Phase-Locked Solutions of a Coupled Pair of Nonidentical Oscillators

We start with a simple example of two coupled phase oscillators. In this example, stable phase-locking occurs if and only if coupling is sufficiently strong. We then add amplitude (radial) variables to the phase oscillators in the most straightforward possible way. For symmetric coupling, stable phase-locking still requires sufficiently strong coupling. For asymmetric coupling, however, stable phase-locking now becomes possible for arbitrarily weak coupling. We also give an exact formula for the common frequency of the two oscillators in the phase-locked state. By examining the degenerate Routh-Hurwitz criterion for Hopf bifurcation, we confirm the presence of periodic solutions. By solving the coupled system in polar coordinates, the conditions for the existence of phase-locked solutions are derived. This leads us to demonstrate various oscillatory scenarios in several parameter ranges. Stability or instability of the phased-locked solutions is justified in some cases. We further develop a new approach which combines the monotone dynamics theory and sequential contracting technique to conclude the attraction of the stable phase-locked solution and locate its basin of attraction. The associations between the findings from the Hopf bifurcation theory and those from the polar-coordinate setting are also addressed.

Authors and Affiliations

1. Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 30010, Taiwan

   Kuan-Wei Chen & Chih-Wen Shih

Contributions

C-KW and S-CW completed the analysis and wrote the main manuscript, and C-KW carried out the numerical simulations. All authors reviewed the manuscript.

Corresponding author

Correspondence to Chih-Wen Shih.

Conflict of interest

The authors declare no competing interests.

Communicated by Anthony Bloch.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

Degenerate Routh-Hurwitz criterion:

Consider ODE system

with equilibrium \(\bar{\textbf{x}}\) at \( \mu =\mu ^*\). Let \(J(\bar{\textbf{x}}; \mu ^*)\) be the Jacobian matrix of \(\textbf{f}\) at \(\bar{\textbf{x}}\) and \(\mu = \mu ^*\), with the characteristic polynomial

Let \(H_j\) be the jth Hurwitz matrix generated from \(\Delta (\lambda )\), cf. Uspensky (1948). It was termed simple Hopf bifurcation when the Hopf bifurcation occurs in the situation that \(J(\bar{\textbf{x}}; \mu ^*)\) has a simple pair of purely imaginary eigenvalues and all other eigenvalues have negative real parts. The condition for the simple Hopf bifurcation has been characterized in Liu (1994):

Theorem 6.1

(Liu (1994)) The simple Hopf bifurcation for system (A.1) occurs at \(\textbf{x}=\bar{\textbf{x}}\) and \(\mu =\mu ^*\) if and only if

We call the condition in (A.3) degenerate Routh-Hurwitz criterion. When \(n=4\), the conditions in Theorem 6.1 can be expressed precisely as

Appendix B

Computation of the terms determining the properties of Hopf bifurcation for system (1.8) under symmetric coupling:

At \(\mu = \mu ^* = c - (c^2-(\omega _1 - \omega _2)^2/4)^{1/2}\),

We apply the change of variables \(\textbf{x}=P\textbf{z}\), where

Then,

The transformed system becomes

where \(\textbf{z}=(z_1,z_2,z_3,z_4)\) and \(\mathbf{F({\textbf{z},\mu })}=P^{-1}{} \textbf{f}({P\textbf{z},\mu })\) with \(\textbf{F}=(F_1,F_2,F_3,F_4)\). At \(\mu =\mu ^*\) and \(\textbf{z}=\textbf{0}\), we define

cf. Hassard et al. (1981). Next, for \(k=3,4\), we set

and let \(w_{11}^{k-2},w_{20}^{k-2}\in {\mathbb {C}}^2\) be the solutions of

where

For \(k=3,4\), let

Then we define

Here, \(g_{11} = g_{02} = g_{20} = 0\), and \(g_{21}=-2\). Thus,

Appendix C

Proof of Theorem 3.1:

(i) We perform the phase line analysis to see the dynamics of \(\phi (t)\). It is clear that \(\phi _a^*\) attracts all points \(\phi (0) \in [0, 2 \pi ]\setminus \{\phi _b^{*}\}\) (where we identify \(2\pi \) with 0), see Fig. 2. On the other hand, \(\phi (t)\) starting near \(\phi _b^{*}\) moves away from \(\phi _b^{*}\). With \(\phi (t)\) known from every initial point \(\phi (0)\), we substitute it into the first equation to acquire the behavior of r(t). It can be shown that \(r(t) \rightarrow (\mu -c+c \cos \phi _a^*)^{1/2}={\bar{r}}_a\) as \(t \rightarrow \infty \), for all \( r(0) >0\), \(\phi (0) \in [0, 2 \pi ]\), and \( \phi (0)\ne \phi _b^{*}\). From these arguments, we also see that \(({\bar{r}}_b, \phi _b^{*})\) is unstable.

