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Different transitions of bursting and mixed-mode oscillations in Liénard system
Int. J. Electron. Commun. (AEÜ) 111 (2019) 152898
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Regular paper
Different transitions of bursting and mixed-mode oscillations in Liénard system S. Dinesh Vijay, S. Leo Kingston, K. Thamilmaran ⇑ Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, Tamil Nadu 620024, India
a r t i c l e
i n f o
Article history: Received 19 June 2019 Accepted 31 August 2019
Keywords: Bursting oscillations Spiking trains Mixed-mode oscillations Transient chaos Liénard system Electronic circuit
a b s t r a c t In this paper, we present the emergence of bursting oscillations, mixed-mode oscillations and transient chaos in the forced Liénard system. The system exhibits intriguing dynamical transitions such as from chaotic bursting oscillations to spiking trains via periodic bursting oscillations, period doubling mixedmode oscillations route to chaos, and intermittent mixed-mode oscillations on variation of the nonlinear damping coefficient as the control parameter. In addition, the system transits from spiking trains to chaotic bursting oscillations through successive period adding sequences of bursting oscillations upon variation of the strength of nonlinearity. Besides the system exhibits transient chaos also. We believe that it is for the first time that these multiplicity of dynamical behaviors and transitions are reported in a single model. We validate our numerical investigations with the help of experimental observations using a simple nonlinear electronic circuit implementation of the model. Ó 2019 Elsevier GmbH. All rights reserved.
1. Introduction Studies on neuronal dynamics and their different dynamical transitions have received great attention in the past few decades. Many researchers have applied nonlinear theory successfully to explore the diversity of excitability, spiking, and bursting mechanism of neurons [1,2]. It has been found that when two neurons communicate information among themselves, electrical pulses called action potential are exchanged between them. The collection of action potentials of the neurons are known as spiking trains [1–3]. The sequence of continuous firing (spiking) states alternating with the quiescent states is called Bursting Oscillations (BOs) [4,5]. Understanding the dynamics of bursting and spiking trains is one of the major interesting areas in the field of nonlinear dynamics, since it has potential applications in neurobiology, biophysics, chemistry and other disciplines [6]. The first work in this field was the Hodgkin-Huxley neuron model [7], but later on different models were put forward to account for the potential mechanism of signal processing and mode transitions of electrical activities of neurons [8–12]. The mode of electrical activities of neurons and its multiple transitions are explained using suitable electromagnetic radiation or force [8]. In the latter case, the mode
⇑ Corresponding author. E-mail addresses: [email protected] (S. Dinesh Vijay), kingston. [email protected] (S. Leo Kingston), [email protected] (K. Thamilmaran). https://doi.org/10.1016/j.aeue.2019.152898 1434-8411/Ó 2019 Elsevier GmbH. All rights reserved.
transitions of neuron under the influence of low–high electromagnetic radiation with white noise have been reported [9]. Similarly, a new physical neuron model is proposed in [12] using charge and flux controlled memristor to describe the time varying electromagnetic field. Further, collective behavior of neuronal networks with different topologies such as, synchronization transitions and pattern selections are used to understand the occurrence of neuronal diseases [13]. In recent years, several oscillator-like neuron models are reported in the literature to understand the different spiking and bursting properties as well as their distinct dynamical transitions [14–20]. Especially, BOs reported in the variety of neuronal models [4,21], single as well as multiple time-scale periodically excited systems, fast-slow subsystems of both autonomous [14,15] and nonautonomous systems [16–18]. In nonautonomous systems, the BOs occur owing to the interaction between the large internal frequency of the system and small external forcing frequency. There are several reports in the literature, mainly focused on the emerging mechanism of BOs. [15–17,21,22]. Yet a clear picture of these transitions and their related mechanisms have not been well understood so far. Similarly, Mixed-Mode Oscillations (MMOs) are another kind of bursting oscillations which appear as an alternate sequence of the large and small amplitude of oscillations. These MMOs are denoted the notation as Ls, where L and s are the large and small amplitude of oscillations, respectively. The existence of MMOs in different
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model systems and their emerging mechanism have been reported in [23–27]. There are few predominant sequences were established for the existence of MMOs in the literature, such as an alternate sequence of periodic MMOs and chaotic state [3,26,28], successive period adding sequence of MMOs [29] and Farey sequences [30]. Besides, another interesting complex phenomenon so-called transient chaos (TC) have been studied both numerically and experimentally in the literature [31,32]. In this dynamics, the system exhibits chaotic state for a finite interval of time and switches over to periodic oscillations in the asymptotic limit [32]. Transient behavior plays a vital in the brain functionality [31] and is also used for understanding the transient and extreme events in nervous systems [33,34]. The experimental observation of transient chaos has been reported such as in nonlinear pendulum [35], optical metamaterials [36], electronic circuits [37,38], closed chemical systems [39], population model [40] and simple Hopfield neural networks [41].
1.1. Motivation From the preceding paragraphs we find that there are plenty of models, each of which could simulate and describe only some particular aspects of the neuronal dynamics and a few of the dynamical transitions shown by the neurons. While these models are by themselves elegant, the following questions arises, namely (i) Is it possible to have a single mathematical model which can adequately describe almost all of these known behaviors? (ii) Will this model lend itself to easy mathematical analysis and numerical simulations? (iii) Can such a model be implemented experimentally so as to observe these behaviors in real time? It is in trying to find out answers to these questions that we chanced upon the Liénard model to describe neural systems. In fact this is a continuation of the earlier work of Leo Kingston et al., Ref. [26], wherein the authors reported for the first time that BOs and MMOs are indeed present in Liénard system and that this system can be implemented experimentally also. The general form of Liénard system proposed by the French mathematical physicist Liénard from the pioneering work of van der Pol model [42]. This Liénard system is a paradigmatic model for the different physical, engineering as well as biological systems, and it exhibits several intriguing complex dynamics [26,29,43,44]. It has been used to understand different bifurcation theories, various biological processes, and signal processing methods [42]. In this present study, we have observed that all the possible electrophysical activities of neurons can be observed in this single system, namely the Liénard system by a proper choice of system
parameters. To the best of our knowledge we believe that, perhaps for the first time such a multiplicity of dynamical behaviours has been observed in a single system. The structure of this paper presented as follows. The proposed model of the forced Liénard system and their detailed stability analysis have been explained in Section 2. The emergence of periodic and chaotic bursting oscillations, spiking trains are explored in Section 3. Section 4 deals with the existence of period doubling mixed-mode oscillations (MMOs) route to chaos as well as intermittent MMOs in the considered model. The system also found to possess transient chaos for the precise choice of system parameters which denoted in Section 5. Experimental circuit realization of the forced Liénard system and the obtained experimental results are given in Section 6. The final results presented in the concluding Section 7. 2. Model system We consider the following form of Liénard system with external sinusoidal forcing,
€x þ ax2 x_ þ cx þ bx3 ¼ AsinðxtÞ:
ð1Þ
For numerical study, the Eq. (1) is written as the following form of coupled two first order equations
x_ ¼ y y_ ¼ ax2 y cx bx3 þ AsinðxtÞ:
ð2Þ
Here, a; c, and b represent the nonlinear damping constant, internal frequency and strength of nonlinearity, whereas A and x are amplitude and frequency of the external sinusoidal forcing term. Eqs. (1) and (2) satisfy the general form of Liénard system [42]. For the suitable choice of the system parameters of Eq. (2), we identified periodic and chaotic BOs, spiking trains, periodic MMOs, intermittent MMOs, and transient chaos. 2.1. Stability analysis In order to find the stability of the system of Eq. (2), we set the external forcing term AsinðxtÞ to zero and fix the values of the system parameters as a = 0.25, b = 0.5 and c = 0.5. For this choice of parameters, the unforced system (Eq. (2)) has two different equilibria such as, stable foci (SF) (blue) on the (1; 0) planes in phase space and a saddle (Sd) (black) at the origin which are shown in Fig. 1(a). We analyzed the stability of bifurcation by perturbing the system. For this we used the standard continuation package XPPAUT AUTO [45]. The bifurcations of equilibrium points were observed for every cycle of oscillation (AsinðxtÞ, modð2pÞ). The
Fig. 1. (a) Equilibrium points of the Liénard system: the saddle (Sd) at the origin and stable foci (SF) on the (1; 0) planes. (b) Stability bifurcation diagram in the (A x) plane: here SF+ and SF denote the stable foci on the planes perpendicular to the x-axis at x ¼ 1 respectively and Sd represent the saddle points. LP1 and LP2 are the turning points for the transitions from saddle to stable foci.
