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Chaos, Solitons and Fractals xxx (2005) xxx–xxx www.elsevier.com/locate/chaos
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H.G. Enjieu Kadji
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, J.B. Chabi Orou b, R. Yamapi
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Accepted 11 November 2005
11 Abstract
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This paper deals with the nonlinear dynamics of the biological system modeled by the multi-limit cycles Van der Pol oscillator. Both the autonomous and non-autonomous cases are considered using the analytical and numerical methods. In the autonomous state, the model displays phenomenon of birhythmicity while the harmonic oscillations with their corresponding stability boundaries are tackled in the non-autonomous case. Conditions under which superharmonic, subharmonic and chaotic oscillations occur in the model are also investigated. The analytical results are validated and supplemented by the results of numerical simulations. 2005 Published by Elsevier Ltd.
20 1. Introduction
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Nonlinear oscillators have been a subject of particular interest in recent years [1–9]. This is due to their importance in many scientific fields ranging from physics, chemistry, biology to engineering. Among these nonlinear oscillators, a particular class contains self-sustained components such as the classical Van der Pol oscillator which serves as a paradigm for smoothly oscillating limit cycle or relaxation oscillations [3]. In the presence of an external sinusoidal excitation, it leads to various interesting phenomena such as harmonic, subharmonic and superharmonic oscillations, frequency entrainment [4], devil’s staircase in the behavior of the winding number [5], chaotic behavior in a small range of control parameters [5–7]. The generalization of the classical Van der Pol oscillator including cubic nonlinear term (so-called Duffing–Van der Pol or Van der Pol–Duffing oscillator) has also been investigated by Venkatesan et al. in Ref. [8]. They have shown that the model exhibits chaotic motion between two types of regular motion, namely periodic and quasiperiodic oscillations in the principal resonance region. They have also obtained a perturbative solution for the periodic oscillations and carried out a stability analysis of such solution to predict the Neimark bifurcation. In this paper, we consider another self-excited model namely a biological system based on the enzymes–substrates reactions in order
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P. Woafo
Laboratory of Nonlinear Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon b Institut de Mathe´matiques et de Sciences Physiques, BP 613, Porto-Novo, Be´nin c Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon
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Nonlinear dynamics and strange attractors in the biological system
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Corresponding author. Tel.: +237 932 93 76; fax: + 237 340 75 69. E-mail addresses:
[email protected] (H.G. Enjieu Kadji),
[email protected] (J.B. Chabi Orou),
[email protected] (R. Yamapi),
[email protected] (P. Woafo). 0960-0779/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.11.063
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to show up the behavior of such a system in the autonomous and non-autonomous states. The paper is organized as follows: in Section 2, we describe the biological model under consideration and derive the equations of motion. Section 3 deals with the harmonic oscillatory states of such a model in the autonomous and non-autonomous states using respectively the Lindsted’s perturbation method [9] and the harmonic balance method [1]. The stability boundaries of the forced harmonic oscillations are investigated using the Floquet theory [1,4]. In Section 4, light is shed on the superharmonic and subharmonic oscillations using the multiple time scales method [1]. Strange attractors and transition from regular to chaotic oscillations are tackled in Section 5 through numerical simulations. Conclusion is given in Section 6.
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40 2. Biological model and equations of motion
41 Coherent oscillations in biological systems are considered here through the case of an enzymatic substrate reaction 42 with ferroelectric behavior in brain waves model [10]. The following suggestions made by Frohlich [11,12] are taken as a 43 physical basis for a theoretical investigation. • When metabolic energy is available, long-wavelength electric vibrations are very strongly and coherently excited in active biological system. • Biological systems have metastable states with a very high electric polarization.