(ii) Suppose the constant solution \(({\bar{r}}_b, \phi _b^{*})\) of system (3.6) exists. Let (r(t), \(\theta _1(t), \theta _2(t))\) be the solution of system (3.5) starting from an arbitrary initial point \((r(0), \theta _1(0), \theta _2(0))\), with \( r(0)> 0, \theta _1(0)- \theta _2(0) \ne \phi _b^*\). Then \(\phi (t)= \theta _1(t)- \theta _2(t) \) satisfies the second equation of system (3.6), and thus approaches \(\phi _a^*\) as \(t \rightarrow \infty \), by (i). Subsequently, r(t) converges to \({\bar{r}}_a\), as \(t \rightarrow \infty \). In addition, adding the last two equations of (3.5) gives

Thus, \(\theta _1(t)+ \theta _2(t)=( \omega _1+\omega _2)t+ \theta _1(0)+ \theta _2(0)\). Therefore,

We conclude that, subject to phase shift \([\theta _1(0)+ \theta _2(0)]/2\), the solution \((r(t), \theta _1(t), \theta _2(t))\) converges to the phase-locked solution \(({\bar{r}}_a, \theta _{1,a}(t), \theta _{2, a}(t))\) as \(t \rightarrow \infty \), where

The phase difference for this phase-locked solution is \(\phi _a^*\). This completes the proof. \(\square \)

Appendix D

Proposition 6.1 and Theorem 6.2:

Proposition 6.1

Within the following indicated parameter ranges, the constant solutions of system (3.2) that exist are

1. (I)

   \(2c>\omega _{\Delta }\) and

   1. (i)

      \(c-(c^2-\omega _{\Delta }^2/4)^{1/2}<\mu <c+(c^2-\omega _{\Delta }^2/4)^{1/2}\): \(({\bar{r}}_a, {\bar{r}}_a, \phi _a^*)\)

   2. (ii)

      \(c+(c^2-\omega _{\Delta }^2/4)^{1/2}<\mu <c+(4c^2-\omega _{\Delta }^2)^{1/2}\): \(({\bar{r}}_a, {\bar{r}}_a, \phi _a^*)\), \(({\bar{r}}_b, {\bar{r}}_b, \phi _b^*)\)

   3. (iii)

      \(\mu >c+(4c^2-\omega _{\Delta }^2)^{1/2}\): \(({\bar{r}}_a, {\bar{r}}_a, \phi _a^*)\), \(({\bar{r}}_b, {\bar{r}}_b, \phi _b^*)\), \(({\bar{r}}_{1, c}, {\bar{r}}_{2, c}, \phi _c^*)\), \(({\bar{r}}^*_{1, c}, {\bar{r}}^*_{2, c}, \phi _c^*)\)

2. (II)

   \(2c<\omega _{\Delta }\) and \(\mu >c\): \(({\bar{r}}_{1, c}, {\bar{r}}_{2, c}, \phi _c^*)\), \(({\bar{r}}^*_{1, c}, {\bar{r}}^*_{2, c}, \phi _c^*)\).

Note that the condition in (I)(i) is equivalent to \(4c^2 -\omega ^2_{\Delta }>4 (c-\mu )^2\). In (I)(iii), we observe that \(\phi _c^*>\phi _b^*\), from their definitions.

Theorem 6.2

Within the indicated parameter range, the phase-locked solutions \((r_1(t), \theta _1(t), r_2(t), \theta _2(t))\) of system (3.1) that exist are

1. (I)

   \(2c>\omega _{\Delta }\) and

   1. (i)

      \(c-(c^2-\omega _{\Delta }^2/4)^{1/2}<\mu <c+(c^2-\omega _{\Delta }^2/4)^{1/2}\): \( \Gamma _a\)

   2. (ii)

      \(c+(c^2-\omega _{\Delta }^2/4)^{1/2}<\mu <c+(4c^2-\omega _{\Delta }^2)^{1/2}\): \( \Gamma _a\), \( \Gamma _b\)

   3. (iii)

      \(\mu >c+(4c^2-\omega _{\Delta }^2)^{1/2}\): \( \Gamma _a\), \( \Gamma _b\), \( \Gamma _c\), \( \Gamma '_c\)

2. (II)

   \(2c< \omega _{\Delta }\) and \(\mu >c\): \( \Gamma _c\), \( \Gamma '_c\).

Appendix E

Proof of Theorem 3.3:

We linearize system (3.2) and obtain the Jacobian matrix

We denote its characteristic polynomial by

According to Theorem 3.2, \(({\bar{r}}_a,{\bar{r}}_a,\phi _a^*)\) exists under \(2c > |\omega _\Delta |\) and (3.9). At \(({\bar{r}}_a,{\bar{r}}_a,\phi _a^*)\), with \(\mu -c = {\bar{r}}_a^2 - (c^2-\omega _{\Delta }^2/4)^{1/2}\), \(\sin \phi _a^* = \omega _{\Delta }/2c\), and \(\cos \phi _a^*=[1-(\omega _{\Delta }/2c)^2]^{1/2}\), the Jacobian matrix (A.9) becomes

Its characteristic polynomial (A.10) has coefficients

Since \(b_1>0\), \(b_3>0\), and

By the Routh-Hurwitz criterion, all the roots of \(\Delta (\lambda )\) have negative real parts.

At \(({\bar{r}}_b,{\bar{r}}_b,\phi _b^*)\), with \(\mu -c = {\bar{r}}_b^2 + (c^2-\omega _{\Delta }^2/4)^{1/2}\), \(\sin \phi _b^* = \omega _{\Delta }/2c\), and \(\cos \phi _b^*=-[1-(\omega _{\Delta }/2c)^2]^{1/2}\), the Jacobian matrix becomes

The coefficients of its characteristic polynomial (A.10) are

Recall that \(({\bar{r}}_b,{\bar{r}}_b,\phi _b^*)\) exists under \(2c > |\omega _\Delta |\) and (3.10), and hence \(b_1\) and \(b_3\) have opposite signs, and they vanish simultaneously. When \(b_1\) and \(b_3\) are nonzero, the Routh-Hurwitz criterion is not met, and there exists a root with positive real part for \(\Delta (\lambda )\). When \(b_1\) and \(b_3\) vanish, there are two positive roots.