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bifurcation diagram obtained is a S-shaped curve as shown in Fig. 1 (b). In this Fig. 1(b), the filled blue circles and hollow black circles represent the stable foci (SF) and saddle (Sd) regions respectively. The limit points LP1 and LP2 are the two turning points in the stability bifurcation diagram. As compared with the earlier work [26], the system exhibits different stable behaviors leading to a rich variety of transitions as the parameters of the system are varied. However, we are interested in the study of the dynamical transitions of the system that occur when the nonlinear damping coefficient (a) and strength of nonlinearity (b) are varied. When these parameters are varied, we find the system transits from saddle equilibrium state (Sd) to stable foci (SF) via stable node (SN). This is shown clearly in the two parameter phase diagram in the (b a) plane shown in Fig. 2. 3. Bursting oscillations and spiking trains
0.1370 6a 61.283. When the nonlinear damping (a) is increased further beyond 1.283, the system transits to spiking trains behavior and finally it exhibits regular periodic motion. The inserts of Fig. 3(a) and (b) show the magnified regions of chaotic BOs. The various dynamical behaviors for the system are shown in Fig. 4. In this figure, the typical time series for the variable yðtÞ for a(i) chaotic BOs (a = 0.12), b(i) periodic BOs (a = 0.25) and c (i) spiking trains (a = 1.5) are shown. The corresponding phase portraits in the (x y) plane are shown in figures a(ii), b(ii) and c(ii) respectively. Among the BOs, the existence of chaotic BOs and periodic BOs are identified by plotting the return map of ISI. The return map of ISI depicted in 5(a) shows the randomly distributed points for chaotic BOs. However, the regularly distributed return map of ISI (Fig. 5(b)) portrays the periodic BOs. 3.1. Emerging mechanism of bursting oscillations
When the system parameters were fixed as b = 0.5, c = 0.5, A = 0.2, x = 0.05 and the nonlinear damping coefficient a was varied in the range 0.0 6a 61.8, we noticed Bursting Oscillations (BOs) and spiking trains. In the BOs region we observed chaotic BOs and periodic BOs. These were followed by spiking trains. Bursting Oscillations refer to alternate sequence of continuous firing and resting states. If this sequence is chaotic, then the behavior is called chaotic BOs, while if the sequence is periodic, it is called as periodic BOs. If continuous firing alone occurs, the behavior is called spiking trains. In order to locate the regions of different bursting oscillations and spiking trains, we plotted inter spike interval (ISI) bifurcation diagram in the (a – ISI) plane as shown in Fig. 3(a). The inter spike interval was calculated by finding the interval between two consecutive spikes. The corresponding Lyapunov spectrum was plotted in the (a k1;2 ) plane as shown in Fig. 3(b). In this Fig. 3 (b), the first Lyapunov exponent is represented by solid blue line while the second Lyapunov exponent is represented by dashed red line. From this diagram, we find the system exhibits chaotic BOs for the lower range of a 2 (0, 0.1369) as evidenced by the positive value for the largest Lyapunov exponent. When the nonlinear damping is increased further, that is for a > 0.1369, the system exhibits periodic bursting oscillations for the range of a as
2
In this subsection, we explain the emergence of bursting oscillations in the forced Liénard system of Eq. (2). Let us assume the external forcing to be denoted by d such that d ¼ A sinðxtÞ. The phase portrait of the Liénard system was obtained in the (d x) plane. This was superimposed on the stability bifurcation diagram in the (A x) plane of Fig. 1(b). The resultant diagram is shown in Fig. 6. Here in this figure, when the trajectory approaches the turning point LP2 , it gets attracted to the stable focus (SFþ ) by fold bifurcation. After evolving for sometime, the trajectory reaches the turning point LP1 and then it switches to the stable focus (SF ). This process continues to exist in the BOs. Furthermore, we observed another interesting transition, that is, from spiking trains to chaotic bursting via periodic BOs for another set of system parameters with the strength of nonlinearity (b) as the control parameter. For more details kindly refer Appendix. 4. Mixed-mode oscillations Mixed-Mode Oscillations (MMOs) refer to the existence of an alternating sequence of large amplitude oscillations and small
SN
α
1.5 SF
Sd
1 0.5 0 0
0.5
1
β
1.5
2
Fig. 2. Two parameter phase diagram in the (b – a) plane for the fixed system parameter c = 0.5 showing the transition from saddle (Sd) to stable foci (SF) via saddle node (SN).
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Fig. 3. (a) Inter spike interval (ISI) bifurcation diagram in the (a – ISI) plane and (b) the corresponding Lyapunov spectra in the (a – k1;2 ) plane.
Fig. 4. (i) Time series of yðtÞ and its corresponding (ii) phase portraits in the (x y) planes. (a) chaotic BOs for a = 0.12, (b) periodic BOs for a = 0.25 and (c) spiking trains for a = 1.5.
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Fig. 5. Inter spike interval return map in the (ISIn – ISInþ1 ) planes: (a) with randomly scattered phase points denoting chaotic BOs for a = 0.12 and (b) regular distribution phase points denoting periodic BOs for a = 0.25.
Fig. 6. The phase portrait of BOs in the (d x) plane superimposed on the S-shaped stability bifurcation diagram of Fig. 1(b) showing BOs.
amplitude oscillations. For example they have been found to occur in BVP oscillator Ref. [46]. Here, the number of large amplitude and small amplitude oscillations are generally denoted by (Ls ), where L stands for the period of the large amplitude oscillations and the superscript s stands for the period of small amplitude oscillations. In this section, we report the existence of Mixed-Mode Oscillations (MMOs) in the Liénard system. In particular, we have found the existence of successive period doubling MMOs leading to chaos and periodic intermittent MMOs. For identifying these, we fix the system parameters as b = 0.33, c = 0.7, A = 0.7 and x = 0.45 and varied the control parameter, namely the nonlinear damping coefficient a in the range a 2 (0.0001, 0.06). The results obtained are explained in the following subsections.
doubling is shown in Fig. 7. While the time series and phase por-
4.1. Period doubling mixed-mode oscillations route to chaos
4.2. Intermittent and chaotic mixed-mode oscillations
The system exhibits a period doubling mixed-mode oscillations route to chaos for the control parameter a in the range 0.06 Pa P0.04548. When the control parameter a is gradually decreased
In addition to the period doubling MMOs route to chaos, the system also exhibits an alternate sequence of periodic MMOs and chaotic states. At the beginning for the value of a = 0.0001, the system is in the chaotic states. When we gradually increase the value of a, it becomes periodic MMOs via saddle-node bifurcation [26]
from 0.06, the 11 MMOs bifurcate to 22 MMOs. This transition of the system from 11 ðLs Þ to 22 ðLs Þ mixed-mode oscillations via period
traits for 11 ðLs Þ mixed-mode oscillations are shown in Fig. 7 a(i) and a(ii), the corresponding diagrams for 22 ðLs Þ mixed-mode oscillations are shown in Fig. 7 b(i) and b(ii) respectively. On further reduction of the a value, the system attains chaotic states via successive period doubling sequence of MMOs. This is shown in the one parameter bifurcation diagram of Fig. 8(a). The insert of Fig. 8(a) is a blown up portion of the bifurcation diagram showing clearly the period doubling MMOs route to chaos. The existence of this period doubling MMOs to chaos is corroborated by the Lyapunov spectrum which is shown in Fig. 8(b). In Fig. 8(b), the largest Lyapunov exponent is represented by solid blue line which shows negative values for periodic MMOs and positive values for chaotic states.
