Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies

Commun Nonlinear Sci Numer Simulat 72 (2019) 16–25

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Research paper

Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies Xiujing Han a,b,c,∗, Yang Liu c,d, Qinsheng Bi a, Jürgen Kurths b,c a

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, PR China Department of Physics, Humboldt University, Berlin 12489, Germany c Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany d Department of Computer Science, Technische Universität Berlin, Berlin 10587, Germany b

a r t i c l e

i n f o

Article history: Received 19 August 2018 Revised 23 September 2018 Accepted 10 December 2018 Available online 11 December 2018 Keywords: Incommensurate excitation frequencies Fast-slow analysis Bursting dynamics Systems with slow excitations

a b s t r a c t This paper aims to report an approximation method, the frequency-truncation fast-slow analysis, for analyzing fast-slow dynamics in parametrically and externally excited systems with two slow incommensurate excitation frequencies (PEESTSIEFs). We obtain truncated, commensurate excitation frequencies, which are approximations of the incommensurate excitation frequencies. Then, we show numerically that bursting behavior in PEESTSIEFs can be approximated in the same systems but with truncated, commensurate excitation frequencies, and therefore bursting dynamics in PEESTSIEFs can be understood by analyzing the same systems with truncated, commensurate excitation frequencies. Based on this, the approximation method for analyzing bursting dynamics in PEESTSIEFs is proposed. The validity of the approach is demonstrated by the Duffing and van der Pol systems, respectively. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Fast-slow systems, i.e., dynamical systems whose variables evolve over two different scales (the fast and slow ones), are ubiquitous in neuroscience [1,2], biology [3,4], chemistry [5,6] and physics [7,8]. Bursting, as a result of mutual influence between different scales, is frequently observed [9–12], and can be understood by a bifurcation analysis of the fast subsystem with respect to the slow variables [13]. The fast subsystem can be in different states (e.g., the rest and active states), which is modulated by the slow variables. Bursting will appear if the slow variables visit the fast subsystem’s different parameter areas where different states exist [14–16]. In the process of modulating the behaviors of the fast subsystem, the slow variables, however, may not get any feedback from the fast variables. That is, the slow variables do not rely on the fast ones, but evolve on their own. For this case, bursting behavior can be described by a singularly perturbed system of the form [17]

x˙ = F (x, u ), u˙ =  G(u ), where  is small (0 <   1), representing the ratio of time scales between the fast variable x ∈ u ∈ Rn . ∗

Corresponding author. E-mail address: [email protected] (X. Han).

https://doi.org/10.1016/j.cnsns.2018.12.007 1007-5704/© 2018 Elsevier B.V. All rights reserved.

(1.1) Rm

and the slow variable

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The most common fast-slow systems within the framework (1.1) are dynamical systems driven by slowly varying control parameters. In studying the case of slow linear variation of control parameters, it was found that bifurcation delay behaviors [18–20], i.e., the delay loss of stability of attractors, may happen when the slow parameters pass through bifurcation points of the system. Such delay behaviors have been found to have a strong influence on bursting behaviors [21–23]. Another interesting topic is a slow periodic variation of control parameters. Bursting dynamics related to this has been investigated intensively. For example, Golubitsky et al. [17] treated the slow excitations as slow variables, and showed that the slow excitations can form a closed path on the plane of unfolding parameters for the case when the excitations have the same frequencies. In particular, a special path related to bifurcations can be designed in order to generate a given bursting pattern. Subsequently, this idea was applied successfully by Osinga et al. [24] and Saggio et al. [25] to investigate dynamical mechanisms and classifications of bursting patterns. In [26], the concept of transformed phase diagram was proposed. This concept uncovers the evolution process of the fast variables with respect to the slow variables, and has been found to be an effective tool for studying bursting dynamics [5,8,23,27,28]. In [29], we proposed the idea of transforming excited systems with rationally related excitation frequencies into fast-slow forms with a single slow variable so that the standard analysis [13,14] based on a bifurcation diagram of the fast subsystem with respect to the slow variable can be used to investigate bursting dynamics. Although much work has been done, bursting behaviors related to incommensurate excitation frequencies are studied little. In particular, an effective method for analyzing bursting dynamics involving incommensurate excitation frequencies is missing and thus needs to be further explored. In the present work, we consider parametrically and externally excited systems with two slow incommensurate excitation frequencies, written in the general form:





x˙ = F x, β1 cos(ω1 t ) + β2 cos(ω2 t ),

(1.2)

where x ∈ describes relatively fast processes, and β 1 cos(ω1 t) and β 2 cos(ω2 t) (0 < ω1,2  1) are the slowly varying parametric and external excitations whose frequencies are incommensurate. Because the fast-slow dynamical characteristics still remain in system (1.2) for incommensurate excitation frequencies, bursting behaviors may also be observed. Here we focus on how to explain the generation of bursting related to incommensurate excitations frequencies. We will show that bursting in system (1.2) can be approximated in the same system but with commensurate excitations frequencies which are the truncations of incommensurate excitation frequencies. Therefore, bursting dynamics in system (1.2) can be understood by a fast-slow analysis towards the same system with truncated, commensurate excitation frequencies. Based on this, an approximation method, i.e., the frequency-truncation fast-slow analysis, is proposed for analyzing bursting dynamics related to incommensurate excitation frequencies. The rest of this paper is organized as follows. In Section 2, some important results given in [29] are briefly summarized. Based on this, in Section 3, the frequency-truncation fast-slow analysis is proposed. Then, in Section 4, two examples related to the systems of Duffing and van der Pol are analyzed, which demonstrate the validity of the proposed approach. Finally, in Section 5, we conclude the paper. Rn

2. Fast-slow analysis for commensurate excitation frequencies We begin our analysis by describing some of our results, given in [29], about fast-slow analysis for parametrically and externally excited systems with two slow commensurate excitation frequencies, which is important for the following analysis about bursting dynamics related to incommensurate excitation frequencies. Typically, system (1.2) is a fast-slow system with two slow variables, i.e., the slow excitations. The basic idea of the fastslow analysis developed in [29] is to transform system (1.2) into the one with one single slow variable. If the excitation frequencies are commensurate, the transformed fast-slow system and its single slow variable can be obtained according to ω m the relation of excitation frequencies by using the de Moivre formula [30]. For example, when ω1 = m1 , in which m1 and 2 2 m2 are integers, the transformed fast-slow system is given by





x˙ = F x, β1 f p∗ g(t )







+ β2 fq∗ g(t ) ,

(2.1)

where g(t ) = cos(ε lt ) is the slow variable. Here the slow excitation frequencies are ω1 = ε m1 and ω2 = ε m2 with ε  1; l is the greatest common divisor of m1 and m2 , which satisfies m1 = pl and m2 = ql, where p and q are two prime numbers; f p∗ (x ) and fq∗ (x ) are the corresponding trigonometric polynomials for cos(ω1 t) and cos(ω2 t), respectively, and they have the general form

fn∗ (x ) = Cn0 xn − Cn2 xn−2 (1 − x2 ) + Cn4 xn−4 (1 − x2 )2 − . . . + imCnm xn−m (1 − x2 ) 2 , m

(2.2)

where i is the imaginary unit, and m is the maximum even number not larger than the integer n. 3. Frequency-truncation fast-slow analysis Because the excitation frequencies considered here are not rationally related, at least one of the excitation frequencies has to be an irrational number. As we know, any irrational number cannot be expressed as the fraction of integers. However, fractions of integers are commonly used to approximate irrational numbers, e.g., 355 113 ( ≈ 3.14159) is an approximate

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π ; (b) ω = 0.01 and ω = Fig. 1. Complex bursting patterns in system (4.1) with incommensurate excitation frequencies. (a) ω1 = 0.01 and ω2 = 100 1 2 √ 3 100

√ 3 ; 100

π and ω = 0.01; (d) ω = (c) ω1 = 100 and ω2 = 0.01. In (a), the insert clearly shows the oscillations; In (c), oscillations of the quasi-static processes of 2 1 bursting are highlighted by red circles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Bursting pattern with incommensurate excitation frequencies (red curve) agrees well with the one with truncated, commensurate excitation frequencies (blue curve). Here (a), (b) and (c) are related to the bursting patterns in Fig. 1(a), (b) and (c), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

fraction of π . From a mathematical point of view, any irrational number cannot be replaced by its approximate fractions. From the application point of view, however, approximate fractions, in particularly high-precision approximate fractions, can solve practical problems well. Therefore, in practical applications, an irrational number can be replaced by its approximate fractions. This is exactly the origin of our approximation method. To give a clear description of this method, here we temporarily assume that, in system (1.2), ω1 is a rational frequency, π . Then, the irrational frequency ω leads to a rational sequence { }: while ω2 is an irrational frequency, e.g., ω2 = 100 n 2