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The number of activated enzyme molecules N can be viewed here as the predator concentration and the substrate molecules S asthe prey population. From Eqs. (4) and (5), we derive the two following steady states (N0, S0) = (0, 0) and n . Perturbing these activated enzymes and substrate molecules around the nontrivial steady state lead ðN 0 ; S 0 Þ ¼ mCc ; mC us to obtain the equations
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These long range interactions may lead to a selective transport of enzymes and thus, rather specific chemical reactions may become possible. For this survey, let us consider a population of enzyme molecules of which N are in the excited polar state and R are not excited. We assume that S is the number of substrate molecules. Both the enzymes and the substrate show long range selective interactions which tend to increase their level by influx. Each transition from the non-polar (or weakly polar) ground state of enzyme to the highly polar excited state leads to the chemical destruction of a substrate molecule. Additionally, there are also spontaneous transitions from the excited to the ground (or weakly polar) state. It is assumed that the rate of increase of the activated enzymes is proportional to their own concentration N, to the rate of the unexcited enzymes R and to the number of the substrate molecules S. Therefore the system can be described by a system of nonlinear differential equations as follows: dN ¼ mNRS nN ; ð1Þ ds dS ¼ cS mNRS; ð2Þ ds dR ¼ nN mNRS kðR CÞ. ð3Þ ds m represents the strength of the nonlinear enzyme–substrate reaction, n the decay rate of excited enzymes to the ground (or weakly polar) state and c the range attraction of the substrate particles due to the autocatalytic reactions. k(R C) also comes from the long range interaction with C the equilibrium concentration of the unexcited enzymes molecules in the absence of the excited enzyme and substrate, i.e., when N = S = 0. One supposes that the equilibrium of the unexcited enzyme concentration is reached fastly in order to simplify the above nonlinear equations. Such a process is also called an adiabatic elimination of the fast variable. Thus both Eqs. (1) and (2) are reduced to the well-known Lotka– Voltera equations [13] dN ¼ mCNS nN ; ð4Þ ds dS ¼ cS mCNS. ð5Þ ds
de ¼ cg þ mCge; ds dg ¼ ne mCge; ds
ð6Þ
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100
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Since an electrical field F interacts with the polarization, it is also important to include its effect which consists of an internal field due to thermal fluctuations and an externally applied field on the excited enzyme. F does not need to be an electrical field necessary and can also represents for example external chemical influences (e.g., an input or an output of enzyme molecules through the transport phenomena). Therefore, adding both the chemical and the dielectric contribution finally lead us to the set of equations de 2 2 ¼ cg þ ðj2 eW e r2 Þe þ mCge þ F ðsÞ; ds ð8Þ dg ¼ ne mCge. ds 2 2 For small values of e and g, if one considers the development in series of the function eW e at the third order to take into account the effects of some nonlinear quantities provided from the excess of concentration of the activated enzymes and uses the following rescaling: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 3 j2 r2 N d t jW; l¼ x0 ¼ nc; ; t ¼ x0 s; x ¼ Ne; N¼ ; EðtÞ ¼ 2 F x0 j2 r2 x0 dt x0 5 7 ðj2 r2 Þ2 ; ðj2 r2 Þ; b¼ a¼ 18j2 162j
101 one have that the biological system is governed by the coming equation 102 104 €x lð1 x2 þ ax4 bx6 Þ_x þ x ¼ E cos Xt;
ð9Þ
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where an overdot denotes time derivative. The quantities a, b are positive parameters, l is the parameter of nonlinearity while E and X are respectively the amplitude and the frequency of the external excitation. The biological system modeled through Eq. (9) has been considered by Kaiser in Ref. [15]. He has emphasized that in the unforced case, the model is a multi-limit cycles oscillator (so-called the multi-limit cycles Van der Pol oscillator (MLC-VdPo)). Since that model has been introduced, just few aspects of his dynamics have been analyzed. Indeed, Kaiser and Eichwald have investigated additionally to the dominating scenarios bifurcation in the superharmonic region [16], the occurrence of a symmetry breaking crisis subsequent type III intermittency [17]. Our aim is to tackle some aspects of its dynamics which remain unsolved, both in the autonomous and non-autonomous cases, using analytical methods and numerical simulations.