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Fig. 7. Time series for the variable xðtÞ showing a(i) 11 mixed-mode oscillations for a = 0.057 and b(i) 22 MMOs for a = 0.05254. The corresponding phase portraits in the (x y) plane are shown in a(ii) and b(ii) respectively.
Fig. 8. (a) One parameter bifurcation diagram in the (a x) plane and its corresponding (b) Lyapunov spectrum in the (a k1;2 ) plane. The largest Lyapunov exponent is represented by solid blue line which shows negative values for periodic MMOs and positive values for chaotic states.
for the value of a = 0.00031. This periodic states continue to exist until a = 0.01797 and becomes chaotic via intermittency route.
5. Transient chaos
Further increasing the value of a the chaotic state transits to 22 MMOs via saddle-node bifurcation for a = 0.01820, this is shown in Fig. 9(a). Upon increasing the control parameter, the system exhibits periodic and chaotic intermittent MMOs for a = 0.01822, and a = 0.01824 respectively as shown in Fig. 9(b) and (c). These
In this section, we explain the existence of Transient Chaos (TC) phenomenon in the forced Liénard system of Eq. (2). Transient chaos refers to a dynamic behavior, wherein the system exhibits chaos for a long interval of time and finally settles down to a periodic state for a fixed choice of system parameters. To observe transient chaos in our system, we set the values of the system parameters as a = 0.005, b = 0.12, c = 0.63, x = 0.053 and A = 0.55. The time series for the variable xðtÞ showing TC for this choice of system parameters is shown in Fig. 10(a). In this dynam-
22 periodic and chaotic intermittent MMOs alternately exist for the range of control parameter 0.01798 6a 60.02214. Presumably these were attributed as arising due to crisis induced intermittency.
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Fig. 9. Poincaré map of the time series xðtÞ of mixed-mode oscillations: (a) periodic MMOs (22 ) for a = 0.01820, (b) intermittent period MMOs (22 ) for a = 0.01822 and (c) intermittent chaotic MMOs for a = 0.01824.
ics, the system transit from aperiodic state to regular steady state. In Fig. 10(a), we have plotted only peaks of the time series for the clear visualization with the blue dots representing the aperiodic oscillations and red dots representing the asymptotic steady state motion. In general, the Poincaré return map is used to characterize the existence of periodic and nonperiodic states of the different complex systems. Here we have taken the state variable xðtÞ and found its return map which is as shown in Fig. 10(b). The widely scattered blue dots illustrate the aperiodic motion of the system, whereas the single large red point depicts the steady state of the system, since the asymptotic steady state of the system is periodic. Further, we confirm the emergence of transient chaos in the forced Liénard system using the statistical tools such as Finite Time Lyapunov Exponent (FTLE) and phase ð/Þ analysis. The FTLEs calculated for the fixed system parameters value are shown in Fig. 11(a).
Here the solid blue line denotes the first Lyapunov exponent and the dashed red line denotes the second Lyapunov exponent of the system for a finite interval of time. It can be noticed from the FTLE measurement that, as the time increases, the first Lyapunov exponent switches from a positive value to a negative value. The positive values of maximum Lyapunov exponents represent the chaotic state and its negative value denotes the periodic state. Hence the FTLE is a statistical measurement that confirms the emergence of transient chaos phenomena in the Liénard system. Besides, the phase (/) of the system state variable xðtÞ was calculated using the Hilbert transformation method Ref. [37] and is shown in Fig. 11(b). The phase ð/Þ of the system gradually increases as time progresses and after some finite interval of time, the phase (/) reaches a constant value. This clearly shows the existence of the chaotic and periodic state in the system.
Fig. 10. (a) Peaks of the time series xðtÞ of transient chaos shows an aperiodic (blue) and periodic (red) states and (b) Poincaré return map in the (xn – xnþ1 ) plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 11. (a) Finite time Lyapunov exponent and (b) phase (/) analysis of transient chaos for the fixed system parameters a = 0.005, b = 0.12, c = 0.63, x = 0.053 and A = 0.55.
Fig. 12. Experimental circuit realization of forced Liénard system.
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Fig. 13. Experimental results: (i) time series of v 2 (t) and its corresponding (ii) phase portraits in the (v 1 – v 2 ) planes, (a) chaotic BOs, (b) periodic BOs and (c) spiking trains.
Fig. 14. Experimental time series
v 1 (t) and its corresponding phase portrait (v 1 v 2 ) plane: (a) 11
6. Circuit realization The analog circuit realization for the forced Liénard system of Eq. (2) is shown in Fig. 12. The circuit consists of linear resistors, capacitors, operational amplifiers, analog device multipliers and external sinusoidal voltage source (f ðtÞ ¼ F sin Xt). The output voltages v 1 and v 2 are measured across the capacitors C 1 and C 2 respectively. Applying the Kirchhoff’s law at the junctions A and B, we get the following circuit equations,
C1
dv 1 0:1v 1 v 22 v 2 0:01v 32 1 þ þ F sin Xt ¼ R1 dt R5 R6 R7
ð3Þ
C2
MMOs and (b) 22 MMOs.
dv 2 v1 ¼ : dt R2
ð4Þ
Combining and rearranging the Eqs. (3) and (4), we get following circuit equations 2
C 1 C 2 R2 R5
0:01
d
v2
dt
2
!
¼ 0:1C 2 R2 v 22
R5 3 R5 v þ F sin Xt: R7 2 R1
dv 2 R5 þ v2 dt R6
ð5Þ
ð6Þ
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Fig. 15. Experimental time series
v 1 (t) of (a) 22
periodic MMOs, (b) 22 periodic intermittent MMOs and (c) chaotic intermittent MMOs.
