Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation

· · · between the two pairs of similar canards. In contrast, the correlation coefﬁcients of the dissimilar canards differ from unity in proportion to the difference between the given bifurcation parameter value and the parameter values at the saddle-node bifurcation points. Approximately, they take values from 0.998 to 0.975. These contrasts are noteworthy. Similar bifurcation phenomena were observed in the 1 / 2-subharmonic entrainment region. We hypothesize that the two pairs of stable and saddle canards are invariant with respect to a slight shift of time at the saddle-node bifurcation points, and we numerically prove that such a property approximately holds at the bifurcation points.


Introduction
Canard phenomena are one of the major discoveries in the field of nonlinear dynamics in the late 20th century and have been extensively studied during the past three decades [1][2][3][4][5][6][7][8][9][10]. Canards can be generated in simple singularly perturbed second-order autonomous differential equations that include a small parameter ε. Let us consider the van der Pol equation in the following form: where ε is assumed to be small. An intersection of the x-nullcline y = −x(1 − x 2 ) and the y-nullcline x = B 0 is an equilibrium; if |B 0 | < 1/ √ 3, it is unstable, and if |B 0 | > 1/ √ 3, it becomes stable. A supercritical Andronov-Hopf bifurcation (AHB) occurs at B 0 = ±1/ √ 3. A remarkable phenomenon is observed when B 0 is chosen to be slightly under 1/ √ 3 (or slightly over −1/ √ 3). The amplitude of the oscillation is extremely sensitive to B 0 . Nonstandard analysis [5] has shown that the amplitude of the oscillation is changed on the order of 1 if |B 0 | decreases on the order of exp(−1/ε). At ε = 0.1 and 0.01, the magnitude of exp(−1/ε) approximates 5 × 10 −5 and 4 × 10 −44 , respectively. As the shape of the oscillation in the phase plane resembles a duck (French canard), it was named a "canard" [5].
We now ask how canards in dynamical systems are influenced by extremely weak periodic perturbations of amplitude on the order of exp(−1/ε). Sekikawa et al. proposed the following nonautonomous dynamics derived from a forced van der Pol oscillator [10]: They discovered that a second entrainment region surrounded by saddle-node bifurcations and containing two coexisting canards is generated inside the fundamental harmonic entrainment region.
Here, a canard is defined as a closed orbit following both attracting and repelling slow manifolds. Such bifurcation structures are observed in several subharmonic and fractional harmonic entrainment regions in the van der Pol equation under such weak periodic perturbation [10].
In this study, we investigate the characteristics of these coexisting canards. The saddle-node bifurcations generate a saddle solution as well as a stable solution. According to our numerical results, we find that the saddle solutions generated by each saddle-node bifurcation are also canard-shaped in the phase plane. The first pair of stable and saddle canards is born from a saddle-node bifurcation that forms the fundamental harmonic entrainment region, while the second pair originates from the second saddle-node bifurcation inside the fundamental harmonic entrainment region. Such nested entrainment regions caused by sequential saddle-node bifurcations represent a newly discovered bifurcation structure in nonautonomous oscillators that we have identified for the first time. In this structure, noteworthy phenomena are observed slightly after the generation of the second saddle-node bifurcation. We found that the stable canard arising from the first saddle-node bifurcation resembles the saddle canard from the second saddle-node bifurcation quite closely, although they differ significantly in phase. Similarly, the saddle canard born of the first saddle-node bifurcation closely resembles the stable canard arising from the second saddle-node bifurcation.
We calculate the correlation coefficients of each y(τ + θ 0 ), where θ 0 is chosen such that y(θ 0 ) = 0 andẏ(θ 0 ) > 0. When B is chosen on the order of exp(−1/ε), the correlation coefficient of the stable canard of the first saddle-node bifurcation and the saddle canard of the second saddle-node bifurcation is 0.9999 · · · if the bifurcation parameter is not close to the saddle-node bifurcation points that generate the dissimilar stable and saddle canards. Similarly, the correlation coefficients between the stable canard arising from the second saddle-node bifurcation and the saddle canard from the first saddle-node bifurcation are consistently 0.9999 · · · . Because they resemble each other quite well, it is practically impossible to differentiate these stable and saddle canards arising from closed orbits projected onto the phase plane. In contrast, according to our numerical results, the correlation coefficients of the dissimilar canards decrease monotonically from unity in proportion to the difference between the given bifurcation parameter value and the parameter values of the saddlenode bifurcation points. We hypothesized that similar stable and saddle canards are generated as a result of a saddle-node bifurcation for which the Jacobian matrix has an eigenvector of 1.0 and 2/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/1/013A02/4794774 by guest on 27 July 2018 attempted to verify this conjecture. Although the hypothesis approximately holds according to our numerical results, we still could not distinguish the saddle-node bifurcations that generate similar stable or dissimilar stable and saddle canards.