At \(({\bar{r}}_{1,c},{\bar{r}}_{2,c},\phi _c^*)\), with \({\bar{r}}_{1,c}^2+{\bar{r}}_{2,c}^2=\mu -c>0\), \(c\cos \phi _c^*=-{\bar{r}}_{1,c}{\bar{r}}_{2,c}\), and \(c\sin \phi _c^*=({\bar{r}}_{1,c}{\bar{r}}_{2,c}~\omega _{\Delta })/(\mu -c)=({\bar{r}}_{1,c}{\bar{r}}_{2,c}~\omega _{\Delta })/({\bar{r}}_{1,c}^2+{\bar{r}}_{2,c}^2)\), the Jacobian matrix (A.9) becomes

where we denote \({\bar{r}}_1={\bar{r}}_{1,c}\), \({\bar{r}}_2={\bar{r}}_{2,c}\) to save notation. The coefficients of its characteristic polynomial (A.10) are

Since \(b_1=0\) and \(b_3>0\), the Routh-Hurwitz criterion does not hold. Moreover, \(\Delta (\lambda )\) has two roots with positive real part. The computation for constant solution \(({\bar{r}}_{1,c}^*,{\bar{r}}_{2,c}^*,\phi _c^*)\) is similar.

Appendix F

Proof of Theorem 3.4:

When \(\beta =\cos \phi _a^*\) and \(\beta =1\), we denote the positive equilibrium of system (3.20) by \(({\check{r}}_0, {\check{r}}_0)\) and \(({\hat{r}}_0, {\hat{r}}_0)\), respectively, see Fig. 3. We compute to obtain

Indeed, they are the positive solutions of \(r_1= r_1 [r_1^2-(\mu -c)]/(c\cos \phi _a^*)\) and \(r_1=r_1[r_1^2-(\mu -c)]/c\), respectively. Note that \({\bar{r}}_a\) was given precisely in (3.8). Accordingly, a solution \((r_1(t), r_2(t), \phi (t))\) of system (3.2), with \(r_1(0), r_2(0)>0\), \(\phi (0) \in [0, \phi _b^*)\), is attracted to

In the following, we shall apply an iterative argument termed sequential contracting (Shih and Tseng 2011, 2013) to construct a sequence of decreasing upper bounds for the \((r_1, r_2)\)-components and a sequence of increasing upper bounds for the \(\phi \)-component of system (3.2). The goal is to show that almost all solutions \((r_1(t), r_2(t), \phi (t))\) of system (3.2) starting from \(\{r_1, r_2 >0, \phi \in [0, \phi ^*_b)\}\) converge to the constant solution \(({\bar{r}}_a, {\bar{r}}_a, \phi _a^*)\), as \(t \rightarrow \infty \). The corresponding solution \((r_1(t), \theta _1(t), r_2(t), \theta _2(t))\) of (3.1) then converges to its phase-locked solution with phase difference \(\phi _a^*\).

For \(r_1, r_2\in [{\check{r}}_0,{\hat{r}}_0]\), we have

and

Thus,

for all \(r_1\), \(r_2\in [{\check{r}}_0,{\hat{r}}_0]\) and \(\phi \) with \(\sin \phi \ge 0\). Let \(\phi ^{(1)} \in (0, \phi _a^*)\) such that

Then \(\omega _{\Delta } - c ({\check{r}}_0/{\hat{r}}_0+{\hat{r}}_0/ {\check{r}}_0)\sin \phi >0\) for \(\phi \in [0, \phi ^{(1)})\). Subsequently, the \(\phi \)-components of the solutions of (3.2) starting from \([{\check{r}}_0, {\hat{r}}_0] \times [{\check{r}}_0, {\hat{r}}_0] \times [0, \phi _a^* ]\) are attracted to \([\phi ^{(1)}, \phi _a^*]\). Then, for \((r_1, r_2) \in [{\check{r}}_0, {\hat{r}}_0] \times [{\check{r}}_0, {\hat{r}}_0] \times [\phi ^{(1)}, \phi _a^*]\), we have

Let \({\hat{r}}^{(1)}:=[\mu +c(\cos \phi ^{(1)}-1)]^{\frac{1}{2}}\), the positive solution of \(r_1=\frac{r_1}{c\cos \phi ^{(1)}}[r_1^2-(\mu -c)]\). Then, the solutions starting from \([{\check{r}}_0, {\hat{r}}_0] \times [{\check{r}}_0, {\hat{r}}_0] \times [0, \phi _a^* ]\) are attracted to \([{\check{r}}_0, {\hat{r}}^{(1)}] \times [{\check{r}}_0, {\hat{r}}^{(1)}] \times [\phi ^{(1)},\phi _a^*]\). For \(r_1, r_2 \in [{\check{r}}_0,{\hat{r}}^{(1)}]\), we have

provided \(\sin \phi \ge 0\). Note that \(({\check{r}}_0/{\hat{r}}^{(1)}+{\hat{r}}^{(1)}/{\check{r}}_0)<({\check{r}}_0/{\hat{r}}_0+{\hat{r}}_0/{\check{r}}_0)\). Let \(\phi ^{(2)} \in (\phi ^{(1)}, \phi _a^*)\) such that