The circuit equation Eq. (5) is equivalent to the normalized equation of the forced Liénard system of Eq. (2). Here the rescaling parameters taken as t ! s = R1 C 1 t, x1 ¼ v 1 ; x2 ¼ v 2 ; a ¼ 0:1 RR15 ; b ¼ 0:01 RR17 , and c ¼ RR16 . For this choice of the circuit parameters, it exhibits periodic BOs, chaotic BOs, spiking trains, periodic MMOs and intermittent MMOs as well as transient chaos. 6.1. Experimental results For all of our experiment studies, we fixed the circuit parameters as C 1 = C 2 = 10 nF and R1 = R2 = R3 = R4 = 10 KX. We obtain different complex dynamics for the suitable choice of the other circuit parameters. In order to obtain the chaotic BOs, periodic BOs and spiking trains, we had chosen R6 = 19.98 KX, R7 = 197 X, F = 0.224 V, X = 56 Hz and varying R5 in the specific range 8.5 KX 6R5 6500 X. For R5 = 8.3 KX, the circuit exhibits chaotic BOs, the time series v 1 (t) and the corresponding phase portrait of chaotic BOs in the (v 1 v 2 ) plane are shown in Fig. 13a(i) and a(ii). On further decreasing the value of R5 (R5 = 4.026 KX), we noticed periodic bursting oscillation, its time series and phase portraits are represented in Fig. 13b(i) and Fig. 13b(ii). Upon decreasing R5 = 628 X the system reveals spiking trains which are shown in Fig. 13c(i) and Fig. 13c(ii). Further, we have experimentally observed the successive period doublings and intermittent mixed-mode oscillations for the suitable choice of fixed circuit parameters. For this we kept R6 = 14.252 KX, R7 = 310 X, F = 0.710 V, X = 498 Hz. For the control parameter value of R5 = 16.656 KX, we obtained 11 MMOs and it bifurcates to 22 MMOs for R5 = 19.103 KX. The experimental time series of MMOs and its phase portraits are shown in Fig. 14(a)
and (b). Besides, we identified 22 periodic MMOs for R5 = 54.942 KX, intermittent periodic MMOs for R5 = 54.823 KX and intermittent chaotic MMOs for R5 = 54.720 KX, its time series are shown in Fig. 15(a)-(c). Finally, we noticed transient chaos experimentally, by fixing the circuit parameters as R5 = 200 KX, R6 = 15 KX, R7 = 833 X, F = 0.536 V and X = 67 Hz. The experimental results of transient chaos so obtained are shown in Fig. 16. It is clear that from the Fig. 16, the system is in the chaotic state at the beginning and it transits to the periodic state after some finite interval of time. The experimental observations are in good agreement with the numerical simulation results.
7. Conclusion In this paper, we have identified two different transitions of bursting oscillations and spiking trains on varying two different control parameters in the forced Liénard system. The emergence of BOs was explained based on its stability analysis. The system exhibits period doubling MMOs route to chaos on gradually decreasing the control parameter values, whereas it shows periodic and intermittent periodic MMOs for a gradual increase of the control parameter. Also, we have observed transient chaos for the suitable choice of the system parameters which is confirmed by finite time Lyapunov exponent, return map, and phase analysis. For an experimental study, the analog circuit designed for the forced Liénard system to validate our numerical simulation results. The experimental observations are in good agree with numerical results. Our investigations would help to understand the different dynamical transitions of neuronal systems. This study can be
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Fig. 16. Experimental time series
11
v 1 (t) of transient chaos.
Fig. 17. (a) ISI bifurcation diagram in the (b – ISI) plane and its corresponding (b) Lyapunov exponents in the (b – k1;2 ) plane.
extended to understand the coupled dynamics using various coupling schemes and different network topologies.
the University Grants Commission (UGC) for the financial assistance through UGC BSR RFSMS scheme. K. Thamilmaran acknowledges DST-PURSE, Govt. of India for financial support.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments S. Dinesh Vijay acknowledges Bharathidasan University for providing University Research Fellowship. S. L. Kingston acknowledges
Appendix A In this appendix, we explain a different transition of BOs with the effect of strength of nonlinearity b of the system. The system parameters are fixed as a = 0.12, c = 0.63, A = 0.55, x = 0.063 and varying b 2 (0.001,2). When the value of b is changed in the range (0.001 6b 62), we identified a transition from spiking trains to chaotic BOs via periodic BOs. To identify the existing regions of
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Fig. 18. Numerical results: (i) time series yðtÞ and (ii) phase portraits in the ðx yÞ planes. (a) spiking trains for b = 0.12, (b) periodic BOs for b = 0.2 and (c) chaotic BOs for b = 1.8.
Fig. 19. Transformed plot of phase portraits of BOs in the (d x) plane which is superimposed with the S-shaped stability bifurcation diagram.
spiking trains and different BOs, we plot the inter spike interval (ISI) bifurcation diagram is shown in Fig. 17(a). It can be obtained from the ISI bifurcation diagram of Fig. 17(a), the system exhibits the spiking trains for the control parameter b value in the range (0.120 6b 60.127). After spiking trains, the system shows successive period adding sequence of BOs for varying b in the range (0.128 6b 61.476). Finally, it reaches to chaotic BOs
for the value of b > 1.477. These transitions are confirmed by using the Lyapunov exponent spectrum in the (b – k1;2 ) plane as shown in Fig. 17(b). In Fig. 17(b), the negative largest Lyapunov exponent values signify the existence of spiking trains and periodic BOs, whereas its positive values reveal that the chaotic BOs state. Fig. 18 shows, (i) the time series of yðtÞ and (ii) its corresponding phase portraits in the (x y) planes for
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spiking trains (Fig. 18(a)), periodic BOs (Fig. 18(b), and chaotic BOs (Fig. 18(c)). The emergence of BOs proved by plotting the transformed plot of phase portraits of BOs in the (d x) plane and its stability bifurcation diagram as shown in Fig. 19. When the trajectory enters into the limit point LP2, it attracted to SF+ by fold bifurcation. The trajectory oscillate in SF+ at some interval of time and switches into SF region through the limit point LP1, this dynamical process is repeated as the dynamical evolution of bursting oscillations.
Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.aeue.2019.152898.
