Infinite number of Wada basins in a megastable nonlinear oscillator

Previous results show that some oscillators possess finite number of Wada basins. Here we find that a nonlinear oscillator can possess a countable infinity of Wada basins and these Wada basins are connected. Infinite number of coexisting attractors and their Wada basins are investigated by the basin cell theorem and generalized basin cell theorem. Infinite number of Wada basins are systematic, which identical basins structure can be identified in each periodic X-axis coordinate interval. This type of Wada basin boundary can lead to a high level of indeterminacy and an extreme sensitive dependence on initial condition.


Introduction
Nonlinear oscillators can evolve in time towards some asymptotic attractors depending on the initial conditions and on system parameters. A basin of attraction is defined as the set of points which lead the system to a specific attractor [1]. A nonlinear oscillator may possess multiple coexisting attractors with own basins of attraction. When there are different coexisting attractors in a given region, different basins of attraction can be separated by a basin boundary. This basin boundary can be smooth or fractal [2,3]. In Hamiltonian systems, no attractors exist and escape basins can be defined in an analogous way. An escape basin is the set of initial conditions that escapes through a certain exit [4,5]. The study of basins of attraction and escape basins can provide much information about the dynamical systems including dynamical evolutions and global analysis.
A basin is called a Wada basin if every open neighborhood of the point on the basin boundary contains at least three basins in a plane [6]. Wada boundaries are usually referred to as those that separate three or more basins at a time in the plane. Wada basins include connected Wada basins and disconnected Wada basins. For disconnected Wada basins, our primary intuition makes us imagine that it is may be possible [7][8][9]. For connected Wada basins, the Wada basin boundaries have intriguing property by a continuous boundary, which is indeed found in dissipative systems [10][11][12][13][14]. Since the first report by Kennedy and Yorke [6], Wada basins have been found in different types of systems, such as discrete maps [10,11], nonlinear oscillators [12][13][14][15], ecological models [16], delayed systems [17], binary black hole systems [18], controlled systems [19,20], mechanical systems [21,22], plasma physics [23] etc. The most interesting feature of Wada basins is a special kind of unpredictability and uncertainty because an arbitrarily J. Wang Á Y. Zhang (&) School of Mathematics and Statistics, Qingdao University, Qingdao 266071, People's Republic of China e-mail: zyx9701@126.com small perturbation of initial conditions on a Wada basin boundary can drive the system to any of the possible attractors.
Over the past three decades, many scholars have developed diverse methods (or algorithms) to detect Wada property. Kennedy and Yorke method need us to find the density of the stable manifold of the accessible saddle, however it is difficult to achieve in practice [6]. Nusse and Yorke developed a more useful method by means of the basin cell theory. It verifies the Wada basins by constructing the basin cell and plotting the unstable manifolds of a saddle which is located on the basin boundary [10]. Very recently, Zhang developed the Nusse-Yorke method by the generalized basin cell and the computation of the unstable manifolds of several saddles [24]. Daza et al. proposed the Grid method [25] and the Merging method [26] to detect the Wada property. The Grid method has high degree of accuracy and can verify the type of Wada basin (connected Wada basins, disconnected Wada basins and partially Wada basins [11]). The Merging method is a quick method, but the degree of accuracy is not very high because the result is reliable up to the resolution of the basin. Ziaukas developed the alternative grid method, which makes Grid method faster [27]. Saunoriene et al. proposed a Wada index, which further optimized the alternative grid method [28]. Wagemakers et al. proposed a Saddle-straddle method, and it does not require the basins of attraction and does rely on the dynamics of the system [29]. Wagemakers et al. commented on these computational methods detecting Wada basins [30]. Daza et al. proposed a basin entropy tool to analyze the uncertainty on different Wada basin boundaries [31] and this method are also used to test for fractal boundaries [32] and explosion of basin boundary [33].