We also verified that a similar bifurcation structure can be observed in the 1/2-subharmonic entrainment region. Figure 1 shows a circuit diagram of a forced van der Pol oscillator. In the figure, C and L denote capacitance and inductance, respectively, E 1 is a DC power supply, and E 2 sin(ω t) is an AC power supply. The term N.C. represents a nonlinear conductance. We assume that the v − i characteristic of the nonlinear conductance i n (v) can be represented by the following third-order polynomial function:

Van der Pol oscillator under weak periodic perturbation and canards
From Kirchhoff's law, the governing equation of the circuit is represented by the following second order nonautonomous ordinary differential equations: We assume that C is small and E 2 is extremely small. Rescaling then produces and the normalized equation represented by Eq. (2). Throughout the discussion, we set ε = 0.1 and B is on the order of exp(−1/ε). Figure 2 shows stable solutions in the absence of perturbation (B = 0). The significant change of the amplitude under slight variation of B 0 is called a "canard explosion".
In our previous works [9,10], we discovered that two canard-shaped attractors coexist in some synchronization regions under such small periodic perturbations. These pairs are born as a consequence of two successive saddle-node bifurcations and, because saddle-node bifurcations generate  a pair of stable and saddle solutions, the saddle solutions must exist. In this section, B and ω are chosen inside the second saddle-node bifurcation curves [10] generated in the fundamental harmonic entrainment region. According to our numerical results, the two saddle solutions in this entrainment region are also canard-shaped in the phase plane. Hereafter, we call the stable and saddle canards arising from the first saddle-node bifurcation stable canard 1 and saddle canard 1, respectively. Similarly, we call the stable and saddle canard arising from the second saddle-node bifurcation stable canard 2 and saddle canard 2, respectively. The two stable solutions and two saddle solutions are illustrated in Fig. 3 as an example. Successive saddle-node bifurcations and their entrainment regions are discussed in detail in the next section. Figure 4 presents the waveforms of the four canards. We see that the waveform of stable canard 1 resembles that of saddle canard 2, although the phases of these two canards are significantly different. Similarly, the waveform of stable canard 2 resembles that of saddle canard 1. Actually, when we superpose Fig. 3(a) (resp. Fig. 3(c)) and Fig. 3(d) (resp. Fig. 3(b)), the two solutions almost overlap. Figure 5 shows a superposed view of the four canards presented in Figs. 3(a), (b), (c), and (d); we cannot distinguish the difference between stable canard 1 and saddle canard 2 (or between stable canard 2 and saddle canard 1) as long as we observe the solutions in the phase plane.   define the stroboscopic Poincaré map as follows:

Analysis of the significant resemblance between the two pairs of stable and saddle canards in the fundamental harmonic entrainment region
where ϕ(τ , u 0 , λ) is the solution, u 0 ≡ (x 0 , y 0 ) is an initial point at τ = 0, and λ is a bifurcation parameter.  Let u 0 be an m-periodic point of T λ . Then, u 0 satisfies The characteristic equation is represented as follows: Let μ 1 and μ 2 (|μ 1 | ≥ |μ 2 |) be the solutions of μ. If 0 < |μ 2 | < |μ 1 | < 1, the m-periodic point is said to be a stable node. If |μ 1 | > 1 > |μ 2 |, the m-periodic point is said to be a saddle.