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[19] Han X, Wei M, Bi Q, Kurths J. Obtaining amplitude-modulated bursting by multiple-frequency slow parametric modulation. Phys Rev E 2018;97 (1):012202. [20] Han X, Yu Y, Zhang C, Xia F, Bi Q. Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings. Int J Nonlin Mech 2017;89:69–74. [21] Izhikevich EM. Simple model of spiking neurons. IEEE T Neural Networ 2003;14(6):1569–72. [22] Bi Q, Li S, Kurths J, Zhang Z. The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures. Nonlinear Dynam 2016;85(2):993–1005. [23] Upadhyay RK, Mondal A, Teka WW. Mixed mode oscillations and synchronous activity in noise induced modified Morris–Lecar neural system. Int J Bifurcat Chaos 2017;27(05):1730019. [24] Babak V, Kouhnavard M, Elbasiouny SM. Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions. Plos One 2017;12 (6):1–20. [25] Desroches M, Krauskopf B, Osinga HM. Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system. Chaos 2008;18 (1):015107. [26] Kingston SL, Thamilmaran K. Bursting oscillations and mixed-mode oscillations in driven Liénard system. Int J Bifurcat Chaos 2017;27 (07):1730025. [27] Brøns M, Kaper TJ, Rotstein HG. Introduction to focus issue: Mixed mode oscillations: Experiment, computation, and analysis. Chaos 2008;18 (1):015101. [28] Chakraborty S, Dana SK. Shilnikov chaos and mixed-mode oscillation in Chua circuit. Chaos 2010;20(2):023107. [29] Kingston SL, Suresh K, Thamilmaran K. Mixed-mode oscillations in memristor emulator based Liénard system. AIP Conf. Proc., vol. 1942. AIP Publishing; 2018. p. 060008. [30] Koper MT, Gaspard P. The modeling of mixed-mode and chaotic oscillations in electrochemical systems. J Chem Phys 1992;96(10):7797–813. [31] Keplinger K, Wackerbauer R. Transient spatiotemporal chaos in the MorrisLecar neuronal ring network. Chaos 2014;24(1):013126. [32] Lai YC, Tél T. Transient chaos: complex dynamics on finite time scales, vol. 173. London: Springer; 2011. [33] Tél T. The joy of transient chaos. Chaos 2015;25(9):097619. [34] Zou HL, Li M, Lai CH, Lai YC. Origin of chaotic transients in excitatory pulsecoupled networks. Phys Rev E 2012;86(6):066214. [35] de Paula AS, Savi MA, Pereira-Pinto FHI. Chaos and transient chaos in an experimental nonlinear pendulum. J Sound Vib 2006;294(3):585–95. [36] Ni X, Lai YC. Transient chaos in optical metamaterials. Chaos 2011;21 (3):033116. [37] Sabarathinam S, Thamilmaran K. Transient chaos in a globally coupled system of nearly conservative Hamiltonian Duffing oscillators. Chaos Solitons Fract 2015;73:129–40. [38] Hoff A, da Silva DT, Manchein C, Albuquerque HA. Bifurcation structures and transient chaos in a four-dimensional Chua model. Phys Lett A 2014;378 (3):171–7. [39] Scott SK, Peng B, Tomlin AS, Showalter K. Transient chaos in a closed chemical system. J Chem Phys 1991;94(2):1134–40. [40] Duarte J, Januário C, Martins N, Sardanyés J. On chaos, transient chaos and ghosts in single population models with Allee effects. Nonlinear Anal Real World Appl 2012;13(4):1647–61. [41] Yang XS, Yuan Q. Chaos and transient chaos in simple hopfield neural networks. Neurocomputing 2005;69(1–3):232–41. [42] Zhang LH, Wang Y. A note on periodic solutions of a forced Liénard-type equation. ANZIAM J 2010;51(3):350–68. [43] Kingston SL, Thamilmaran K, Pal P, Feudel U, Dana SK. Extreme events in the forced Liénard system. Phys Rev E 2017;96(5):052204. [44] Mishra A, Hens C, Bose M, Roy PK, Dana SK. Chimeralike states in a network of oscillators under attractive and repulsive global coupling. Phys Rev E 2015;92 (6):062920. [45] Ermentrout B. Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. Philadelphia: SIAM; 2002. [46] Shimizu K, Saito Y, Sekikawa M, Inaba N. Complex mixed-mode oscillations in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation. Physica D 2012;241(18):1518–26.
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Regular paper
Different transitions of bursting and mixed-mode oscillations in Liénard system S. Dinesh Vijay, S. Leo Kingston, K. Thamilmaran ⇑ Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, Tamil Nadu 620024, India
a r t i c l e
i n f o
Article history: Received 19 June 2019 Accepted 31 August 2019
Keywords: Bursting oscillations Spiking trains Mixed-mode oscillations Transient chaos Liénard system Electronic circuit
a b s t r a c t In this paper, we present the emergence of bursting oscillations, mixed-mode oscillations and transient chaos in the forced Liénard system. The system exhibits intriguing dynamical transitions such as from chaotic bursting oscillations to spiking trains via periodic bursting oscillations, period doubling mixedmode oscillations route to chaos, and intermittent mixed-mode oscillations on variation of the nonlinear damping coefficient as the control parameter. In addition, the system transits from spiking trains to chaotic bursting oscillations through successive period adding sequences of bursting oscillations upon variation of the strength of nonlinearity. Besides the system exhibits transient chaos also. We believe that it is for the first time that these multiplicity of dynamical behaviors and transitions are reported in a single model. We validate our numerical investigations with the help of experimental observations using a simple nonlinear electronic circuit implementation of the model. Ó 2019 Elsevier GmbH. All rights reserved.
1. Introduction Studies on neuronal dynamics and their different dynamical transitions have received great attention in the past few decades. Many researchers have applied nonlinear theory successfully to explore the diversity of excitability, spiking, and bursting mechanism of neurons [1,2]. It has been found that when two neurons communicate information among themselves, electrical pulses called action potential are exchanged between them. The collection of action potentials of the neurons are known as spiking trains [1–3]. The sequence of continuous firing (spiking) states alternating with the quiescent states is called Bursting Oscillations (BOs) [4,5]. Understanding the dynamics of bursting and spiking trains is one of the major interesting areas in the field of nonlinear dynamics, since it has potential applications in neurobiology, biophysics, chemistry and other disciplines [6]. The first work in this field was the Hodgkin-Huxley neuron model [7], but later on different models were put forward to account for the potential mechanism of signal processing and mode transitions of electrical activities of neurons [8–12]. The mode of electrical activities of neurons and its multiple transitions are explained using suitable electromagnetic radiation or force [8]. In the latter case, the mode
⇑ Corresponding author. E-mail addresses: [email protected] (S. Dinesh Vijay), kingston. [email protected] (S. Leo Kingston), [email protected] (K. Thamilmaran). https://doi.org/10.1016/j.aeue.2019.152898 1434-8411/Ó 2019 Elsevier GmbH. All rights reserved.
transitions of neuron under the influence of low–high electromagnetic radiation with white noise have been reported [9]. Similarly, a new physical neuron model is proposed in [12] using charge and flux controlled memristor to describe the time varying electromagnetic field. Further, collective behavior of neuronal networks with different topologies such as, synchronization transitions and pattern selections are used to understand the occurrence of neuronal diseases [13]. In recent years, several oscillator-like neuron models are reported in the literature to understand the different spiking and bursting properties as well as their distinct dynamical transitions [14–20]. Especially, BOs reported in the variety of neuronal models [4,21], single as well as multiple time-scale periodically excited systems, fast-slow subsystems of both autonomous [14,15] and nonautonomous systems [16–18]. In nonautonomous systems, the BOs occur owing to the interaction between the large internal frequency of the system and small external forcing frequency. There are several reports in the literature, mainly focused on the emerging mechanism of BOs. [15–17,21,22]. Yet a clear picture of these transitions and their related mechanisms have not been well understood so far. Similarly, Mixed-Mode Oscillations (MMOs) are another kind of bursting oscillations which appear as an alternate sequence of the large and small amplitude of oscillations. These MMOs are denoted the notation as Ls, where L and s are the large and small amplitude of oscillations, respectively. The existence of MMOs in different
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model systems and their emerging mechanism have been reported in [23–27]. There are few predominant sequences were established for the existence of MMOs in the literature, such as an alternate sequence of periodic MMOs and chaotic state [3,26,28], successive period adding sequence of MMOs [29] and Farey sequences [30]. Besides, another interesting complex phenomenon so-called transient chaos (TC) have been studied both numerically and experimentally in the literature [31,32]. In this dynamics, the system exhibits chaotic state for a finite interval of time and switches over to periodic oscillations in the asymptotic limit [32]. Transient behavior plays a vital in the brain functionality [31] and is also used for understanding the transient and extreme events in nervous systems [33,34]. The experimental observation of transient chaos has been reported such as in nonlinear pendulum [35], optical metamaterials [36], electronic circuits [37,38], closed chemical systems [39], population model [40] and simple Hopfield neural networks [41].