So far, finite number of connected Wada basins can coexist in dynamical systems. However, whether infinite number of connected Wada basins can exist is an open question. If the connected Wada basins can exist, their mutual organization law is still unclear. Another motivation of this study comes from the study of infinitely many coexisting attractors or extreme multistability, which is an intriguing phenomenon [34][35][36]. Coexistence of infinitely many attractors has been a fundamental study since 1974 through the works of Newhouse [37]. He predicted the coexistence of infinitely many attractors in a two-dimensional diffeomorphism. More than twenty years later, Chawanya reported the coexistence of infinitely many periodic and chaotic attractors in game dynamical systems with networks of heteroclinic orbits [38,39]. Recently, homoclinic tangencies were found responsible for the coexistence of infinitely many stable periodic solutions in two-dimensional discrete maps [40,41]. The extension of similar concepts can also be found in continuous dynamical systems in the past years [42][43][44][45][46][47]. This paper provides an illustration of such continuous systems via the technique of basins of attractions. We used basin cell method and generalized basin cell method to verify the existence of Wada basins for countable infinitely many coexisting attractors. We verify the coexistence of infinite number of basin cells and generalized basin cells in a megastable oscillator. We also found the regularity on Wada basins in the periodic X-axis coordinate interval. Wada basin boundaries and their organization laws are also investigated by the manifold method. This shows that the phenomenon of infinite coexistence is a mathematically plausible scenario. When infinitely many coexisting attractors possess Wada basins, it is not advantageous in real-world applications as the system can switch its states in a tiny perturbation to the initial conditions and hence, the system can develop faults. The advantageous fact of this infinite coexistence is that it can be controlled using techniques from control theory [48] which can prevent fault in real-world systems.
The rest of this paper is organized as follows. In Sect. 2, we give the mathematical model of the megastable oscillator and theoretical methods on Wada basins. Four Theorems on infinitely many Wada basins are presented. In Sect. 3, we discuss infinitely many Wada basins and their organizations by two examples. Infinite number of basin cells and generalized basin cells are found. In Sect. 4, we summarize our main conclusions.

Mathematical model and theoretical methods
Consider a megastable oscillator, which is thoroughly investigated in [49], Here, we discuss the following megastable system: where a,b are two parameters. system (2) has an infinite number of equilibrium points located at (kp=a,0), where k is any integer number. Adding a periodic forcing function to system (2), we obtain system (3): In the oscillator, a suitable Poincaré section is called a stroboscopic section and Poincaré's trick allows the ordered discrete values (ðxðt k Þ; yðt k ÞÞ, t k ¼ 2kp x . We simply write ðx k ; y k Þ as a discrete time mapping, The mapping is also time-2p=x map and F is a twodimensional diffeomorphism. Let p be a saddle with stable manifold W s ðpÞ and unstable manifold W u ðpÞ. Let q be another saddle with stable manifold W s ðqÞ and the unstable manifold W u ðqÞ. In order to better understand the basin boundary, we give the following two Lemmas [50][51][52]. For limited number of coexisting attractors, their Wada basins can be verified by two main theorems [10,24]. The first theorem is based on a theorem by Nusse-Yorke [10]. If a periodic saddle generated a basin cell and the saddle's unstable manifold passes through at least three basins, then the basin is a Wada basin. The second theorem popularized the Nusse-Yorke method by Zhang [24]. If some periodic saddles generated a generalized basin cell and the saddles' unstable manifold passes through at least three basins, then the basin is a Wada basin. Here, these two theorems can be generalized to countable infinitely many basins. In this case, it is very difficult to verify the numerically verifiable conditions of infinitely many basins. When infinite number of coexisting attractors have the same shape but with different positions in phase space, extreme mulitstability is called ''homogeneous'', otherwise it is called ''heterogeneous'' (with different shape) [34,53]. The same class of attractors can be called homogeneous attractors in contrast to heterogeneous attractors. For countable infinitely many homogeneous attractors and heterogeneous attractors, the following theorems can be used to verify the Wada basins. It is assumed that a nonlinear system has countable infinitely many coexisting attractors A i (i 2 N,N are natural numbers). Infinite number of basins of attraction are denoted by B i ði 2 NÞ. The saddle p on the basin A boundary is accessible from basin A. The saddle q on the basin B boundary is accessible from basin B. Similarly, the same class of saddles can be called homogeneous saddles in contrast to heterogeneous saddles. If a basin cell (a generalized basin cell), has the same shape but different location, the basin cell (generalized basin cell) is also called homogeneous. Heterogeneous basin cells (generalized basin cells) have different shapes. It is noted that heterogeneous basin cells (generalized basin cells) may be topologically equivalent. For the coexistence of infinitely many homogeneous and heterogeneous attractors, the following theorems will be used to verify the Wada basins for coexistence of a countable infinity of attractors.