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ð7Þ
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de 2 2 ¼ ðj2 eW e r2 Þe. ds
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where e and g are respectively the excess concentrations of activated enzymes and substrate molecules beyond their equilibrium values N0 and S0. From Frohlich ideas, we may suppose that in large regions of the system of proteins, substrates, ions and structured water are activated by the chemical energy available from substrate enzyme reactions. pffiffiffiffiffi Thus, chemical oscillations in the number of substrate and activated enzyme molecules with a very low frequency nc might be carried out around the equilibrium state [14]. This oscillation also represents an electric oscillation through the high dipole moment of the excited enzyme. The electric dipole moment of the excited enzyme is partially screened by the ions and the remaining polarization causes the system to display a tendency towards a ferroelectric instability. On the other hand, electric resistances against the system’s tendency to become ferroelectric also have to be accounted for and thus, give a contribution r2P viewed as a relaxation term. Assuming the macroscopic polarization P to be proportional to the time dependent number e of the excited enzyme molecules, a nonlinear dielectric contribution is obtained and given as follows:
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114 3. Harmonic oscillatory states 115 3.1. Autonomous oscillatory states 116 117 118 119 120 121 123
We consider in this subsection the case where the model is not influenced by an external force (E = 0) and our purpose is to find the amplitudes and frequencies of the limit cycles. Therefore, an appropriate analytical tool is Lindsted’s perturbation method [9]. In order to permit an interaction between the frequency and the amplitude, it is interesting to set s = xt where x is an unknown frequency. We assume that the periodic solution of Eq. (9) can be performed by an approximation having the form xðsÞ ¼ x0 ðsÞ þ lx1 ðsÞ þ l2 x2 ðsÞ þ ;
ð10Þ
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124 where the functions xn(s) (n = 0, 1, 2, . . .) are periodic functions of s of period 2p. Moreover, the frequency x can be 125 represented by the following expression: 126 128 x ¼ x0 þ lx1 þ l2 x2 þ ; ð11Þ
134 Order l1 135 137 x20€x1 þ x1 ¼ x0 ð1 x20 þ ax40 bx60 Þ_x0 2x0 x1€x0 þ x0 x40 x_ 0 ða bx20 Þ. 2
138 Order l
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129 where xn (n = 0, 1, 2, . . .) are unknown constants at this point. Substituting the expressions (10) and (11) in Eq. (9) and 130 equating the coefficients of l0, l1 and l2 to zero, we obtain the following equations at different order of l: order l0 131 133 x20€x0 þ x0 ¼ 0. ð12Þ ð13Þ
x20€x2 þ x2 ¼ x0 ½ð1 x20 Þ_x1 2x0 x_ 0 x1 2x0 x1€x1 ðx21 þ 2x0 x2 Þ€x0 x1 ð1 x20 Þ_x0 þ x1 ða bx20 Þx40 x_ 1 þ x0 ½ða bx20 Þx40 x_ 1 þ ð4a 6bx20 Þx30 x_ 0 x1 .
140
ð14Þ
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146 Solving Eq. (12) and using conditions (15), it comes 147 149 x0 ¼ A cos s; 150 152 x0 ¼ 1;
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141 Making use of the property x(s + 2p) = x(s) and the initial condition x_ ð0Þ ¼ 0 to determine the unknown quantities in 142 the above equations, we get 143 145 xn ðs þ 2pÞ ¼ xn ðsÞ; x_ n ð0Þ ¼ 0; n ¼ 0; 1; 2. ð15Þ ð16Þ ð17Þ
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153 where A is the amplitude of the limit cycle. In virtue of the solution (16) and the relation (17), Eq. (13) leads to 154 5b 6 a 4 1 4 9b 7 3a 5 1 3 €x1 þ x1 ¼ A A þ A 1 A sin s þ 2x1 A cos s þ A A þ A sin 3s 64 8 4 64 16 4 5b 7 a 5 b 7 A A sin 5s þ A sin 7s. ð18Þ þ 156 64 16 64 157 From this latter equation, the secularity conditions (so called the solvability conditions) lead us to the following: 158 5b 6 a 4 1 2 A A þ A 1 ¼ 0; ð19Þ 160 64 8 4 163
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161 and x1 ¼ 0.
ð20Þ
164 Thus, a general expression for a periodic solution of Eq. (18) can be written as follows: x1 ¼ C cos s þ sin s þ W1 sin 3s þ W2 sin 5s þ W3 sin 7s;
167 where 169
W1 ¼
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1 9b 7 3a A þ A3 ; 32 16 4
W2 ¼
1 5b aA5 ; 384 4
ð21Þ
W3 ¼
b A7 3072
172
¼
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170 the initial condition x_ n ð0Þ, one now obtains 219 1 3 bA7 aA5 þ A3 . 3072 12 32
173 The value of C remains undetermined for the moment and will be determined in the next step. The secularity condition 174 for the solution x2(s) yields the following solutions: 176
C ¼ 0;
ð22Þ
177 and 179
x2 ¼
2 1580b 12 738ab 10 72a þ 309b 8 64a 219b 6 16a þ 3 4 3 A A þ A A þ A A2 . 393; 216 99; 024 768 6144 384 64
ð23Þ
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180 Therefore, the solution of Eq. (9) can be approximated by 182
xðtÞ ¼ A cos xt þ lð sin xt þ W1 sin 3xt þ W2 sin 5xt þ W3 sin 7xtÞ þ Oðl2 Þ;
183 where the frequency x is given by 184 186 x ¼ 1 þ l2 x2 þ Oðl3 Þ.