1.1. Motivation From the preceding paragraphs we find that there are plenty of models, each of which could simulate and describe only some particular aspects of the neuronal dynamics and a few of the dynamical transitions shown by the neurons. While these models are by themselves elegant, the following questions arises, namely (i) Is it possible to have a single mathematical model which can adequately describe almost all of these known behaviors? (ii) Will this model lend itself to easy mathematical analysis and numerical simulations? (iii) Can such a model be implemented experimentally so as to observe these behaviors in real time? It is in trying to find out answers to these questions that we chanced upon the Liénard model to describe neural systems. In fact this is a continuation of the earlier work of Leo Kingston et al., Ref. [26], wherein the authors reported for the first time that BOs and MMOs are indeed present in Liénard system and that this system can be implemented experimentally also. The general form of Liénard system proposed by the French mathematical physicist Liénard from the pioneering work of van der Pol model [42]. This Liénard system is a paradigmatic model for the different physical, engineering as well as biological systems, and it exhibits several intriguing complex dynamics [26,29,43,44]. It has been used to understand different bifurcation theories, various biological processes, and signal processing methods [42]. In this present study, we have observed that all the possible electrophysical activities of neurons can be observed in this single system, namely the Liénard system by a proper choice of system
parameters. To the best of our knowledge we believe that, perhaps for the first time such a multiplicity of dynamical behaviours has been observed in a single system. The structure of this paper presented as follows. The proposed model of the forced Liénard system and their detailed stability analysis have been explained in Section 2. The emergence of periodic and chaotic bursting oscillations, spiking trains are explored in Section 3. Section 4 deals with the existence of period doubling mixed-mode oscillations (MMOs) route to chaos as well as intermittent MMOs in the considered model. The system also found to possess transient chaos for the precise choice of system parameters which denoted in Section 5. Experimental circuit realization of the forced Liénard system and the obtained experimental results are given in Section 6. The final results presented in the concluding Section 7. 2. Model system We consider the following form of Liénard system with external sinusoidal forcing,
€x þ ax2 x_ þ cx þ bx3 ¼ AsinðxtÞ:
ð1Þ
For numerical study, the Eq. (1) is written as the following form of coupled two first order equations
x_ ¼ y y_ ¼ ax2 y cx bx3 þ AsinðxtÞ:
ð2Þ
Here, a; c, and b represent the nonlinear damping constant, internal frequency and strength of nonlinearity, whereas A and x are amplitude and frequency of the external sinusoidal forcing term. Eqs. (1) and (2) satisfy the general form of Liénard system [42]. For the suitable choice of the system parameters of Eq. (2), we identified periodic and chaotic BOs, spiking trains, periodic MMOs, intermittent MMOs, and transient chaos. 2.1. Stability analysis In order to find the stability of the system of Eq. (2), we set the external forcing term AsinðxtÞ to zero and fix the values of the system parameters as a = 0.25, b = 0.5 and c = 0.5. For this choice of parameters, the unforced system (Eq. (2)) has two different equilibria such as, stable foci (SF) (blue) on the (1; 0) planes in phase space and a saddle (Sd) (black) at the origin which are shown in Fig. 1(a). We analyzed the stability of bifurcation by perturbing the system. For this we used the standard continuation package XPPAUT AUTO [45]. The bifurcations of equilibrium points were observed for every cycle of oscillation (AsinðxtÞ, modð2pÞ). The
Fig. 1. (a) Equilibrium points of the Liénard system: the saddle (Sd) at the origin and stable foci (SF) on the (1; 0) planes. (b) Stability bifurcation diagram in the (A x) plane: here SF+ and SF denote the stable foci on the planes perpendicular to the x-axis at x ¼ 1 respectively and Sd represent the saddle points. LP1 and LP2 are the turning points for the transitions from saddle to stable foci.
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bifurcation diagram obtained is a S-shaped curve as shown in Fig. 1 (b). In this Fig. 1(b), the filled blue circles and hollow black circles represent the stable foci (SF) and saddle (Sd) regions respectively. The limit points LP1 and LP2 are the two turning points in the stability bifurcation diagram. As compared with the earlier work [26], the system exhibits different stable behaviors leading to a rich variety of transitions as the parameters of the system are varied. However, we are interested in the study of the dynamical transitions of the system that occur when the nonlinear damping coefficient (a) and strength of nonlinearity (b) are varied. When these parameters are varied, we find the system transits from saddle equilibrium state (Sd) to stable foci (SF) via stable node (SN). This is shown clearly in the two parameter phase diagram in the (b a) plane shown in Fig. 2. 3. Bursting oscillations and spiking trains
0.1370 6a 61.283. When the nonlinear damping (a) is increased further beyond 1.283, the system transits to spiking trains behavior and finally it exhibits regular periodic motion. The inserts of Fig. 3(a) and (b) show the magnified regions of chaotic BOs. The various dynamical behaviors for the system are shown in Fig. 4. In this figure, the typical time series for the variable yðtÞ for a(i) chaotic BOs (a = 0.12), b(i) periodic BOs (a = 0.25) and c (i) spiking trains (a = 1.5) are shown. The corresponding phase portraits in the (x y) plane are shown in figures a(ii), b(ii) and c(ii) respectively. Among the BOs, the existence of chaotic BOs and periodic BOs are identified by plotting the return map of ISI. The return map of ISI depicted in 5(a) shows the randomly distributed points for chaotic BOs. However, the regularly distributed return map of ISI (Fig. 5(b)) portrays the periodic BOs. 3.1. Emerging mechanism of bursting oscillations
When the system parameters were fixed as b = 0.5, c = 0.5, A = 0.2, x = 0.05 and the nonlinear damping coefficient a was varied in the range 0.0 6a 61.8, we noticed Bursting Oscillations (BOs) and spiking trains. In the BOs region we observed chaotic BOs and periodic BOs. These were followed by spiking trains. Bursting Oscillations refer to alternate sequence of continuous firing and resting states. If this sequence is chaotic, then the behavior is called chaotic BOs, while if the sequence is periodic, it is called as periodic BOs. If continuous firing alone occurs, the behavior is called spiking trains. In order to locate the regions of different bursting oscillations and spiking trains, we plotted inter spike interval (ISI) bifurcation diagram in the (a – ISI) plane as shown in Fig. 3(a). The inter spike interval was calculated by finding the interval between two consecutive spikes. The corresponding Lyapunov spectrum was plotted in the (a k1;2 ) plane as shown in Fig. 3(b). In this Fig. 3 (b), the first Lyapunov exponent is represented by solid blue line while the second Lyapunov exponent is represented by dashed red line. From this diagram, we find the system exhibits chaotic BOs for the lower range of a 2 (0, 0.1369) as evidenced by the positive value for the largest Lyapunov exponent. When the nonlinear damping is increased further, that is for a > 0.1369, the system exhibits periodic bursting oscillations for the range of a as
2
In this subsection, we explain the emergence of bursting oscillations in the forced Liénard system of Eq. (2). Let us assume the external forcing to be denoted by d such that d ¼ A sinðxtÞ. The phase portrait of the Liénard system was obtained in the (d x) plane. This was superimposed on the stability bifurcation diagram in the (A x) plane of Fig. 1(b). The resultant diagram is shown in Fig. 6. Here in this figure, when the trajectory approaches the turning point LP2 , it gets attracted to the stable focus (SFþ ) by fold bifurcation. After evolving for sometime, the trajectory reaches the turning point LP1 and then it switches to the stable focus (SF ). This process continues to exist in the BOs. Furthermore, we observed another interesting transition, that is, from spiking trains to chaotic bursting via periodic BOs for another set of system parameters with the strength of nonlinearity (b) as the control parameter. For more details kindly refer Appendix. 4. Mixed-mode oscillations Mixed-Mode Oscillations (MMOs) refer to the existence of an alternating sequence of large amplitude oscillations and small
SN
α
1.5 SF
Sd
1 0.5 0 0
0.5
1
β
1.5
2
Fig. 2. Two parameter phase diagram in the (b – a) plane for the fixed system parameter c = 0.5 showing the transition from saddle (Sd) to stable foci (SF) via saddle node (SN).