Then \(\omega _{\Delta } - c ({\check{r}}_0/{\hat{r}}^{(1)}+{\hat{r}}^{(1)}/{\check{r}}_0)\sin \phi >0\) for \(\phi \in [\phi ^{(1)}, \phi ^{(2)})\). Subsequently, the \(\phi \)-components of the solutions of (3.2) starting from \([{\check{r}}_0, {\hat{r}}^{(1)}] \times [{\check{r}}_0, {\hat{r}}^{(1)}] \times [\phi ^{(1)},\phi _a^*]\) are attracted to \([\phi ^{(2)}, \phi _a^*]\). For \(\phi \in [\phi ^{(2)}, \phi _a^*]\), we have

Let \({\hat{r}}^{(2)}:=[\mu +c(\cos \phi ^{(2)}-1)]^{\frac{1}{2}}\), the positive solution of \(r_1=\frac{r_1}{c\cos \phi ^{(2)}}[r_1^2-(\mu -c)]\). Then, the solutions starting from \([{\check{r}}_0, {\hat{r}}^{(1)}] \times [{\check{r}}_0, {\hat{r}}^{(1)}] \times [\phi ^{(1)},\phi _a^*]\) are attracted to \([{\check{r}}_0, {\hat{r}}^{(2)}] \times [{\check{r}}_0, {\hat{r}}^{(2)}] \times [\phi ^{(2)},\phi _a^*]\).

Continuing the process, we can define \(\phi ^{(3)},\ldots ,\phi ^{(k)},\ldots \) and \({\hat{r}}^{(3)},\ldots ,{\hat{r}}^{(k)}, \ldots \), successively, with \( \phi ^{(k+1)}> \phi ^{(k)}\), \({\hat{r}}^{(k+1)} <{\hat{r}}^{(k)}\), where

Now let us show that \(\phi ^{(k)}\rightarrow \phi _a^*\) as \(k\rightarrow \infty \). Let

for \(\rho >0\). Then \(h({\hat{r}}^{(k-1)})={\hat{r}}^{(k)}\) and \(h({\check{r}}_0)={\check{r}}_0\). We compute

Thus \(h'(\rho )>0\) for all \(\rho >{\check{r}}_0\) and \(h'({\check{r}}_0)=0\). Recall that \({\hat{r}}^{(k)}<{\hat{r}}^{(k-1)}\) for all \(k\in {\mathbb {N}}\) and \({\check{r}}_0\) is a fixed point of h. By the graphical or cobweb analysis, \({\check{r}}_0\) attracts the sequence \(\{ {\hat{r}}^{(k)}\}\). We thus conclude that \({\hat{r}}^{(k)}\rightarrow {\bar{r}}_a\) as \(k\rightarrow \infty \), recalling that \({\check{r}}_0={\bar{r}}_a\). Subsequently, \(\phi ^{(k)}\rightarrow \phi _a^*\) as \(k\rightarrow \infty \).

Appendix G

Proof of Lemma 4.1:

For the asymmetric-coupling case, we consider the following upper-lower systems

where \(\rho _1, \rho _2 \ge 0\), and \(\beta >0\). System (A.11) is still cooperative, for each \(\beta \). This family of systems (A.11) provides a sequence of upper-lower-bound systems for (4.1) on \(\{r_1>0, r_2>0, \phi \in [0, {\hat{\phi }}^{(0)}]\}\), for \(\beta \in [ \cos {\hat{\phi }}^{(0)}, 1]\). It is clear that \((\rho _1, \rho _2 )=(0, 0)\) is always an equilibrium of (A.11), whereas the existence of positive equilibrium \(({\bar{\rho }}_1, {\bar{\rho }}_2)\) requires some conditions. Such conditions can be derived from considering the intersection of the two nullclines of (A.11). Let

Then the graph of \(\rho _2=h_1(\rho _1)\) gives the nullcline \(\Gamma _1^\beta :=\{(\rho _1, \rho _2): f_1(\rho _1, \rho _2; \beta ) =0\}\), whereas the reflection of the graph of \(\rho _2=h_2(\rho _1)\) about \(\{\rho _1=\rho _2\}\) gives the nullcline \(\Gamma _2^\beta :=\{(\rho _1, \rho _2): f_2(\rho _1, \rho _2; \beta ) =0\}\), cf. Fig. 13. Each of \(h_1\) and \(h_2\) is a cubic polynomial with 0 as a root; \(h_1\) (resp., \(h_2\)) has a positive root \((\mu -c_1)^{1/2}\) (resp., \((\mu -c_2)^{1/2}\)), provided \(\mu >c_1\) (resp., \(\mu >c_2\)). In addition,

The positive equilibrium \(({\bar{\rho }}_1, {\bar{\rho }}_2)\) of system (A.11) is given by the intersection of \(\Gamma _1^\beta \) and \(\Gamma _2^\beta \). From the properties of functions \(h_1\) and \(h_2\), such intersection exists if and only if

i.e., correspondingly, \(c_1 < \mu \), or \(c_2< \mu \), or \(c_1, c_2>\mu \) and \((c_1-\mu )(c_2-\mu )<c_1c_2\beta ^2\). Notably, \(c_1, c_2>\mu \) implies \((c_1-\mu )(c_2-\mu )<c_1c_2\). According to these observations, we thus derive the conditions for the existence of positive equilibrium \(({\bar{\rho }}_1, {\bar{\rho }}_2)\) of system (A.11) for all \(\beta \in [ \cos {\hat{\phi }}^{(0)}, 1]\), and nonexistence of \(({\bar{\rho }}_1, {\bar{\rho }}_2)\) for \(\beta \in [ \cos {\hat{\phi }}^{(0)}, 1)\), as presented in Lemma 4.1.