{ n } =

 πn  100

, n ∈ N,

(3.1)

in which π n denotes the 10−n -grade truncated π , i.e., an approximation of π (accurate to n decimal places). For example, π3 = 3.141 and π6 = 3.141592, which are illustrated by π

 3  π = 3 .141 592 653 · · · . π6

(3.2)

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Fig. 3. One parameter bifurcation diagram of the fast subsystem (4.3) with respect to γ . The numbers from 1 to 208 mean fold bifurcations (red points) of the equilibria. Here larger number indicates larger bifurcation value. The equilibrium branches are stable unless they are between the two fold points 2n − 1 and 2n (n = 1, 2, 3, · · · , 104). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Bursting trajectory of the bursting pattern in Fig. 1(a) is overlayed with the bifurcation diagram in Fig. 3. This is a local diagram, showing the local evolution (from γ = 1 to γ = 0.95) of the bursting. The equilibrium branches are highlighted with thick blue curves. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Based on (3.1), we obtain the following sequence of parametrically and externally excited systems ({Dn }, n ∈ N) with truncated, commensurate excitation frequencies ω1 and n :





Dn : x˙ = F x, β1 cos(ω1 t ) + β2 cos(n t ).

(3.3)

Here Dn gives the approximation of system (1.2) varying in degrees, and in particular, larger n means larger accuracy. Bursting dynamics in system (1.2) can be understood by analyzing system (3.3) under the condition that the bursting behavior in system (3.3) agrees with this in system (1.2). On the other hand, note that the excitation frequencies in system (3.3) are rationally related, and therefore the bursting therein can be explained by our method given in [29] (see Section 2). Based on the above analysis, we conclude that bursting dynamics in excited systems with incommensurate excitation

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Fig. 5. A local one parameter bifurcation diagram of the fast subsystem (4.5) with respect to γ (a, b) and its overlay with the trajectory of the bursting in Fig. 1(b) (c, d). The red numbers 6 and 14 indicate the number of fold points in (a) and (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. The same as in Fig. 5 for the bursting pattern in Fig. 1(c). Here the slow variable is g(t ) = cos(0.0 0 02t ) and the bifurcation diagrams are obtained ∗ ∗ (γ )]x + x3 = β2 f50 (γ ), where γ is the control parameter and the other parameters are the same as in based on the fast subsystem x¨ + δ x˙ − [b + β1 f 157 Fig. 1. The red circles in (b) highlight the fact that the stable equilibrium branches now show multiple ups and downs, leading to distinct oscillations in the quasistatic processes of the bursting in Fig. 1(c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

frequencies may be uncovered by analyzing the same systems with truncated, commensurate excitation frequencies that are the approximations of the original incommensurate excitation frequencies. Remark: Our numerical results show that, in general, small values of the positive integer n (e.g., n = 2 and 3, as shown in Section 4) can lead bursting behaviors in system (3.3) to agree well with those in system (1.2). How to theoretically determine the values of n such that bursting behaviors in system (3.3) agree well with those in system (1.2) is an interesting problem, and needs to be further explored. 4. Examples In this section, we focus on the equations of Duffing and van der Pol. In these two prototypical examples, bursting dynamics involving incommensurate excitation frequencies will be explored by using the proposed approximation method. 4.1. Duffing equation We consider the parametrically and externally excited Duffing equation, given by

x¨ + δ x˙ − [b + β1 cos(ω1 t )]x + x3 = β2 cos(ω2 t ),

(4.1)

where β 1 cos(ω1 t) and β 2 cos(ω2 t) (0 < ω1,2  1) are two slow excitations. We find that complex bursting behaviors are ubiquitous when the excitation frequencies are not rationally related. Several typical examples of complex bursting patterns with different values of incommensurate excitation frequencies are shown in Fig. 1, where the system parameters are the same as in Ref. [29], i.e., δ = 0.1, b = 0.5 and β1 = β2 = 1. We analyze the bursting pattern in Fig. 1(a), in which the incommensurate excitation frequencies are ω1 = 0.01 and π . Bursting patterns related to the truncated, commensurate excitation frequencies are shown in Fig. 2(a), where the ω2 = 100 bursting from Fig. 1(a) is also superimposed to give a clear view of the anastomosis of bursting patterns. It is seen that

