where $V_2^4(R^2)$ represents the linear space of the second-order homogeneous polynomials in four variables $(x_1,x_2,\mu _1,\mu _2)$ with coefficients in $R^2$. Then, it is easy to verify that one may choose the decomposition $V_2^4(R^2)=\mathrm {Im}(M_2^1)\oplus \mathrm {Im}(M_2^1)^c$. The base of $V_2^4(R^2\times \mathrm {Ker} \pi )$ is composed by the following 20 elements $(i=1,2)$:

$$\begin{aligned}&\left( \begin{array}{c} x_1^2\\ 0\end{array}\right) , \left( \begin{array}{c} x_2^2\\ 0\end{array}\right) , \left( \begin{array}{c} x_1x_2\\ 0\end{array}\right) , \left( \begin{array}{c} x_1\mu _i\\ 0\end{array}\right) , \left( \begin{array}{c} x_2\mu _i\\ 0\end{array}\right) , \left( \begin{array}{c} \mu _i^2\\ 0\end{array}\right) ,\\&\quad \left( \begin{array}{c} \mu _1\mu _2\\ 0\end{array}\right) , \left( \begin{array}{c} 0\\ x_1^2\end{array}\right) , \left( \begin{array}{c} 0\\ x_2^2\end{array}\right) , \left( \begin{array}{c} 0\\ x_1x_2\end{array}\right) , \left( \begin{array}{c} 0\\ x_1\mu _i\end{array}\right) , \left( \begin{array}{c} 0\\ x_2\mu _i\end{array}\right) ,\\&\quad \left( \begin{array}{c} 0\\ \mu _i^2\end{array}\right) , \left( \begin{array}{c} 0\\ \mu _1\mu _2\end{array}\right) , \end{aligned}$$

and images of there elements under $M_2^1$ are

$$\begin{aligned}&\left( \begin{array}{c} 2x_1x_2\\ 0\end{array}\right) , \left( \begin{array}{c} 0\\ 0\end{array}\right) , \left( \begin{array}{c} x_1^2\\ 0\end{array}\right) , \left( \begin{array}{c} x_2\mu _i\\ 0\end{array}\right) , \left( \begin{array}{c} -x_1^2\\ 2x_1x_2\end{array}\right) , \left( \begin{array}{c} -x_1x_2\\ x_2^2\end{array}\right) , \\&\quad \left( \begin{array}{c} -x_2^2\\ 0\end{array}\right) , \left( \begin{array}{c} -x_1\mu _i\\ x_2\mu _i\end{array}\right) , \left( \begin{array}{c} -x_2\mu _i\\ 0\end{array}\right) . \end{aligned}$$

Therefore, a basis of $\mathrm {Im}(M_2^1)^c$ can be taken as the set composed by the elements

$$\begin{aligned}&\left( \begin{array}{c} 0\\ x_1^2\end{array}\right) , \left( \begin{array}{c} 0\\ x_1x_2\end{array}\right) , \left( \begin{array}{c} 0\\ x_1\mu _i\end{array}\right) , \left( \begin{array}{c} 0\\ x_2\mu _i\end{array}\right) ,\\&\quad \left( \begin{array}{c} 0\\ \mu _i^2\end{array}\right) , \left( \begin{array}{c} 0\\ \mu _1\mu _2\end{array}\right) , \end{aligned}$$

Consequently, the normal form of (13) on the center manifold associated with the origin equilibrium near $(\mu _1,\mu _2)=(0,0)$ has the form

$$\begin{aligned} \frac{\mathrm {d}x}{\mathrm {d}t}=Nx+\frac{1}{2}g_2^1(x,0,\mu _1,\mu _2)+\mathrm {h.o.t.}, \end{aligned}$$

where $g_2^1$ is the function giving the quadratic terms in $(x_1,x_2,\mu _1,\mu _2)$ for $v_{t}=0$ and is determined by $g_2^1(x,0,\mu _1,\mu _2)=\mathrm {Proj}_{(\mathrm {Im}(M_2^1))^c}\times f_2^1(x,0,\mu _1,\mu _2)$, where $f_2^1(x,0,\mu _1,\mu _2)$ is the function giving the quadratic terms in $(x,\mu _1,\mu _2)$ for $v_{t}=0$ defined by the first equation of (31). Thus, the normal form, truncated at the quadratic-order terms, is given by

$$\begin{aligned} \left\{ \begin{aligned}&\dot{x}_1=x_2,\\&\dot{x}_2=\lambda _1x_1+\lambda _2x_2+d_1x_1^2+d_2x_1x_2, \end{aligned} \right. \end{aligned}$$

(32)

where $\lambda _1=\frac{2\tau _c}{J}\mu _2$, $\lambda _2=-2B\mu _1-\frac{4\tau _c}{3J}\mu _2$, $d_1=\frac{2\tau _c(b-pK_m)}{J}$, $d_2=\frac{4\tau _c(b-pK_m)}{3J}-\frac{4b\tau _c}{J}$.