Let us show that all solutions of (A.11) are bounded in forward time. We consider the case \(c_1>c_2\). Set \(V(\rho _1, \rho _2)=(\rho _1^2 +\rho _2^2)/2\) and \(\textbf{f}=(f_1, f_2)\). Then

which is negative for \((\rho _1, \rho _2)\) satisfying \((\rho _1^2-A/2)^2 + (\rho _2^2-A/2)^2 >A^2/2\), where \(A:=\mu -c_2+\beta (c_1+c_2)/2\). Thus, every solution starting from the interior of \({{\mathbb {R}}}^2_+\) is bounded.

By applying the theorem for cooperative system in Jiang (1994); Smith (1995), we conclude that every solution starting from the interior of \({{\mathbb {R}}}^2_+\) converges to the positive equilibrium \(({\bar{\rho }}_1, {\bar{\rho }}_2)\), when it exists. If \(({\bar{\rho }}_1, {\bar{\rho }}_2)\) does not exist, then the only equilibrium is (0, 0), and so every solution starting from \({{\mathbb {R}}}^2_+\) converges to the origin.

Appendix H

Proof of Theorem 4.3:

Consider the case \(c_1>c_2\). The arguments are similar for the other case \(c_1 <c_2\). Let \(({\check{r}}_1^{(0)}, {\check{r}}_2^{(0)})\) and \(({\hat{r}}_1^{(0)}, {\hat{r}}_2^{(0)})\) be the constant solutions of system (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\) and \(\beta =1\), respectively. That is, \({\hat{r}}_1^{(0)}={\hat{r}}_2^{(0)}=\sqrt{\mu }\). The nullclines and their intersections for system (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\) and \(\beta =1\), plotted on \((r_1, r_2)\)-plane, are illustrated in Fig. 13, where we denote

From the graphical analysis on the vector field of system (4.17), we can see that \({\check{r}}_1^{(0)}< {\check{r}}_2^{(0)}\) and \({\check{r}}_i^{(0)}<{\hat{r}}_i^{(0)}\), for \(i=1,2\), cf. Fig. 13. According to Lemma 4.1, every solution of (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\) (resp., \(\beta =1\)) converges to \(({\check{r}}_1^{(0)}, {\check{r}}_2^{(0)})\) (resp., \(({\hat{r}}_1^{(0)}, {\hat{r}}_2^{(0)})\)) as \(t \rightarrow \infty \). By the monotone dynamics theory and the Kamke’s Theorem (Smith and Waltman 1995), the solution of system (4.1) lies between the solution of the lower-bound system and the solution of the upper-bound system, if these solutions all start from the same initial point. Therefore, \(\lbrace (r_1,r_2,\phi ): r_1, r_2 >0, \phi \in [0, \phi ^u)\rbrace \) is attracted to \([{\check{r}}_1^{(0)}, {\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)}, {\hat{r}}_2^{(0)}] \times [0, {\hat{\phi }}^{(0)}]\). In the following, we shall construct a sequence of finer lower-bound and upper-bound systems which bound the solutions of system (4.1) successively.

Nullclines and their intersections for system (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\) and \(\beta =1\), plotted on \((r_1, r_2)\)-plane, for the case \(c_1>c_2\)

Full size image

Note that

for \((r_1, r_2) \in [{\check{r}}_1^{(0)},{\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)},{\hat{r}}_2^{(0)}]\). We first consider the case

By analyzing the graph of function \(c_1 /\xi + c_2 \xi \) with \(\xi =r_1/r_2\), we have

see Fig. 14. Thus,

for all \((r_1, r_2) \in [{\check{r}}_1^{(0)},{\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)},{\hat{r}}_2^{(0)}]\), and \(\phi \) with \(\sin \phi \ge 0\). Let \(\phi ={\hat{\phi }}^{(1)}, {\check{\phi }}^{(1)}\in (0,{{\hat{\phi }}}^{(0)})\), which satisfy

respectively. It is clear that \(0<{\check{\phi }}^{(1)}<{\hat{\phi }}^{(1)}<{{\hat{\phi }}}^{(0)}\). It follows that

and thus all solutions of system (4.1) are attracted to \( [{\check{\phi }}^{(1)}, {\hat{\phi }}^{(1)}]\). Subsequently,

for \(\phi \in [{\check{\phi }}^{(1)}, {\hat{\phi }}^{(1)}]\). Let \(({\check{r}}_1^{(1)}, {\check{r}}_2^{(1)})\) and \(({\hat{r}}_1^{(1)}, {\hat{r}}_2^{(1)})\) be the constant solutions of system (4.17) with \(\beta =\cos {\hat{\phi }}^{(1)}\) and \(\beta =\cos {\check{\phi }}^{(1)}\), respectively. From the configurations of curves \(\Gamma _1^\beta , \Gamma _2^\beta \), which are summarized in Appendix I, we see that \({\check{r}}_i^{(0)}< {\check{r}}_i^{(1)}<{\hat{r}}_i^{(1)}<{\hat{r}}_i^{(0)}\), \(i=1,2\), and \({\check{r}}_1^{(1)}<{\check{r}}_2^{(1)}\), \({\hat{r}}_1^{(1)}<{\hat{r}}_2^{(1)}\), cf. Fig. 15. By applying Lemma 4.1 and the Kamke’s Theorem, we conclude that solutions starting from \([{\check{r}}_1^{(0)}, {\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)}, {\hat{r}}_2^{(0)}] \times [0, {\hat{\phi }}^{(0)}]\) are attracted to \([{\check{r}}_1^{(1)}, {\hat{r}}_1^{(1)}] \times [{\check{r}}_2^{(1)}, {\hat{r}}_2^{(1)}] \times [{\check{\phi }}^{(1)}, {\hat{\phi }}^{(1)}]\).