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π ; (b) ω = π and ω = 0.01; Fig. 7. Complex bursting patterns in system (4.6) with incommensurate excitation frequencies. (a) ω1 = 0.01 and ω2 = 100 1 2 100 √

√

2 (c) ω1 = 0.01 and ω2 = 1002 ; (d) ω1 = 100 and ω2 = 0.01. The insert in (a) gives a clear view that the envelope (red curve) of the active phase of bursting oscillates. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. The same as in Fig. 2 for the bursting in Fig. 7(a).

the bursting pattern with 10−2 -grade truncated frequency agrees well with the one in Fig. 1(a). Therefore, the bursting in Fig. 1(a) can be explained by analyzing the one with 10−2 -grade truncated frequency 2 . For this purpose, we explore the following excited Duffing system with truncated, commensurate excitation frequencies (ω1 and 2 ):

x¨ + δ x˙ − [b + β1 cos(ω1 t )]x + x3 = β2 cos(2 t ),

(4.2)

.14 where ω1 = 0.01, 2 = 3100 = 0.0314 and the system parameters are the same as in Fig. 1(a). Note that the excitation frequencies in system (4.2) are rationally related. So, according to the results given in Section 2, system (4.2) can be transformed into a fast-slow form with one single slow variable. The slow variable is g(t ) = cos(0.0 0 02t ), and the fast subsystem is ∗ ∗ (γ )]x + x3 = β2 f157 ( γ ), x¨ + δ x˙ − [b + β1 f50

∗ (γ f50

(4.3)

∗ (γ f157

where γ is the control parameter, ) and ), decided by (2.2), are the corresponding trigonometric polynomials for cos(ω1 t) and cos(2 t), respectively. The fast subsystem (4.3) exhibits within the parameter interval −1 ≤ γ ≤ 1 a fairly complex bifurcation behavior with 208 fold bifurcation points, which divide the whole equilibrium curve into 209 branches (see Fig. 3). The equilibrium branches are stable unless they are between the two fold points 2n − 1 and 2n (n = 1, 2, 3, · · · , 104). When the slow variable g(t ) = cos(0.0 0 02t ) is switched on, the trajectory switches among a large number of stable equilibrium branches by fold bifurcations (e.g., see Fig. 4). This is exactly the reason for the generation of the complex bursting pattern in Fig. 1(a). As the second example, we analyze briefly the bursting pattern in Fig. 1(b). Fig. 2(b) shows that this bursting pattern is .732 well approximated by the one with 10−3 -grade truncated, commensurate excitation frequencies ω1 = 0.01 and 3 = 1100 = 0.01732. Therefore, the bursting in Fig. 1(b) can be understood by analyzing the following system

x¨ + δ x˙ − [b + β1 cos(ω1 t )]x + x3 = β2 cos(3 t ).

(4.4)

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Fig. 9. A whole bifurcation diagram of the fast subsystem (4.8) with respect to γ . Each of the active states is highlighted by red number. In all, 26 active states are exhibited within −1 ≤ γ ≤ 1, and they appear and disappear by supercritical Hopf (supH) bifurcations. Here the active states show distinct amplitude modulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Then, we have the slow variable g(t ) = cos(0.0 0 0 04t ) and the fast subsystem: ∗ ∗ (γ )]x + x3 = β2 f433 ( γ ), x¨ + δ x˙ − [b + β1 f250

(4.5)