ð25Þ
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The amplitudes An (n = 1, 2, 3) of the limit cycles and their related frequencies x(An) are obtained by solving respectively Eqs. (19) and (25) via the Newton–Raphson algorithm. Depending to the values of the parameters a and b, Eq. (19) can give birth to one to one or three positive amplitudes which correspond respectively to the amplitudes of one or three limit cycles. In the case of three limit cycles, two are stable and one is unstable. For instance, with a = 0.144 and b = 0.005, the stable limit cycles have the following characteristics A1 = 2.6390 with the frequency x(A1) = 1.0011 and A2 = 4.8395 with x(A2) = 1.0545 while the unstable limit cycle is given by A3 = 3.9616 with x(A3) = 1.0114. Such a coexistence of two stable limit cycles with different amplitudes and frequencies (or periods) separated by an unstable limit cycle for a given set of parameters refer to as birhythmicity [18]. Therefore, birhythmicity provides the capability of switching back and forth, upon appropriate perturbation or parameter change, between two distinct types of stable oscillations characterized by markedly different periods (or frequencies) and amplitudes. Such a phenomenon is used to model glycotic oscillations in yeast and muscle [19,20]. The unstable limit cycle represents the separatrix between the basins of attraction of the two stable limit cycles. From Eq. (19), a map showing some regions where one or three limit cycles can be found has been constructed as shown in Fig. 1. The above stable limit cycles and their corresponding attraction basins are obtained from a direct numerical simulation of Eq. (9) using the fourth-order Runge–Kutta algorithm (see Fig. 2). The evolution of the amplitude of oscillations versus the parameter a for different values of the parameter b has also been drawn from Eq. (19) as shown in Fig. 3. It appears from that figure the occurrence of jump phenomenon which disappear with increasing b. Such situations can illustrate the explanation of the existence of multiple frequency and intensity windows in the reaction of biological systems when they are irradiated with very low weak electromagnetic fields [21–24].
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ð24Þ
Fig. 1. A limit cycles map showing some regions where one or three limit cycles can be found.
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Fig. 2. Phases portrait of the two stable limit cycles (a) and their corresponding basins of attraction (b) for l = 0.1, a = 0.144 and b = 0.05.
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206 3.2. Forced harmonic oscillatory states
207 3.2.1. Harmonic oscillatory states 208 Assuming that the fundamental component of the solutions has the period of the external excitation, we use the har209 monic balance method [1] to derive the amplitude of the forced harmonic oscillatory states (E 5 0) of Eq. (9). For this 210 purpose, we express its solution xs as 211 213 xs ¼ a1 cos Xt þ a2 sin Xt. ð26Þ
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70 (a) (b) (c) (d)
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0 0
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Fig. 3. Behavior of the amplitude of autonomous oscillatory states versus a for different values of b. (a) b = 104, (b) b = 103, (c) b = 5 · 103, (d) b = 0.1.
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214 Inserting Eq. (26) in Eq. (9) and equating the cosine and sine terms separately, we obtain 215 1 aX 4 5b 6 A A a2 ¼ E; ð1 X2 Þa1 lX 1 A2 þ 4 8 64 1 2 aX 4 5b 6 A A a1 þ ð1 X2 Þa2 ¼ 0; lX 1 A þ 217 4 8 64 218 where tan / ¼
lXð1 14 A2 þ aX A4 5b A6 Þ 64 8 X2 1
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A2 ¼ a21 þ a22 ;
ð27Þ
225 226 227 228 229 230 231 232 233 234 235
E2 ¼ 0. l2 X2
ð28Þ
We find the behavior of A when the frequency of the external excitation X is varied and the results are represented in Fig. 4 where the comparison between analytical and numerical response frequency curves A(X) of the model is shown. We provide in Fig. 5 the effects of E on the multi-limit cycles and it appears that as soon as the amplitude of the external excitation is different from zero, one of the two stable limit-cycles collapses. The physiological importance of such a situation can be explained by the fact that the membrane has stored a lot of energy which created the destruction of one of the two stale limit cycles. The remaining stable limit cycle displays resonance peaks which disappears as the amplitude E increases for some range of parameters. The phenomenon of destruction of one of the two stable limit cycles under the effect of an external field is of capital importance in biology. It is for example the case where one of the cycles is a pathological cycle whereas the second is a physiological limit cycle. In such a situation, the destruction of the pathological limit cycle under the effects of E, which can be represented by external chemical influences (e.g., an input or an output of enzyme molecules via the transport phenomena) is of utility.