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Fig. 3. (a) Inter spike interval (ISI) bifurcation diagram in the (a – ISI) plane and (b) the corresponding Lyapunov spectra in the (a – k1;2 ) plane.
Fig. 4. (i) Time series of yðtÞ and its corresponding (ii) phase portraits in the (x y) planes. (a) chaotic BOs for a = 0.12, (b) periodic BOs for a = 0.25 and (c) spiking trains for a = 1.5.
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Fig. 5. Inter spike interval return map in the (ISIn – ISInþ1 ) planes: (a) with randomly scattered phase points denoting chaotic BOs for a = 0.12 and (b) regular distribution phase points denoting periodic BOs for a = 0.25.
Fig. 6. The phase portrait of BOs in the (d x) plane superimposed on the S-shaped stability bifurcation diagram of Fig. 1(b) showing BOs.
amplitude oscillations. For example they have been found to occur in BVP oscillator Ref. [46]. Here, the number of large amplitude and small amplitude oscillations are generally denoted by (Ls ), where L stands for the period of the large amplitude oscillations and the superscript s stands for the period of small amplitude oscillations. In this section, we report the existence of Mixed-Mode Oscillations (MMOs) in the Liénard system. In particular, we have found the existence of successive period doubling MMOs leading to chaos and periodic intermittent MMOs. For identifying these, we fix the system parameters as b = 0.33, c = 0.7, A = 0.7 and x = 0.45 and varied the control parameter, namely the nonlinear damping coefficient a in the range a 2 (0.0001, 0.06). The results obtained are explained in the following subsections.
doubling is shown in Fig. 7. While the time series and phase por-
4.1. Period doubling mixed-mode oscillations route to chaos
4.2. Intermittent and chaotic mixed-mode oscillations
The system exhibits a period doubling mixed-mode oscillations route to chaos for the control parameter a in the range 0.06 Pa P0.04548. When the control parameter a is gradually decreased
In addition to the period doubling MMOs route to chaos, the system also exhibits an alternate sequence of periodic MMOs and chaotic states. At the beginning for the value of a = 0.0001, the system is in the chaotic states. When we gradually increase the value of a, it becomes periodic MMOs via saddle-node bifurcation [26]
from 0.06, the 11 MMOs bifurcate to 22 MMOs. This transition of the system from 11 ðLs Þ to 22 ðLs Þ mixed-mode oscillations via period
traits for 11 ðLs Þ mixed-mode oscillations are shown in Fig. 7 a(i) and a(ii), the corresponding diagrams for 22 ðLs Þ mixed-mode oscillations are shown in Fig. 7 b(i) and b(ii) respectively. On further reduction of the a value, the system attains chaotic states via successive period doubling sequence of MMOs. This is shown in the one parameter bifurcation diagram of Fig. 8(a). The insert of Fig. 8(a) is a blown up portion of the bifurcation diagram showing clearly the period doubling MMOs route to chaos. The existence of this period doubling MMOs to chaos is corroborated by the Lyapunov spectrum which is shown in Fig. 8(b). In Fig. 8(b), the largest Lyapunov exponent is represented by solid blue line which shows negative values for periodic MMOs and positive values for chaotic states.
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Fig. 7. Time series for the variable xðtÞ showing a(i) 11 mixed-mode oscillations for a = 0.057 and b(i) 22 MMOs for a = 0.05254. The corresponding phase portraits in the (x y) plane are shown in a(ii) and b(ii) respectively.
Fig. 8. (a) One parameter bifurcation diagram in the (a x) plane and its corresponding (b) Lyapunov spectrum in the (a k1;2 ) plane. The largest Lyapunov exponent is represented by solid blue line which shows negative values for periodic MMOs and positive values for chaotic states.
for the value of a = 0.00031. This periodic states continue to exist until a = 0.01797 and becomes chaotic via intermittency route.
5. Transient chaos
Further increasing the value of a the chaotic state transits to 22 MMOs via saddle-node bifurcation for a = 0.01820, this is shown in Fig. 9(a). Upon increasing the control parameter, the system exhibits periodic and chaotic intermittent MMOs for a = 0.01822, and a = 0.01824 respectively as shown in Fig. 9(b) and (c). These
In this section, we explain the existence of Transient Chaos (TC) phenomenon in the forced Liénard system of Eq. (2). Transient chaos refers to a dynamic behavior, wherein the system exhibits chaos for a long interval of time and finally settles down to a periodic state for a fixed choice of system parameters. To observe transient chaos in our system, we set the values of the system parameters as a = 0.005, b = 0.12, c = 0.63, x = 0.053 and A = 0.55. The time series for the variable xðtÞ showing TC for this choice of system parameters is shown in Fig. 10(a). In this dynam-
22 periodic and chaotic intermittent MMOs alternately exist for the range of control parameter 0.01798 6a 60.02214. Presumably these were attributed as arising due to crisis induced intermittency.
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Fig. 9. Poincaré map of the time series xðtÞ of mixed-mode oscillations: (a) periodic MMOs (22 ) for a = 0.01820, (b) intermittent period MMOs (22 ) for a = 0.01822 and (c) intermittent chaotic MMOs for a = 0.01824.
ics, the system transit from aperiodic state to regular steady state. In Fig. 10(a), we have plotted only peaks of the time series for the clear visualization with the blue dots representing the aperiodic oscillations and red dots representing the asymptotic steady state motion. In general, the Poincaré return map is used to characterize the existence of periodic and nonperiodic states of the different complex systems. Here we have taken the state variable xðtÞ and found its return map which is as shown in Fig. 10(b). The widely scattered blue dots illustrate the aperiodic motion of the system, whereas the single large red point depicts the steady state of the system, since the asymptotic steady state of the system is periodic. Further, we confirm the emergence of transient chaos in the forced Liénard system using the statistical tools such as Finite Time Lyapunov Exponent (FTLE) and phase ð/Þ analysis. The FTLEs calculated for the fixed system parameters value are shown in Fig. 11(a).
Here the solid blue line denotes the first Lyapunov exponent and the dashed red line denotes the second Lyapunov exponent of the system for a finite interval of time. It can be noticed from the FTLE measurement that, as the time increases, the first Lyapunov exponent switches from a positive value to a negative value. The positive values of maximum Lyapunov exponents represent the chaotic state and its negative value denotes the periodic state. Hence the FTLE is a statistical measurement that confirms the emergence of transient chaos phenomena in the Liénard system. Besides, the phase (/) of the system state variable xðtÞ was calculated using the Hilbert transformation method Ref. [37] and is shown in Fig. 11(b). The phase ð/Þ of the system gradually increases as time progresses and after some finite interval of time, the phase (/) reaches a constant value. This clearly shows the existence of the chaotic and periodic state in the system.
Fig. 10. (a) Peaks of the time series xðtÞ of transient chaos shows an aperiodic (blue) and periodic (red) states and (b) Poincaré return map in the (xn – xnþ1 ) plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 11. (a) Finite time Lyapunov exponent and (b) phase (/) analysis of transient chaos for the fixed system parameters a = 0.005, b = 0.12, c = 0.63, x = 0.053 and A = 0.55.
Fig. 12. Experimental circuit realization of forced Liénard system.
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Fig. 13. Experimental results: (i) time series of v 2 (t) and its corresponding (ii) phase portraits in the (v 1 – v 2 ) planes, (a) chaotic BOs, (b) periodic BOs and (c) spiking trains.
