$\mathbf{x}(t)=(x(t), y(t))$

$$x(t+T)=x(t), \quad y(t+T)=y(t) \quad \text { for all } t$$

To give an example to demonstrate the bifurcation of two limit cycles, we take $\rho=1.397661 \cdots$, which gives $A=\bar{A}=0.208536 \cdots, C=0.5$, and $B=B_{h}=$ $0.132553 \cdots$, resulting in $v_{0}=v_{1}=0$ and $v_{2}=-0.000222 \cdots<0$, as expected. Then, we perturb the parameters $A$ and $B$ such that $v_{1}>0, v_{0}<0$ and $\left|v_{0}\right| \ll v_{1} \ll$ $\left|v_{2}\right|$, and so two limit cycles can be obtained. More precisely, let $A=0.208536+\varepsilon_{1}$ and $B=B_{h}+\varepsilon_{2}$, where $\varepsilon_{1}=10^{-3}$ and $\varepsilon_{2}=-7 \times 10^{-6}$, which yield $A=$ $0.209536, \quad B=0.132444$ and
$$v_{0} \mu=-0.731036 \times 10^{-6}, \quad v_{1}=0.293971 \times 10^{-4}, \quad v_{2}=-0.229863 \times 10^{-3}$$
Note that $\varepsilon_{2}<0$ again implies that $B$ is decreasing to pass the Hopf critical point $\boldsymbol{B}{\boldsymbol{h}}$ since $v{0} \approx-0.672163<0$. Thus, the truncated normal form equation $v_{0} \mu+v_{1} r^{2}+$ $v_{2} r^{4}=0$ has two real roots: $r_{1} \approx 0.1839$ and $r_{2} \approx 0.3067$, which approximate the amplitudes of the two limit cycles. Since $v_{2}<0$, the outer limit cycle is stable while the inner one is unstable, and the equilibrium solution at this critical point is a stable focus.
The simulation, shown in Fig. 5, takes the exact parameter values:
$$A=0.2095365226, \quad C=0.5, \quad D=B=0.1324446775 \text {. }$$
The simulated phase portrait is shown in Fig. 5a where the stable (the larger one) and unstable (the smaller one) limit cycles are denoted by the red and blue curves, respectively. Analytic predictions based on the normal form are also shown in Fig. 5a as the green curves. It indeed indicates a good agreement between the simulations and the analytic predictions. Figure $5 \mathrm{~b}$ depicts the time history of the stable (outer) limit cycle. Note that the simulation for the unstable limit cycle (or the unstable periodic motion) is obtained by using a negative time step in a fourth-order Runge-Kutta integration scheme.

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