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221 After some algebraic manipulations of Eq. (27), it comes that the amplitude A satisfies the following nonlinear algebraic 222 equation: ! 2 2 25b2 14 5abX 12 2a X þ 5b 10 2aX þ 5b 8 4aX þ 1 6 1 4 ð1 X2 Þ2 2 A þ A A þ A A þ 1þ A A 256 128 32 16 2 4096 l 2 X2
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Fig. 4. Comparison between analytical and numerical frequency–response curve A(X) with the parameters a = 0.10, b = 0.2, l = 0.1 and E = 1.0.
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Fig. 5. Effects of the external excitation F on the amplitude of oscillations with the parameters of Fig. 4. (a) E = 0.3, (b) E = 1.0, (c) E = 3.0.
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236 3.2.2. Stability of the harmonic oscillations 237 To examine local stability of the harmonic solution, we derive the linear variational equation from Eq. (9) around 238 the oscillatory states and obtain 239 €f lð1 x2 þ ax4 bx6 Þf_ þ ½1 þ lð2xs 4ax3 þ 6bx5 Þ_xs f ¼ 0; 241 ð29Þ s s s s s where xs is the oscillatory states defined by Eq. (26). The oscillatory states are stable if f tends to zero when the time goes up. With regard to the evolution of the biological system represented by xs, f can indicate an exogenic hormonal agent which forms the basis of the theurapeutic action. The therapy will consist to seek temporal laws to be followed by the exogenic hormone so that after a transitional phase, the state of the system remains more close to physiological behavior [25]. The appropriate analytical tool to examine the stability condition is the Floquet theory [1,4]. Setting t ¼ 2s , the variational Eq. (29) can be rewritten as X €f þ ðsÞf_ þ CðsÞf ¼ 0;
251 where
253
CðsÞ ¼
2l ½H þ I cosð4s 2/Þ þ J cosð8s 4/Þ þ K cosð12s 6/Þ; x
ð30Þ
4 ½1 þ L sinð4s 2/Þ þ M sinð8s 4/Þ þ Q sinð12s 6/Þ; x2
254 with
a 3b J ¼ A4 A6 ; 8 16 3lb 6 A. Q¼ 16
K¼
b 6 A; 32
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1 3a 5b 1 a 15b 6 H ¼ 1 A2 þ A4 A6 ; I ¼ A2 þ A4 A; 2 8 16 2 2 32 15b 4 lX 4 3b A ; A a A2 ; M¼ L ¼ lXA2 2 þ 2aA2 8 2 2
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ðsÞ ¼
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264
q¼
l 5b 6 3a 4 1 2 A A þ A 1 . X 16 8 2
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262 where
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257 To discuss further the stability process, we first transform Eq. (30) into the standard form by introducing a new variable 258 K as follows: 259 Z 1 s 0 0 fðsÞ ¼ KðsÞ exp ðs Þ ds 2 0 l J K I sinð4s 2/Þ þ sinð8s 4/Þ þ sinð12s 6/Þ ; ð31Þ ¼ KðsÞ exp qs þ 261 4x 2 3
269 with
ð32Þ
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1 l2 l2 4 2 2 2 2 ð2H 4 þ I þ J þ K Þ ; j ¼ ð2HI þ IJ þ JKÞ; j4s ¼ 2 ðL lXIÞ; 4c 2 2 2 X X X l2 I 2 4 l2 þ 2HJ þ IK ; j8s ¼ 2 ðM 2lXIÞ; j8c ¼ 2 j12c ¼ 2 ð2HK þ IJ Þ; X 2 X X 4 l2 J 2 l2 l2 þ IK ; j20c ¼ 2 JK; j12s ¼ 2 ðN 3lXKÞ; j16c ¼ 2 j24c ¼ 2 K 2 . X 2 X 2X X j0 ¼
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265 By coupling both Eqs. (30) and (31), it comes that K satisfies the following equation: 266 € þ ½j0 þ j4c cosð4s 2/Þ þ j4s sinð4s 2/Þ þ j8c cosð8s 4/Þ þ j8s sinð8s 4/Þ þ j12c cosð12s 6/Þ K þ j12s sinð12s 6/Þ þ j16c cosð16s 8/Þ þ j20c cosð20s 10/Þ þ j24c cosð24s 12/ÞK ¼ 0; 268
272 The Floquet theory [1,4] now leads us to seek a particular solution of Eq. (32) in the following form: 273 n¼þ1 X K ¼ expðsÞwðsÞ ¼ wn expðn sÞ; n ¼ þ 2jn; 275 n¼1
ð33Þ
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276 where w(s) = w(s + p) replaces the Fourier series and is the complex number while n is a constant. Substituting Eq. 277 (33) into Eq. (32) and equating each of the coefficients of exponential functions to zero yields the following homoge278 neous equation for wm: 279 1 1 1 ½2m þ j0 wm þ ½j4c jj4s expð2j/Þwm2 þ ½j4c þ jj4s expð2j/Þwmþ2 þ ½j8c jj8s expð4j/Þwm4 2 2 2