∗ (γ ) and f ∗ (γ ) are the corresponding trigonometric polynomials for cos(ω t) where γ is the control parameter, and f250 1 433 and cos(3 t), respectively. Bifurcation behaviors of the fast subsystem (4.5) are shown in Fig. 5(a, b), where 20 fold bifurcation points are observed in the parameter interval 0.995 ≤ γ ≤ 1. Note that this parameter interval is local, and accounts for merely 0.25% of the whole parameter interval −1 ≤ γ ≤ 1. Therefore, complex bifurcation behaviors, showing many fold bifurcations within −1 ≤ γ ≤ 1, can be predicted. As a result, a complex bursting pattern is created [see Figs. 1(b) and 5(c, d)]. For other bursting patterns, the associated dynamical mechanisms can be analyzed similarly. For example, Fig. 2(c) shows that the bursting in Fig. 1(c) can be well approximated by the one with 10−2 -grade truncated, commensurate excitation .14 frequencies 2 = 3100 = 0.0314 and ω2 = 0.01. Based on this, the slow variable and the fast subsystem, related to the para.14 metrically and externally excited Duffing system with truncated, commensurate excitation frequencies (2 = 3100 = 0.0314 and ω2 = 0.01), can be obtained. Then, we compute the bifurcation diagram of the fast subsystems [Fig. 6(a, b)], and overlay it with the bursting trajectory [Fig. 6(c, d)] to explain the generation of the bursting. We find that, the bursting trajectory agrees well with the bifurcation diagram, and this indicates that the bursting pattern in Fig. 1(c) follow the same dynamical mechanism as discussed for the ones in Fig. 1(a, b). That is, all these bursting patterns are generated because the trajectory switches by fold bifurcations among many equilibrium branches. However, the bursting pattern in Fig. 1(c) shows some different features compared to those in Fig. 1(a, b), e.g., distinct oscillations appear in the quasi-static processes of the bursting [see Fig. 1(c)]. The reason for this is that the stable branches related to the bursting pattern in Fig. 1(c) now become ones with twists and turns, characterized by multiple extreme points appearing on the stable branches [see Fig. 6(b)].

4.2. van der Pol equation We now analyze the following parametrically and externally excited van der Pol equation

x¨ + [1 + β1 cos(ω1 t )](x2 − 1 )x˙ + x = β2 cos(ω2 t ),

(4.6)

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Fig. 10. The same as in Fig. 4 for the amplitude-modulated bursting in Fig. 7(a), and this gives a local evolution of the bursting from γ = 1 to γ = 0.85.

Fig. 11. The same as in Fig. 8 for the bursting in Fig. 7(d).

where β 1 cos(ω1 t) and β 2 cos(ω2 t) (0 < ω1,2  1) are the slow excitations. As in [29], here we fix β1 = 1.5 and β2 = 0.5, and explore bursting behaviors related to incommensurate excitation frequencies. As we found in the Duffing system, equation (4.6) also exhibits rich and complex bursting behaviors when the excitation frequencies are incommensurate. Fig. 7 shows several typical examples of complex bursting patterns with incommensurate excitation frequencies. We see that some of the bursting patterns show interesting dynamical characteristics. For example, as shown in Fig. 7(a), distinct oscillations are observed in the envelope of the active phase (i.e., large-amplitude oscillations) of bursting. This class of bursting rhythms is known as amplitude-modulated bursting [31], a novel bursting pattern reported recently. For other bursting patterns, however, no significant oscillations are found in the envelope [e.g., see Fig. 7(d)]. For this case, a common bursting pattern is obtained. To begin with, we analyze the amplitude-modulated bursting represented by that in Fig. 7(a). Bursting patterns with different truncated, commensurate excitation frequencies are shown in Fig. 8, where the amplitude-modulated bursting in Fig. 7(a) is also superimposed. We see that the amplitude-modulated bursting is well approximated by the one with 10−2 .14 grade truncated, commensurate excitation frequencies ω1 = 0.01 and 2 = 3100 = 0.0314 [Fig. 8(b)]. Therefore, in order to reveal the dynamical mechanism of the bursting in Fig. 7(a), we explore the following van der Pol equation with 10−2 -grade truncated, commensurate excitation frequencies:

x¨ + [1 + β1 cos(ω1 t )](x2 − 1 )x˙ + x = β2 cos(2 t ).

(4.7)

Then, according to the results given in Section 2, we obtain the same slow variable g(t ) = cos(0.0 0 02t ) as discussed for the Duffing equation. Based on this, we have the following fast subsystem related to system (4.7): ∗ ∗ (γ )](x2 − 1 )x˙ + x = β2 f157 ( γ ), x¨ + [1 + β1 f50

∗ (γ f50

(4.8) ∗ (γ f157

where γ is the control parameter, and ) and ) are the corresponding trigonometric polynomials for cos(ω1 t) and cos(2 t), respectively. A whole bifurcation diagram, within the parameter interval −1 ≤ γ ≤ 1, of the fast subsystem (4.8) with respect to γ is shown in Fig. 9, where complex bifurcation behaviors showing 26 active states, which alternate with the rest states by

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Fig. 12. The same as in Figs. 5 and 6 for the bursting in Fig. 7(d).