Relative positions of \({\check{r}}_1^{(0)}/{\hat{r}}_2^{(0)}\) and \({\hat{r}}_1^{(0)}/{\check{r}}_2^{(0)}\), in the graph of function \(c_1 /\xi + c_2 \xi \) with \(\xi =r_1/r_2\), under \(c_1>c_2\)

Full size image

Nullclines and their intersections for systems (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\), \(\beta =1\), \(\beta =\cos {\hat{\phi }}^{(1)}\), \(\beta =\cos {\check{\phi }}^{(1)}\), respectively, drawn on \((r_1, r_2)\)-plane, under \(c_1>c_2\)

Full size image

Continuing this process, we obtain sequences \(\{{\check{\phi }}^{(k)}\}\) and \(\{{\hat{\phi }}^{(k)}\}\) which satisfy

respectively. We thus obtain sequences \(\{({\check{r}}_1^{(k)},{\check{r}}_2^{(k)})\}\) and \(\{({\hat{r}}_1^{(k)},{\hat{r}}_2^{(k)})\}\), where \((r_1, r_2)= ({\check{r}}_1^{(k)},{\check{r}}_2^{(k)}), ({\hat{r}}_1^{(k)},{\hat{r}}_2^{(k)})\) satisfy

respectively. In addition, \({\check{\phi }}^{(k-1)}<{\check{\phi }}^{(k)}<{\hat{\phi }}^{(k)}<{\hat{\phi }}^{(k-1)}\), \({\check{r}}_i^{(k-1)}< {\check{r}}_i^{(k)}<{\hat{r}}_i^{(k)}<{\hat{r}}_i^{(k-1)}\), \(i=1,2\), and \({\check{r}}_1^{(k)}<{\check{r}}_2^{(k)}\), \({\hat{r}}_1^{(k)}<{\hat{r}}_2^{(k)}\), for all \(k \in {{\mathbb {N}}}\). Moreover, solutions starting from \([{\check{r}}_1^{(0)}, {\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)}, {\hat{r}}_2^{(0)}] \times [0, {\hat{\phi }}^{(0)}]\) are attracted to \([{\check{r}}_1^{(k)}, {\hat{r}}_1^{(k)}] \times [{\check{r}}_2^{(k)}, {\hat{r}}_2^{(k)}] \times [{\check{\phi }}^{(k)}, {\hat{\phi }}^{(k)}]\).

Recall that \(({\bar{r}}_{1, a}, {\bar{r}}_{2,a}, \phi _a^*)\) is a constant solution of system (4.1), with \(\phi ^*_a \in (0, \pi /2)\). From the above construction, we see that \(\phi ^*_a< {\hat{\phi }}^{(0)}\), as \(\phi ^*_a\) satisfies \(\omega _{\Delta } =(c_1{\bar{r}}_{2,a}/{\bar{r}}_{1, a} +c_2 {\bar{r}}_{1, a}/{\bar{r}}_{2,a}) \sin \phi ^*_a\), whereas \({\hat{\phi }}^{(0)}\) satisfies \(\omega _{\Delta } =2\sqrt{c_1c_2}\sin {\hat{\phi }}^{(0)}\). As \(({\check{r}}_1^{(0)}, {\check{r}}_2^{(0)})\) and \(({\hat{r}}_1^{(0)}, {\hat{r}}_2^{(0)})\) are obtained as constant solutions of (4.17) with \(\beta =\cos {\hat{\phi }}^{(0)}\) and \(\beta =1=\cos 0\) respectively, and \(0<\phi ^*_a<{\hat{\phi }}^{(0)}\), from the graphical analysis, we can see that \({\check{r}}_1^{(0)}<{\bar{r}}_{1, a}<{\hat{r}}_1^{(0)}, {\check{r}}_2^{(0)}<{\bar{r}}_{2, a}< {\hat{r}}_2^{(0)}\). From the above successive formulation, we can also confirm that \({\check{\phi }}^{(k)}<\phi ^*_a<{\hat{\phi }}^{(k)}\), \({\check{r}}_1^{(k)}<{\bar{r}}_{1, a}<{\hat{r}}_1^{(k)}, {\check{r}}_2^{(k)}<{\bar{r}}_{2, a}< {\hat{r}}_2^{(k)}\), for all \(k \in {{\mathbb {N}}}\).