Theorem 1 Let P k (k 2 N) be countable infinitely many hyperbolic homogeneous saddles and every P k can generate a homogeneous basin cell C k (k 2 N). Let B k (k 2 N) be countable infinitely many disjoint basins, where B k is the basin of C k . If there exists a saddle P 1 and the unstable manifold W u ðP 1 Þ intersects at least three basins, then oB 1 is a Wada basin boundary w. r. t. B k (k 2 N).

Remark 1
The proof idea of Theorem 1 is the same as that of theorem 2 [10] for a basin B 1 . The numerically verifiable conditions are easy in Theorem 1 because of the homogeneous properties of coexistence attractors and basin cells. Nusse and Yorke point out that a basin cell determines both the structure of its basin and the global structure of the corresponding basin boundary [51]. It has been proved that the basin boundary oB 1 is equal to the closure of the stable manifold of the saddle P 1 . The stable manifold of the saddle P 1 is divided into two parts: the boundary of a basin cell C 1 and diverging curves. Although every homogeneous basin cell is in a different phase plane position, the limit set of each diverging curve can be the same [52]. When the unstable manifold W u ðP 1 Þ intersects at least three basins (e.g.,B 1 ,B 2 and B 3 ), then Lemma 2). Assume that oB n = oB nþ1 is true, then oB nþ1 = oB nþ2 according to the similar method above. By mathematical induction, all basins can share the common boundary.
Theorem 2 Let P j k (k 2 N,1 j n, n is the number of the different class of saddles) be countable infinitely many hyperbolic heterogeneous saddles and every P j k can generate a basin cell C j k (k 2 N, 1 j n). For each fixed j;P j k is countable infinitely many homogeneous saddles and C j k is countable infinitely many homogeneous basin cells. Let B j k (k 2 N,1 j n) be countable infinitely many disjoint basins, where B j k is the basin of C j k . If for every j(1 j n), there exists a saddle P j 1 and the unstable manifold W u ðP j 1 Þ intersects at least three basins, then oB j 1 is a Wada basin boundary w. r. t.B j k (k 2 N,1 j n). Remark 2 The proof idea of Theorem 2 is the same as that of theorem 2 [10] for the limit basins B j (1 j n). For the countable infinitely many basins with coexisting homogeneous basin cells, the result is the same as that of theorem 1. Assume that for every j(1 j n), the unstable manifold W u ðP j 1 Þ intersects these n different basins. The existence of a basin cell C j 1 implies that there is only one saddle P j 1 on the basin boundary oB j 1 , therefore W s ðP j 1 Þ = oB j 1 . Then for every i,j(1 i; j n and i 6 ¼ j), the manifolds W s ðP i 1 Þ and W u ðP j 1 Þ intersect transversally at some points (heteroclinic point). Since i,j are arbitrarily chosen, we Lemma 2). This can explain why the unstable manifold of saddles are similar in some numerical experiments because the time is limited in the numerical experiment. If for every j(1 j n), the unstable manifold W u ðP j 1 Þ intersects these n basins, then Theorem 3 Let GC k (k 2 N) be countable infinitely many homogeneous generalized basin cells and each generalized basin cell is generated by saddle-hyperbolic periodic orbits P m k (k 2 N,1 m L). Let B k (k 2 N) be countable infinitely many disjoint basins, where B k is the basin of GC k . If there exists saddles P m 1 (1 m L) and the unstable manifold W u ðP m 1 Þ intersects at least three basins, then oB 1 is a Wada basin boundary w. r. t.B k (k 2 N).