If |μ 1 | > |μ 2 | > 1, the m-periodic point is said to be completely unstable. When |μ 1 | = 1, a bifurcation occurs. We analyze the case of μ 1 = 1, in which a saddle-node bifurcation set is produced. To draw the bifurcation diagram, we adopt the shooting algorithm in Ref. [11]. Figure 6 shows a bifurcation diagram of the fundamental harmonic entrainment region surrounded by the two saddle-node bifurcation curves G 1 (μ 1 = 1). In this fundamental harmonic entrainment region, we find another entrainment region surrounded by another set of saddle-node bifurcation curves, which are denoted by G 2 [10]. To the best of our knowledge, the nested entrainment regions caused by the two saddle-node bifurcations G 1 and G 2 have not yet been observed in a forced oscillator. The four canards discussed in Figs   To investigate how the two pairs of stable and saddle canards resemble each other, we vary ω to observe the two pairs of stale and saddle canards projected onto the x-y phase plane.
We can analyze the nested bifurcation structures caused by the successive saddle-node bifurcations by tracing the bifurcation diagram of the fundamental harmonic entrainment region shown in  Figure 9(a) shows stable canard 1 and saddle canard 1 at A 1 immediately after the first saddle-node bifurcation point in Fig. 7. As A 1 is close to the curve G 1 , these two canards nearly overlap in the phase plane. Because A 2 is significantly separated from G 1 in Fig. 9(b), we can detect the difference between stable canard 1 and saddle canard 1. Note that the fixed point of stable canard 1 on the Poincaré section (•) moves to some extent, while that of saddle canard 1 (•) moves little as ω is separated from G 1 .
At the bifurcation point Q 2 in Fig. 7, the second saddle-node bifurcation occurs and another pair of stable canard 2 and saddle canard 2 is born, as illustrated in Fig. 9(c). The solid blue circle and the light green open circle denote the fixed points of stable canard 2 and the saddle canard 2 on the Poincaré section, respectively. In contrast to the former case, the light green open circle corresponding to saddle canard 2 begins to move significantly, while the blue solid circle corresponding to stable canard 2 moves only slightly as ω increases. This discrepancy is noteworthy.
Note that a qualitative change emerges slightly after the second saddle-node bifurcation. As shown in Figs. 9(d)-(i), following generation of the second saddle-node bifurcation at Q 2 , a significant resemblance is observed between stable canard 1 and saddle canard 2 and between stable canard 2 and saddle canard 1. The fixed points of stable canard 1 (red) and saddle canard 2 (light green) are considerably separated, i.e., the phases of these two canard solutions are different, as shown in Fig. 4.
By increasing ω further, stable canard 1 and saddle canard 2 disappear through the saddle-node bifurcation at Q 3 ; similarly, stable canard 2 and saddle canard 1 disappear through the saddle-node bifurcation at Q 4 .
To evaluate the similarities of the canard solutions, the correlation coefficients are defined as follows: Y. Nagata et al. Fig. 9. Four canard solutions: stable canard 1 (red), stable canard 2 (blue), saddle canard 1 (purple), and saddle canard 2 (light green) generated in the fundamental harmonic entrainment region for parameters A 1 -A 10 in Fig. 7. In (a) and (b), only stable canard 1 and saddle canard 1 are shown owing to the bifurcation structure shown in Fig. 7  where y and y are the y-components of two of the four canard solutions in Eq. (2), y (y ) is the mean value of y (y ), and y i = y( 2π i 2 20 ω ). To calculate the coefficients of resemblance, phase θ is selected such that is satisfied. Figure 10(a) shows a graph of the correlation coefficients that are obtained along L 1 (B = 5×10 −5 ) and Fig. 10(b) shows a magnified view of Fig. 10 Fig. 7. At these points, the correlation coefficients between stable canard 1 and saddle canard 1, stable canard 2 and saddle canard 2, and saddle canard 1 and stable canard 2 are 1, because each stable canard coincides with the corresponding saddle canard. As seen in these figures, the correlation coefficients between stable canard 1 and saddle canard 2 and saddle canard 1 and stable canard 2 are close to 1; however, the given parameter value ω is very close to Q 1 and Q 2 , which are saddle-node bifurcation points that generate dissimilar canards. In contrast, the correlation coefficients between stable canard 1 and saddle canard 1 and stable canard 2 and the saddle canard 2 decrease proportionally as the bifurcation parameter ω varies from the saddle-node bifurcation points Q 1 and Q 2 . Table 1 shows the calculated correlation coefficients for A 4 in Fig. 7, which is located at a slight distance of 0.001 from the saddle-node bifurcation points Q 1 and Q 2 . These results indicate that the close respective correlations of stable canards 1 and 2 and saddle canards 2 and 1 are maintained over a wide range of parameters. If we trace the fixed-point manifold from right to left in Fig. 7, an attractor and saddle solution with a similar shape but with a nontrivial time shift are born at Q 4 , whereas a second similar saddle-node bifurcation occurs at Q 3 , i.e., Q 3 and Q 4 are saddle-node bifurcation points that generate similar canards.