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1 1 1 þ ½j8c þ jj8s expð4j/Þwmþ4 þ ½j12c jj12s expð6j/Þwm6 þ ½j12c þ jj12s expð6j/Þwmþ6 2 2 2 1 1 1 1 þ j16c expð8j/Þwm8 þ j16c expð8j/Þwmþ8 þ j20c expð10j/Þwm10 þ j20c expð10j/Þwmþ10 2 2 2 2 1 1 þ j24c expð12j/Þwm12 þ j24c expð12j/Þwmþ12 ¼ 0. 2 2
281 282 283 284 285
ð34Þ
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1 ð2 þ j0 Þ 4 þ 2ðj0 þ 4Þ2 þ ðj0 4Þ2 ðj24c þ j24s Þ ¼ 0. 4
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For nontrivial solutions, the determinant of the matrix in Eq. (34) must vanish. But since the determinant is infinite, convergence considerations are taking into account by dividing Eq. (34) by j0 4m2. Considering only the central rows and columns of the Hill determinant [1], approximate solutions are obtained through the following approximate characteristic equation:
1
ð 2jÞ2 þ j0 0 ðj4c þ jj4s Þ expð2j/Þ
2
DðÞ ¼
ð35Þ
¼ 0; 0 2 þ j 0 0
2 1
ðj4c jj4s Þ expð2j/Þ 0 ð þ 2jÞ þ j0 287 2
291 Since the characteristic exponents are the solutions of D() = 0, there are three following possibilities of solutions as 292 follows:
296 297 298 299 300 301
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where D ¼ 16j0 þ 14 ðj24c þ j24s Þ. Once are known and either imaginary ¼ j or real ¼ ( real positive). We note that the stability of the harmonic solution (26) depends exclusively on the exponent of coefficient q (see Eq. (31)). Generally, if the real part of the quantity q is negative, the variation f goes to zero when the time goes up and therefore, the harmonic solution (26) is stable. If it is positive, the solution is unstable and therefore, f never tends to zero when the time increases, but has a bounded oscillatory behavior or goes to infinity. Let us consider the two following possibilities:
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pffiffiffi • ¼ j j0 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi • ¼ j j0 þ 4 þ D, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi • ¼ ðj0 þ 4Þ þ D,
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308
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302 • if ¼ , then the solution for f can be stable if q > 0 and q2 > 2, otherwise it is unstable. 303 • if ¼ j, the solution described by Eq. (26) is stable if q > 0 (see Eq. (31)) and unstable if q < 0. 304 305 Before discussing the form of instability in the case ¼ j, let us consider the first harmonic component in the Fou306 rier series of the function w(t) as follows: wðtÞ ¼ t1 cos Xt þ t2 sin Xt.
ð36Þ
309 Consequently, the general solution for f(t) within the unstable region can be written as 311 312 313 314 315
fðtÞ ¼ eqt ft1 cos½ðX þ Þt þ u1 þ t2 sin½ðX Þt þ u2 g;
ð37Þ
where t1, t2, u1 and u2 are constants. Therefore, the form of instability defined by ¼ j (for q < 0) results in a buildup of new harmonic components with the frequencies X þ and X , which are in general incommensurate with the frequency X of the periodic solution (26). Where as for q < 0 the solution is unstable and so that q = 0 is the boundary of the instability and stability region (see Fig. 4). That kind of instability can be interpreted as a Neimark instability
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CO
which gives rise to a Neimark bifurcation. It should be noticed that the Neimark bifurcation is expected at the frequency where the stable branch of the resonance curve crosses the critical boundary Ac = 1.282 and in this survey, it occurs for X = 0.47 and X = 1.33. As we have mentioned before, the solution described by Eq. (26) is stable if A > Ac and unstable if A < Ac. These results are observed in Fig. 4 following the comparison between analytical and numerical frequency–response curves A(X). The corresponding numerical critical boundary is An = 1.30 for X = 0.474 and X = 1.331. A very good agreement is obtained between analytical and numerical results. The behavior of the model in the Neimark instability region, at the boundary where the Neimark bifurcation occurs and in the domain of stable oscillatory states is reported in Fig. 6. Such kind of instability has been also obtained for the Van der Pol–Duffing oscillator [8].