Note that each of \(\{{\check{\phi }}^{(k)}\}\), \(\{{\hat{\phi }}^{(k)}\}\), \(\{{\check{r}}_i^{(k)}\}\), \(\{{\hat{r}}_i^{(k)}\}\), \(i=1,2\), is a convergent sequence, as they are bounded monotone sequences. We denote \({\hat{r}}_1^{(k)},\rightarrow {\hat{r}}_1^*\), \({\check{r}}_1^{(k)}\rightarrow {\check{r}}_1^*\), \({\hat{r}}_2^{(k)}\rightarrow {\hat{r}}_2^*\), \({\check{r}}_2^{(k)}\rightarrow {\check{r}}_2^*\), \({\hat{\phi }}^{(k)}\rightarrow {\hat{\phi }}^*\), and \({\check{\phi }}^{(k)}\rightarrow {\check{\phi }}^*\), as \(k\rightarrow \infty \). According to the way these sequences are constructed, \(({\hat{r}}_1^*,{\hat{r}}_2^*,{\check{\phi }}^*,{\check{r}}_1^*,{\check{r}}_2^*,{\hat{\phi }}^*)\) satisfies

All the considered solutions are attracted to \(\Lambda ^*:=[{\check{r}}_1^*, {\hat{r}}_1^*] \times [{\check{r}}_2^*, {\hat{r}}_2^*] \times [{\check{\phi }}^*, {\hat{\phi }}^*]\) which contains \(({\bar{r}}_{1, a}, {\bar{r}}_{2,a}, \phi _a^*)\). If \(\Lambda ^*\) does not degenerate into single point \(({\bar{r}}_{1, a}, {\bar{r}}_{2,a}, \phi _a^*)\), we consider the case \({\check{r}}_1^*<{\hat{r}}_1^*, {\check{r}}_2^*< {\hat{r}}_2^*, {\check{\phi }}^*< {\hat{\phi }}^*\) so that \(\Lambda ^*\) is a rectangular box and none of its vertices is an equilibrium of system (4.1). Note that for \((r_1, r_2) \in [{\check{r}}_1^*, {\hat{r}}_1^*] \times [{\check{r}}_2^*, {\hat{r}}_2^*] {\setminus } \{({\check{r}}_1^*, {\hat{r}}_2^*)\}\),

The term in (A.16) is zero at \(\phi = {\check{\phi }}^*\). In addition, we have \({\dot{\phi }} \ge 0\) at \(({\check{r}}_1^*, {\hat{r}}_2^*, {\check{\phi }}^*)\). Similarly, we can see, for \((r_1, r_2) \in [{\check{r}}_1^*, {\hat{r}}_1^*] \times [{\check{r}}_2^*, {\hat{r}}_2^*] {\setminus } \{({\hat{r}}_1^*, {\check{r}}_2^*)\}\),

which is zero at \(\phi = {\hat{\phi }}^*\) and \({\dot{\phi }} \le 0\) at \(({\hat{r}}_1^*, {\check{r}}_2^*, {\hat{\phi }}^*)\). As \(({\check{r}}_1^*, {\hat{r}}_2^*, {\check{\phi }}^*)\) and \(({\hat{r}}_1^*, {\check{r}}_2^*, {\hat{\phi }}^*)\) are not equilibria of system (4.1), the \(\phi \)-components of the solutions enter into the interior of \([{\check{\phi }}^*, {\hat{\phi }}^*]\). If so, then one can construct a rectangle smaller than \([{\check{r}}_1^*, {\hat{r}}_1^*] \times [{\check{r}}_2^*, {\hat{r}}_2^*]\) which attracts the \((r_1, r_2)\)-components of the solutions of system (4.1), by the above process. Set \({\bar{\Lambda }} \subset \Lambda ^*\) as the minimal rectangular box in \((r_1, r_2, \phi )\)-space, which attracts all the considered solutions of system (4.1). Then we argue that \({\bar{\Lambda }}=\{({\bar{r}}_1, {\bar{r}}_2, \phi ^*_a)\}\). If not, we can construct smaller rectangular boxes by the above process to yield a contradiction to the minimality of \({\bar{\Lambda }}\). We thus conclude the convergence to \(({\bar{r}}_{1, a}, {\bar{r}}_{2,a}, \phi _a^*)\) as \(t \rightarrow \infty \).

The situation that (A.13) does not hold, i.e., \({\hat{r}}_1^{(0)}/{\check{r}}_2^{(0)} \ge \sqrt{c_1/c_2}\), is apt to occur when \(c_1\) is close to \(c_2\), since \({\hat{r}}_1^{(0)}/{\check{r}}_2^{(0)} >1\). The arguments indeed share a similarity with the symmetric-coupling case. Recall that the contracting arguments go only one-sided in the symmetric-coupling case in Theorem 3.4. Herein, the contraction also proceeds one-sided in the first few steps. More precisely, in (A.14), the lower bound of \(c_1r_2/r_1+c_2r_1/r_2\) is still \(2\sqrt{c_1c_2}\), and only the right inequality holds. This is due to