Remark 3
The proof idea of Theorem 3 is the same as that of theorem 1 [24] for a basin B 1 . The numerically verifiable conditions are also easy in Theorem 3 because generalized basin cells are homogeneous and the unstable manifold W u ðP m 1 Þ doesn't have to traverse all basins. Each generalized basin cell determines the centrebody of the corresponding basin. Because the shape of each generalized basin cell is exactly the same, the structure of the basin has the same comments as Remark 1.
Theorem 4 Let GC j k (k 2 N,1 j n) be countable infinitely many generalized basin cells with n different shapes. Each generalized basin cell with different shapes is generated by saddle-hyperbolic periodic orbits P j m k (k 2 N,1 m L). For each fixed j;GC j k (k 2 N) is countable infinitely many homogeneous generalized basin cells. Let B j k (k 2 N) be countable infinitely many disjoint basins, where B j k is the basin of GC j k . If for every j(1 j n), there exist saddles P j m 1 (1 m L) and the unstable manifold W u ðP j m 1 Þ intersects at least three basins, then oB j 1 (1 j n) is a Wada basin boundary w. r. t.B j k (k 2 N,1 j n).

Remark 4
The proof idea of Theorem 4 is the same as that of theorem 1 [24] for the limit basins B j (1 j n). For the countable infinitely many homogeneous generalized basin cells, the results on Wada basin boundary are the same as that of theorem 1. We omit similar comments on the basin boundary structure as Remark 1 and Remark 2.
Remark 5 At present, the methods of detecting Wada basins are mainly based on various numerical algorithms, such as Grid method [25], the Merging method [26], Saddle-straddle method [29]. These methods are very effective in a given limit regions. When a nonlinear system has countable infinitely many coexisting attractors, we can not understand the global basin boundary structure by the above numerical algorithms, for example, the basin boundary of attractors outside the regions may intrude into the given region. The proposed method's advantages mainly lie in two aspects. On the one hand, we can understand the main structure of the basin boundary through basin cells and generalized basin cells. The presented idea of homogeneous basin cells and generalized basin cells is very useful for expanding the number of Wada basins. On the other hand, the presented method can explain some numerical results, for example, it can explain why the unstable manifold of saddles and the stable manifold of saddles are similar in some numerical results. It can also explain why the basin boundary points will increase with the expansion of the given region.

Infinitely many Wada basins and their organizations
Firstly, we plot all the basins and their corresponding attractors for the given parameters. We selected a grid of 640 9 480 points as initial conditions in the x À y plane in order to numerically generate the basins of attraction. The basins of attraction for different resolutions have no effect on generally the predictions (initial condition probability belonging to each attractor) in the Wada regions [54]. We use different colors to plot the basins of attraction that corresponds to the attractor.

Infinite number of coexisting attractors, generalized basin cells and Wada basins
In the absence of a periodic forcing function A sin xt, the system (3) has an infinite number of coexisting equilibrium points located at (kp=a,0), where k is any integer number and the results can be obtained by the theoretical solution. When the periodic forcing function is applied to the system ( Tables 1  and 2. Table 1 shows three coexisting period-1 attractors in the X-axis coordinate interval [-10 p, 10 p]. All attractors are shown in Table 2    . It can be seen that the structure of the basin P1A 0 (P1B 0 , P1C 0 ) is the same as that of P1A 1 (P1B 1 , P1C 1 ), respectively. Similarly, every basin of P1A n (P1B n , P1C n ) (n 2 Z) has the same basin structure. In fact, the structure of the three basins is identical to those of the three basins in each X-axis coordinate interval [10(2 n-1)p, 10(2 n ? 1)p], n 2 Z. Therefore, all coexisting attractors are heterogeneous, but all P1A n (P1B n , P1C n ) (n 2 Z) are homogeneous.