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316 317 318 319 320 321 322 323 324
RR
Fig. 6. Behavior of the model during the stability process with the parameters of Fig. 5 and the initial conditions ðx0 ; x_ 0 Þ ¼ ð0; 0:1Þ. (a) in the Neimark instability region for X = 0.30, (b) at the first Neimark’s bifurcation boundary for X = 0.474, (c) in the region of stable oscillatory states for X = 0.70 (d) at the second Neimark’s bifurcation boundary for X = 1.33.
325 4. Superharmonic and subharmonic oscillations 326 327 328 329 330 331
Such types of oscillations are of interest in the problem of interaction between biological systems and electromagnetic waves. Therefore, depending on the frequency of coherent electromagnetic field which should be applied in order to influence the physico-chemical basis of biological function and order, subharmonic or superharmonic frequencies can be needed. Particularly, there is a great effect of the superharmonic oscillations on the homoclinic bifurcation. Indeed, adding a superharmonic lead first to the elimination of the homoclinic bifurcation and then as a consequence, to the elimination of unwanted scattered chaotic attractors [25]. Such oscillations often occur when the frequency of the
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332 external excitation is too far or close to the natural frequency of the device. It should be stressed that harmonic, super333 harmonic and subharmonic oscillations take place at different time scales. Thus the best tool to perform them is the 334 multiple time scales method [1]. In such a case, an approximate solution is generally seeking under the form 335 337 xðt; lÞ ¼ x0 ðT 0 ; T 1 Þ þ lx1 ðT 0 ; T 1 Þ þ ; ð38Þ
OO F
338 where the fast scale T0 and the slow scale T1 are associated respectively to the unperturbated system and to the ampli339 tude and phase modulations induced by the global first order perturbation. The time derivatives now becomes 340 d d2 ¼ D0 þ lD1 þ ; ¼ D20 þ 2lD0 D1 þ ð39Þ 342 dt dt2 343 with D0 ¼
o ; oT 0
D1 ¼
o ; oT 1
T n ¼ ln ;
n ¼ 0; 1; 2; . . . .
PR
345
346 Inserting expressions (38) and (39) into Eq. (9) and equating coefficients of like power of l, one obtains 347 Order l0 348 350 D20 x0 þ x0 ¼ E cos XT 0 . 351 Order l 352 354 D20 x1 þ x1 ¼ 2D0 D1 x0 þ ð1 x20 þ ax40 bx60 ÞD0 x0 .
D
1
ð40Þ
ð41Þ
355 After solving Eq. (40), we obtain the following general solution:
1 Þ expðjT 0 Þ þ N expðjXT 0 Þ; x0 ¼ AðT 1 Þ expðjT 0 Þ þ N expðjXT 0 Þ þ AðT
represents the complex conjugate of A and 358 where A N¼
E 2ð1 X2 Þ
ð42Þ
EC
360
TE
357
361 Substituting the general solution x0 into Eq. (41) yields 362 D20 x1 þ x1 ¼ jfC1 expðjT 0 Þ þ C2 exp½jXT 0 þ C3 exp½3jXT 0 þ C4 exp½jð2 XÞT 0 þ C5 exp½3jT 0 þ C6 exp½jð2
RR
þ XÞT 0 þ C7 exp½jð1 þ 2XÞT 0 þ C8 exp½jð1 2XÞT 0 Þ þ C9 exp½5jT 0 þ C10 exp½jð4 þ XÞT 0 þ C11 exp½jð4 XÞT 0 þ C12 exp½jð1 þ 4XÞT 0 þ C13 expð5jXT 0 Þ þ C14 exp½jð1 4XÞT 0 þ C15 exp½jð2 þ 3XÞT 0 þ C16 exp½jð3 þ 2XÞT 0 þ C17 exp½jð2 3XÞT 0 þ C18 exp½jð1 þ 6XÞT 0
CO
þ C19 exp½jð3 þ 4XÞT 0 þ C20 exp½jð5 2XÞT 0 þ C21 exp½jð6 XÞT 0 þ C22 exp½jð3 4XÞT 0 þ C23 exp½jð2 þ 5XÞT 0 þ C24 exp½jð4 5XÞT 0 þ C25 exp½jð4 þ 3XÞT 0 þ C26 exp½jð3 þ 2XÞT 0 þ C27 exp½jð4 3XÞT 0 þ C28 exp½jð5 þ 2XÞT 0 þ C29 exp½jð6 þ XÞT 0 þ C30 exp½jð2 5XÞT 0 þ C31 exp½jð1 6XÞT 0 þ C32 exp½7jT 0 þ C33 exp½7jXT 0 g þ C C;
ð43Þ
UN
364
365 where C Æ C denotes the complex conjugate of the previous terms while the coefficients Cl (l varies from 1 to 33) are given 366 in Appendix A. Many types of resonance occur from Eq. (43) but we are focussing our analysis on two particular cases: 367 the superharmonic and the subharmonic resonances which are displayed whenever X 13 and X 3 respectively. 368 4.1. Superharmonic resonances 369 In order to express the closeness of X to the internal (natural) frequency, we introduce the detuning parameter r0 370 according to 371 373 ð44Þ 3X ¼ 1 þ lr0 .