which follows from

and \({\hat{r}}_1^{(0)}{\hat{r}}_2^{(0)}-{\check{r}}_1^{(0)}{\check{r}}_2^{(0)}>0\), \(c_1{\hat{r}}_2^{(0)}{\check{r}}_2^{(0)}-c_2{\hat{r}}_1^{(0)}{\check{r}}_1^{(0)}>0\), as \(c_1>c_2\), \({\hat{r}}_i^{(0)}>{\check{r}}_i^{(0)}\), \(i=1,2\), and \({\check{r}}_2^{(0)}>{\check{r}}_1^{(0)}\), see Fig. 13. We note that \({\hat{r}}_1^{(0)}={\hat{r}}_2^{(0)}=\sqrt{\mu }\). Subsequently, only the first inequality in (A.15) holds. We can then obtain \({\check{\phi }}^{(1)}< {\hat{\phi }}^{(0)}\). In turn, \({\hat{r}}_1^{(1)}\) and \({\hat{r}}_2^{(1)}\) can be defined, and \({\hat{r}}_i^{(1)}<{\hat{r}}_i^{(0)}, i=1, 2\). By graphical analysis, we also see that \({\hat{r}}_1^{(1)} < {\hat{r}}_2^{(1)}\). If \({\hat{r}}_1^{(1)}/{\check{r}}_2^{(0)} \) is still greater than \( \sqrt{c_1/c_2}\), we continue to construct \({\check{\phi }}^{(2)}\) and then \({\hat{r}}_1^{(2)}\) and \({\hat{r}}_2^{(2)}\). We thus obtain an increasing sequence \(\{{\check{\phi }}^{(k)}\}\) and decreasing sequences \(\{{\hat{r}}_i^{(k)} \}\), \(i=1,2\). In addition,

continues to hold. We continue this process until there is an \(k \in {\mathbb {N}}\) such that \({\hat{r}}_1^{(k)}/{\check{r}}_2^{(0)} < \sqrt{c_1/c_2}\). Thereafter, the contracting construction can go two-sided. We thus obtain increasing sequences \(\{{\check{\phi }}^{(k)}\}\), \(\{{\check{r}}_i^{(k)} \}\), \(i=1,2\), and decreasing sequences \(\{{\hat{\phi }}^{(k)}\}\), \(\{{\hat{r}}_i^{(k)} \}\), \(i=1,2\). The convergence to \(({\bar{r}}_{1, a}, {\bar{r}}_{2,a}, \phi _a^*)\) as \(t \rightarrow \infty \) then follows the same arguments as above. This completes the proof. \(\square \)

Remark: The above process for constructing finer attracting sets can also be argued as follows: Consider an initial point in \([{\check{r}}_1^{(0)}, {\hat{r}}_1^{(0)}] \times [{\check{r}}_2^{(0)}, {\hat{r}}_2^{(0)}] \times [0, {\hat{\phi }}^{(0)}]\). For a small \(\varepsilon >0\), there exists a \(T_1>0\) such that the solution lies in \(({\check{r}}_1^{(1)}-\varepsilon , {\hat{r}}_1^{(1)}+\varepsilon ) \times ({\check{r}}_2^{(1)}-\varepsilon , {\hat{r}}_2^{(1)}+\varepsilon ) \times ({\check{\phi }}^{(1)}-\varepsilon , {\hat{\phi }}^{(1)}+\varepsilon )\), for all \(t >T_1\). Successively, using these estimates, we then argue that the solution lies in \(({\check{r}}_1^{(k)}-\varepsilon , {\hat{r}}_1^{(k)}+\varepsilon ) \times ({\check{r}}_2^{(k)}-\varepsilon , {\hat{r}}_2^{(k)}+\varepsilon ) \times ({\check{\phi }}^{(k)}-\varepsilon , {\hat{\phi }}^{(k)}+\varepsilon )\), for all \(t >T_k\), for some \(T_k >T_{k-1}\).

Appendix I

The configurations of \(\Gamma ^{\beta }_{1}, \Gamma ^{\beta }_{2}\):

Recall the nullclines of system (A.11):

\(\Gamma ^{\beta }_{1}\) is concave upward and \(\Gamma ^{\beta }_{2}\) is concave downward in \({{\mathbb {R}}}^2_+\). \(\Gamma ^{\beta }_{1}\) is strictly increasing if \(\mu \le c_1\). If \(\mu >c_1\), \(\Gamma ^{\beta }_{1}\) has a local minimum at \(\rho _1=\sqrt{(\mu -c_1)/3}\) and is strictly increasing for \(\rho _1 \in [\sqrt{(\mu -c_1)/3}, \infty )\). \(\Gamma ^{\beta }_{2}\) is strictly increasing for \(\rho _1 \in [0, \infty )\). The slope of \(\Gamma ^{\beta }_{1}\) tends to infinity as \(\rho _1 \rightarrow \infty \), whereas the slope of \(\Gamma ^{\beta }_{2}\) tends to 0 as \(\rho _1 \rightarrow \infty \). When \(\mu \le c_1\), it is obvious that \(\Gamma ^{\beta _1}_{1}>\Gamma ^{\beta _2}_{1}\) if \(\beta _1<\beta _2\). When \(\mu > c_1\), \(\Gamma ^{\beta _1}_{1}<\Gamma ^{\beta _2}_{1}\) for \(\rho _1\in (0,\sqrt{\mu -c_1})\) and \(\Gamma ^{\beta _1}_{1}>\Gamma ^{\beta _2}_{1}\) for \(\rho _1\in (\sqrt{\mu -c_1}, \infty )\), if \(\beta _1<\beta _2\). Thus, if \(\beta _1<\beta _2\), and \((\rho _1, \rho _2^a) \in \Gamma ^{\beta _1}_{1}\), \((\rho _1, \rho _2^b) \in \Gamma ^{\beta _2}_{1}\), and \((\rho _1, \rho _2^a) \) and \((\rho _1, \rho _2^b)\) lie on the part \(\Gamma ^{\beta _1}_{1}>\Gamma ^{\beta _2}_{1}\), we certainly have \(\rho _2^a > \rho _2^b\). On the other hand, \(\Gamma ^{\beta _1}_{2}<\Gamma ^{\beta _2}_{2}\) if \(\beta _1<\beta _2\).

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