supercritical Hopf bifurcations supH, are observed. In particular, the active states are of amplitude-modulated type, i.e., they show amplitudes that alternate between increases and decreases as the control parameter γ varies. This thus leads to distinct oscillations in the envelope of the active phases of bursting (e.g., see Fig. 10). As a result, the amplitude-modulated bursting in Fig. 7(a) is generated. Next, we briefly explore the common bursting pattern with the example in Fig. 7(d). Note that this bursting pat.414 tern agrees well with the one with 10−3 -grade truncated, commensurate excitation frequencies 3 = 1100 = 0.01414 and ω2 = 0.01 (see Fig. 11), so it can be understood by analyzing the following equation

x¨ + [1 + β1 cos(3 t )](x2 − 1 )x˙ + x = β2 cos(ω2 t ).

(4.9)

Treating g(t ) = cos(0.0 0 0 02t ) as the slow variable leads to the following fast subsystem related to system (4.9): ∗ ∗ (γ )](x2 − 1 )x˙ + x = β2 f500 ( γ ), x¨ + [1 + β1 f707

∗ (γ f707

(4.10)

∗ (γ f500

where γ is the control parameter, and ) and ), determined by (2.2), are the corresponding trigonometric polynomials for cos(3 t) and cos(ω2 t), respectively. Typical bifurcation behaviors within the parameter interval 0.995 ≤ γ ≤ 1 of this fast subsystem are presented in Fig. 12(a–c). It is seen that the active states are now very common, because they do not show distinct amplitude modulation. This accounts for the generation of a common bursting pattern as shown in Fig. 7(d) [see Fig. 12(d–f)]. 5. Conclusion Dynamical systems with incommensurate excitation frequencies show rich bursting behaviors. How to understand bursting dynamics related to incommensurate excitation frequencies is an important problem. Because the excitation frequencies are incommensurate, the excitation frequency ratio cannot be expressed as quotient of two integers. Therefore, for this case, the system cannot be transformed into the standard fast-slow form with one single slow variable. Our numerical results show that bursting behaviors related to incommensurate excitation frequencies can be well approximated by those with truncated, commensurate excitation frequencies. Therefore, bursting dynamics related to incommensurate excitation frequencies can be understood by analyzing the same system with truncated, commensurate excitation frequencies. This way, an approximation method, i.e, the frequency-truncation fast-slow analysis, for analyzing bursting dynamics with incommensurate excitation frequencies is proposed. Although our results are based on parametrically and externally excited systems, we believe that our analytical treatment may be extended to other types of excited systems, e.g., dynamical systems with two slow external forcings, where incommensurate excitation frequencies are relevant. We have found that bursting behaviors related to incommensurate excitation frequencies often exhibit complex dynamical characteristics (e.g., see Figs. 1 and 7). In fact, all of our numerical results show that bursting patterns related to incommensurate excitation frequencies can be well approximated by 10−n -grade truncated, commensurate excitation frequencies. Here the positive integral n, as noted in Section 3, is usually relatively small (e.g., see Figs. 2, 8 and 11). However, the ratio of the truncated, commensurate excitation frequencies is often expressed as the quotient of large numbers, e.g., the ratio is ω1 : 2 = 50 : 157 for the bursting pattern in Fig. 1(a) [see Fig. 2(a)], is ω1 : 3 = 250 : 433 for the one in Fig. 1(b) [see Fig. 2(b)], and is 3 : ω2 = 707 : 500 for the one in Fig. 7(d) (see Fig. 11). Note that the ratio of large numbers means that the trigonometric polynomial fn∗ (x ) has a complex expression, which thus complicates bifurcation behaviors of the fast subsystem, and finally leads to complex bursting dynamics related to incommensurate excitation frequencies.

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Compliance with ethical standards Conflict of Interest: The authors declare that they have no conflict of interest. Acknowledgments The authors express their gratitude to the anonymous reviewers whose comments and suggestions have helped improve this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. (11572141, 11632008, 11772161 and 11502091) and the Training Project for Young Backbone Teacher of Jiangsu University. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cnsns.2018.12. 007. References [1] Izhikevich EM, Desai NS, Walcott EC, Hoppensteadt FC. Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 2003;26:161–7. [2] Channell P, Cymbalyuk G, Shilnikov A. Origin of bursting through homoclinic spike adding in a neuron model. Phys Rev Lett 2007;98:134101. [3] Schuster S, Knoke B, Marhl M. 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