Next, we will verify that the three coexisting basins are Wada basins in the X-axis coordinate interval [-10 p, 10 p]. Accessible periodic saddles play an important role in describing basin boundaries. On three basin boundaries, we obtain six accessible period saddles (Table 3) and three generalized basin cells (the orange region) formed by the stable and unstable manifolds of these six saddles (Fig. 2a). Three associated generalized basin cells can be found on three basins. The structure of these generalized basin cells is as follows. As shown in Fig. 2a, the three generalized basin cells respectively correspond to magenta basin, cyan basin, and light gray basin, which are heterogeneous. The left generalized basin cell containing the attractor P1A 0 is generated by the period-2 saddles S2A 0 and S2A 0 0 . The middle generalized basin cell containing the attractor P1B 0 is generated by the period-2 saddles S2B 0 and S2B 0 0 . The right generalized basin cell containing the attractor P1C 0 is generated by the period-2 saddles S2C 0 and S2C 0 0 . These three generalized basin cells are topologically equivalent to the cell (the orange region) shown in Fig. 2b. There are infinitely many homogeneous generalized basin cells with these three categories (Fig. 2a). And Fig. 3a-c show the unstable manifolds of the typical saddles we have plotted. We can find that the unstable manifolds of the saddle can cross up to six basins (P1A 0 , P1B 0 P1C 0 , P1A 1 , P1B 1 and P1C 1 ). Thus, three basins (P1A 0 , P1B 0 and P1C 0 ) are Wada basins. Similarly, the generalized basin cells are formed by the saddles (Table 4) in each X-axis coordinate interval [10(2 n-1)p, 10(2 n ? 1)p] (n 2 Z) and the structure of generalized basin cells is the same as Fig. 2a. Figure 4 shows that the unstable manifold of saddles (P1A 1 , P1B 1 and P1C 1 )  Table 2) are investigated in all X-axis coordinate intervals [10(2 n-1)p, 10(2  Table 5. Four coexisting attractors (P1A 1 , P1B 1 , P4A 1 and P4B 1 ) exist in the X-axis coordinate interval [0, 20 p] and the abscissa of the attractors is shifted by 20 p to the right compared with the abscissa of the attractors (P1A 0 , P1B 0 ,P4A 0 and P4B 0 ), see the Table 5.  Fig. 5c. Table 6 shows all attractors in each X-axis coordinate interval [20(n-1)p, 20 np] (n 2 Z). In each X-axis coordinate interval [20 np-4 p, 20 np ? 4 p] (n 2 Z), there are no attractors but there exist complex basin boundaries. Thus, all coexisting attractors are heterogeneous, but all P1A n (or P1B n , P4A n and P4B n ) in Tables 5 and 6 (n 2 Z) are homogeneous. Next, we will verify that these four basins of attraction are Wada basins in the X-axis coordinate interval [-20   p,0]. We obtain some typical accessible periodic orbits on basin boundaries (see Table 7). All saddles are investigated in Table 8. Thus, all coexisting saddles are heterogeneous, but for every n, S3A n (S3B n , S4A n and S4B n ) in Tables 7 and 8 (n 2 Z) are homogeneous.
We also obtain four typical basin cells formed by their stable and unstable manifolds, which are heterogeneous. The structure of these basin cells is as follows.    Fig. 6c. Figure 6d shows four basin cells in the X-axis coordinate interval [0, 20 p] and we found the same basin cell structure as Fig. 6a. The difference is that the abscissa of the saddles (S3A 1 , S3B 1 , P4A 1 -p 4 and S4B 1 -p 4 ) forming these four basin cells is shifted by 20 p to the right compared with the abscissa of the saddles (S3A 0 ,S3B 0 , P4A 0 -p 4 and S4B 0 -p 4 ). In each X-axis coordinate interval [20(n-1)p, 20 np] (n 2 Z), period-3 saddles S3A n (S3B n ) can generate a basin cell which is topologically equivalent to the cell shown in Fig. 6b. Thus, there are infinitely many homogeneous basin cells generated by the homogeneous saddles S3A n , S3B n , S4A n and S4B n . We have plotted the unstable manifolds of the typical saddles in Fig. 7. It is evident that the unstable manifolds (four accessible saddles from four basins) cross all four basins. Thus, four basins in Fig. 5a are Wada basins according to the Theorem 2. Similarly, four basins in Fig. 5b are also Wada basins. These unstable manifolds of saddles are also similar according to Remark 2. Theorem 2 shows why Fig. 5c and Fig. 5d have the same basin structure.