PTEP 2018, 013A02 Y. Nagata et al.    Figure 11 shows a bifurcation diagram that is obtained by tracing the fixed point of T 2 λ with the shooting algorithm developed in Ref. [11]. Figure 12 shows an associated schematic illustration of the bifurcation diagram of the 1/2-subharmonic entrainment region. A bifurcation structure similar to that of the fundamental harmonic entrainment region is observed in this entrainment region, i.e., two entrainment regions (one between the two G 1 and the other between the two G 2 ) exist, and two pairs of stable and saddle canards are present inside the second entrainment region between the two G 2 . Figure 13 shows periodic-point manifolds of T λ obtained numerically.
In a similar manner, we call the stable and saddle canards arising from the first saddle-node bifurcation denoted by Q 1 stable canard 1 and saddle canard 1 , respectively. Similarly, we call the stable and saddle canards arising from the second saddle-node bifurcation denoted by Q 2 stable canard 2 and saddle canard 2 . Figure 14  Y. Nagata et al. . 15. Four canard solutions: stable canard 1 (red), stable canard 2 (blue), saddle canard 1 (purple), saddle canard 2 (light green) generated in the 1/2-subharmonic entrainment region for B1-B10 in Fig. 12. In (a) and (b), only stable canard 1 and saddle canard 1 are shown owing to the bifurcation structure shown in Fig. 12

Discussion
Finally, we discuss the mechanism underlying the above-described formation of two pairs of similarly shaped stable and saddle canards. We hypothesize that the saddle-node bifurcations denoted by Q 3 and Q 4 occur approximately with the following properties at the bifurcation points: where DT λ (x Q , y Q ) is a Jacobian matrix of T λ on (x Q , y Q ) at the saddle-node bifurcation point. If Eq. (11) approximately holds, the stable and saddle canards arising from the saddle-node bifurcation are nearly invariant for a slight local shift of time. Table 2 shows data obtained at the four saddle-node bifurcation points. (x Q , y Q ) indicates the coordinates of the fixed point of T λ at the saddle-node bifurcation points. (V x , V y ) represents the eigenvector of DT λ at (x Q , y Q ), and p and q representẏ/ẋ and Vy/Vx, respectively. As expected, p is close to q, and |p − q|/q assumes very small values at the saddle-node bifurcation points Q 3 and Q 4 . However, |p − q|/q is also small at the saddle-node bifurcation points Q 1 and Q 2 . Thus, the resemblance of the stable and saddle canards generated at the saddle-node bifurcation points Q 3 and  Q 4 can be locally explained to some extent. However, we cannot not distinguish Q 1 and Q 2 , which generate dissimilar canards, from Q 3 and Q 4 , which generate similar canards.

Conclusion
We investigated the resemblances of two pairs of stable and saddle canards in a van der Pol oscillator with extremely weak periodic perturbation. We found that there are two pairs of stable and saddle canards for parameter values in the region surrounded by the second saddle-node bifurcation curves. More explicitly, we clarified that stable canard 1 resembles saddle canard 2 and stable canard 2 resembles saddle canard 1 quite well, although their phases are significantly different, whereas stable canard 1 (stable canard 2) does not resemble saddle canard 1 (saddle canard 2) as closely. In this case, stable canard 1 and saddle canard 1 originated from the first saddle-node bifurcation, while stable canard 2 and saddle canard 2 originated from the second saddle-node bifurcation. We confirmed this resemblance both for the fundamental harmonic entrainment region and the 1/2subharmonic entrainment region. We investigated the local bifurcation structure at the saddle-node bifurcations, and demonstrated the mechanism causing the resemblance of the two pairs of canards.