Fig. 14. Experimental time series
v 1 (t) and its corresponding phase portrait (v 1 v 2 ) plane: (a) 11
6. Circuit realization The analog circuit realization for the forced Liénard system of Eq. (2) is shown in Fig. 12. The circuit consists of linear resistors, capacitors, operational amplifiers, analog device multipliers and external sinusoidal voltage source (f ðtÞ ¼ F sin Xt). The output voltages v 1 and v 2 are measured across the capacitors C 1 and C 2 respectively. Applying the Kirchhoff’s law at the junctions A and B, we get the following circuit equations,
C1
dv 1 0:1v 1 v 22 v 2 0:01v 32 1 þ þ F sin Xt ¼ R1 dt R5 R6 R7
ð3Þ
C2
MMOs and (b) 22 MMOs.
dv 2 v1 ¼ : dt R2
ð4Þ
Combining and rearranging the Eqs. (3) and (4), we get following circuit equations 2
C 1 C 2 R2 R5
0:01
d
v2
dt
2
!
¼ 0:1C 2 R2 v 22
R5 3 R5 v þ F sin Xt: R7 2 R1
dv 2 R5 þ v2 dt R6
ð5Þ
ð6Þ
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Fig. 15. Experimental time series
v 1 (t) of (a) 22
periodic MMOs, (b) 22 periodic intermittent MMOs and (c) chaotic intermittent MMOs.
The circuit equation Eq. (5) is equivalent to the normalized equation of the forced Liénard system of Eq. (2). Here the rescaling parameters taken as t ! s = R1 C 1 t, x1 ¼ v 1 ; x2 ¼ v 2 ; a ¼ 0:1 RR15 ; b ¼ 0:01 RR17 , and c ¼ RR16 . For this choice of the circuit parameters, it exhibits periodic BOs, chaotic BOs, spiking trains, periodic MMOs and intermittent MMOs as well as transient chaos. 6.1. Experimental results For all of our experiment studies, we fixed the circuit parameters as C 1 = C 2 = 10 nF and R1 = R2 = R3 = R4 = 10 KX. We obtain different complex dynamics for the suitable choice of the other circuit parameters. In order to obtain the chaotic BOs, periodic BOs and spiking trains, we had chosen R6 = 19.98 KX, R7 = 197 X, F = 0.224 V, X = 56 Hz and varying R5 in the specific range 8.5 KX 6R5 6500 X. For R5 = 8.3 KX, the circuit exhibits chaotic BOs, the time series v 1 (t) and the corresponding phase portrait of chaotic BOs in the (v 1 v 2 ) plane are shown in Fig. 13a(i) and a(ii). On further decreasing the value of R5 (R5 = 4.026 KX), we noticed periodic bursting oscillation, its time series and phase portraits are represented in Fig. 13b(i) and Fig. 13b(ii). Upon decreasing R5 = 628 X the system reveals spiking trains which are shown in Fig. 13c(i) and Fig. 13c(ii). Further, we have experimentally observed the successive period doublings and intermittent mixed-mode oscillations for the suitable choice of fixed circuit parameters. For this we kept R6 = 14.252 KX, R7 = 310 X, F = 0.710 V, X = 498 Hz. For the control parameter value of R5 = 16.656 KX, we obtained 11 MMOs and it bifurcates to 22 MMOs for R5 = 19.103 KX. The experimental time series of MMOs and its phase portraits are shown in Fig. 14(a)
and (b). Besides, we identified 22 periodic MMOs for R5 = 54.942 KX, intermittent periodic MMOs for R5 = 54.823 KX and intermittent chaotic MMOs for R5 = 54.720 KX, its time series are shown in Fig. 15(a)-(c). Finally, we noticed transient chaos experimentally, by fixing the circuit parameters as R5 = 200 KX, R6 = 15 KX, R7 = 833 X, F = 0.536 V and X = 67 Hz. The experimental results of transient chaos so obtained are shown in Fig. 16. It is clear that from the Fig. 16, the system is in the chaotic state at the beginning and it transits to the periodic state after some finite interval of time. The experimental observations are in good agreement with the numerical simulation results.
7. Conclusion In this paper, we have identified two different transitions of bursting oscillations and spiking trains on varying two different control parameters in the forced Liénard system. The emergence of BOs was explained based on its stability analysis. The system exhibits period doubling MMOs route to chaos on gradually decreasing the control parameter values, whereas it shows periodic and intermittent periodic MMOs for a gradual increase of the control parameter. Also, we have observed transient chaos for the suitable choice of the system parameters which is confirmed by finite time Lyapunov exponent, return map, and phase analysis. For an experimental study, the analog circuit designed for the forced Liénard system to validate our numerical simulation results. The experimental observations are in good agree with numerical results. Our investigations would help to understand the different dynamical transitions of neuronal systems. This study can be
S. Dinesh Vijay et al. / Int. J. Electron. Commun. (AEÜ) 111 (2019) 152898
Fig. 16. Experimental time series
11
v 1 (t) of transient chaos.
Fig. 17. (a) ISI bifurcation diagram in the (b – ISI) plane and its corresponding (b) Lyapunov exponents in the (b – k1;2 ) plane.
extended to understand the coupled dynamics using various coupling schemes and different network topologies.
the University Grants Commission (UGC) for the financial assistance through UGC BSR RFSMS scheme. K. Thamilmaran acknowledges DST-PURSE, Govt. of India for financial support.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments S. Dinesh Vijay acknowledges Bharathidasan University for providing University Research Fellowship. S. L. Kingston acknowledges
Appendix A In this appendix, we explain a different transition of BOs with the effect of strength of nonlinearity b of the system. The system parameters are fixed as a = 0.12, c = 0.63, A = 0.55, x = 0.063 and varying b 2 (0.001,2). When the value of b is changed in the range (0.001 6b 62), we identified a transition from spiking trains to chaotic BOs via periodic BOs. To identify the existing regions of
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Fig. 18. Numerical results: (i) time series yðtÞ and (ii) phase portraits in the ðx yÞ planes. (a) spiking trains for b = 0.12, (b) periodic BOs for b = 0.2 and (c) chaotic BOs for b = 1.8.
Fig. 19. Transformed plot of phase portraits of BOs in the (d x) plane which is superimposed with the S-shaped stability bifurcation diagram.
spiking trains and different BOs, we plot the inter spike interval (ISI) bifurcation diagram is shown in Fig. 17(a). It can be obtained from the ISI bifurcation diagram of Fig. 17(a), the system exhibits the spiking trains for the control parameter b value in the range (0.120 6b 60.127). After spiking trains, the system shows successive period adding sequence of BOs for varying b in the range (0.128 6b 61.476). Finally, it reaches to chaotic BOs
for the value of b > 1.477. These transitions are confirmed by using the Lyapunov exponent spectrum in the (b – k1;2 ) plane as shown in Fig. 17(b). In Fig. 17(b), the negative largest Lyapunov exponent values signify the existence of spiking trains and periodic BOs, whereas its positive values reveal that the chaotic BOs state. Fig. 18 shows, (i) the time series of yðtÞ and (ii) its corresponding phase portraits in the (x y) planes for
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spiking trains (Fig. 18(a)), periodic BOs (Fig. 18(b), and chaotic BOs (Fig. 18(c)). The emergence of BOs proved by plotting the transformed plot of phase portraits of BOs in the (d x) plane and its stability bifurcation diagram as shown in Fig. 19. When the trajectory enters into the limit point LP2, it attracted to SF+ by fold bifurcation. The trajectory oscillate in SF+ at some interval of time and switches into SF region through the limit point LP1, this dynamical process is repeated as the dynamical evolution of bursting oscillations.
Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.aeue.2019.152898.
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