In each X-axis coordinate interval [20(n-1)p, 20 np] (n 2 Z), the structure of basins is identical. In each Xaxis coordinate interval [20 np-4 p, 20 np ? 4 p] (n 2 Z), the structure of basin boundaries is also identical. Finally, we investigated the Wada basin boundary in the X-axis coordinate interval [-4 p,4 p]. In this Xaxis coordinate interval, no attractors exist but basin boundaries are intertwined. Kennedy and Yorke introduce an important Proposition [6], which states that, let the sets B 1 ,::::::,B k contained in R 2 be any open invariant sets. Let p be a point not in B i (1 i k) whose unstable manifold W u ðpÞ intersects all the sets B 1 ,::::::,B k , then each x in the stable manifold W s ðpÞ is in the boundary oB k for each i. This Proposition is very useful to investigate the basin boundaries. If we plot six basins of attraction (P1B 0 , P4A 0 ,P4B 0 , P1A 1 , P4A 1 and P4B 1 ) in Fig. 5c, these six basins can possess the same Wada boundary because the unstable manifold of the accessible saddle on each basin boundary intersects all six basins, see Fig. 8. If we plot more basins in a larger X-axis coordinate interval, the boundaries of other basins will also accumulate this common boundary. For example, the red basin (P1A 0 ) boundary can be added to this boundary although the proportion is very small in Fig. 9a. For clarity, Fig. 9a retains only four basins of attraction (P1A 0 , P1B 0 , P4A 0 and P4B 0 ) and the red basin boundary is added to this common boundary. In Fig. 9a, the unstable manifold (in a short time) of the accessible saddle from the red basin can cross four basins. If we plot an unstable manifold of the saddle for a longer time in the X-axis coordinate interval [-150, 50] (see Fig. 9b), it can cross twelve basins of attractors (P1A -1 , P1B -1 , P4A -1 , P4B -1 , P1A 0 , P1B 0 , P4A 0 , P4B 0 , P1A 1 , P1B 1 , P4A 1 and P4B 1 ). Thus, we found that more basins of attraction can be added to this common boundary. Remark 1 can explain why more and more basins share the common boundary when more basins are plotted.

Conclusion
In this work, we have verified the existence of infinite number of connected basins in a nonlinear oscillator. We have investigated infinite number of coexisting attractors in this oscillator. We have given two numerical evidences that the oscillator can possess at least three Wada basins in every X-axis coordinate interval. We have investigated the basin cells and the generalized basin cells for each basin. In each X-axis coordinate interval, the structure of the basin cell (or the generalized basin cell) is quite identical. We applied the basin cell method and the generalized basin cell method to verify the Wada property. In each X-axis coordinate interval, we have verified that all basins can possess the same basin boundary. We found that the unstable manifold of the accessible saddle from each basin can cross more and more basins when the time tends to infinity. An open question is whether infinite number of Wada basins can share the common boundaries. For countable infinitely many homogeneous attractors or limited heterogeneous attractors, the presented theorems can verify this property. This implies that the unstable manifold of the saddle may cross infinite number of basins if the time tends to infinity. One of the main consequences of the existence of Wada basins is the difficulty in predicting the final state. If there are any small perturbation of an initial condition on the Wada boundary, one encounters a high level of indeterminacy and an extreme sensitive dependence on initial conditions. As soon as infinite number of Wada basins can share the same boundary, it can lead the system to any of its possible outcomes (attractors). We recommend that infinite number of basins and their basin structure can help to understand megastability. Recently researchers have found the coexistence of infinitely many attractors in memristors [55], memristive neuron systems [48] and engineering applications [49]. It can be expected to be found in physical systems and remains a challenging problem to be undertaken in the future. Fig. 7 The unstable manifolds of saddles (four accessible saddles from four basins) in the x-y plane. a S3A 0 -p 3 ; b S3B 0 -p 1 ; c S4B 0 -p 1 ; d S4A 0 -p 4 Fig. 8 The unstable manifolds of saddles (six accessible saddles from six basins) in the x-y plane. a S4A 0 -p 4 ; b S4B 0 -p 3 ; c S3B 0 -p 1 ; d S4B 1 -p 1 ; e S4A 1 -p 2 ; f S3A 1 -p 3