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13
374 Thus, additionally to the terms proportional to exp(±jT0), the one proportional to exp(±3jXT0) also contribute. There375 fore, the solvability condition is defined as 376 dA 2N2 A þ að12A2 AN 2 þ 2A3 A2 4XA2 AN 2 þ 6AN4 Þ b½20AN6 þ ð86 46XÞA2 AN 4 2 þ A A2 A dT 1 3 Þ XN3 b½9XN7 þ 88XAAN 5 þ ð78X AÞA2 AN 2 þ 5A4 A3 þ faXð3N5 þ 10AAN 3 g expðjr0 T 1 Þ ¼ 0. þ 60A3 AN ð45Þ
OO F
378 379 In polar coordinates, we let 380 1 A ¼ q exp½jhðT 1 Þ; 382 2
ð46Þ
PR
383 where q and h are real quantities and standing respectively for the amplitude and phase of the oscillator. After injecting 384 the expression (46) into Eq. (45), we separate real and imaginary parts and it comes the two following flows: 385 dq ¼ 11 q þ 12 q3 þ 13 q5 þ 14 q7 þ ð15 þ 16 q2 þ 17 q4 Þ cos U; dT 1 ð47Þ dU 1 þ 16 q2 17 q4 sin U; ¼ r0 5 387 dT 1 q 388 where
D
1 1 1 43 12 ¼ þ að3 XÞ b ð1 3aN4 þ 10bN6 Þ; X N2 N2 ; 2 8 2 2 5b 5a 2 4 3 ; 15 ¼ Xð1 3aN þ 9bN ÞN ; 22bN2 N3 ; 16 ¼ X 14 ¼ 128 2 390
TE
11 ¼
U ¼ r0 T 1 h.
13 ¼
1 ða 30bN2 Þ; 16
b 17 ¼ ð2 39XÞN3 ; 8
393
EC
391 For the steady-state motions, amplitude and phase are varied very slowly. Thus, one must have dq dU ¼ ¼ 0; dT 1 dT 1
RR
394 which corresponds to the singular points of Eq. (47) and yields 11 q þ 12 q3 þ 13 q5 þ 14 q7 ¼ ð15 þ 16 q2 þ 17 q4 Þ cos U; 396
r0 q ¼ ð15 þ 16 q2 þ 17 q4 Þ sin U.
ð48Þ
400
CO
397 Squaring and adding these equations give us the following nonlinear equation: 398 124 q14 þ 213 14 þ ð123 þ 212 14 Þq10 þ ð211 14 þ 212 14 126 Þq8 þ ð122 þ 211 13 216 17 Þq6 þ ð211 12 215 17 126 Þq4 þ ð121 215 16 þ r20 Þq2 125 ¼ 0.
ð49Þ
403 404 405 406 407 408
UN
401 Thus, the motion of the superharmonic oscillatory states is described by the following equation: xðtÞ ¼ q cosð3Xt þ UÞ þ N cos Xt þ OðlÞ.
ð50Þ
Eq. (49) is an implicit equation for the amplitude of the response q as a function of the detuning parameter r0 and the amplitude of the forcing term E: it is called the frequency–response equation. The Newton–Raphson algorithm is used to solve it in order to obtain the amplitude response curves q(E) which are plotted in Figs. 7 and 8 for three different values of the parameters a and b respectively. In both cases, the hysteresis phenomenon is observed and one can notice that the parameters a and b have a real effect on the amplitude of such oscillatory states.
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3.5
3
OO F
RHO
2.5
2
1.5
0.5
0 0
2
4
PR
1
6
E
8
10
TE
D
Fig. 7. Effects of the parameter a on the superharmonic amplitude–response curves for b = 0.5 and r0 = 0.05. (a) a = 0.1, (b) a = 2, (c) a = 5.
5
4 3.5
RR
3
RHO
(a) (b) (c)
EC
4.5
2.5
2
CO
1.5
1
0.5
UN
0 0
1
2
3
4
5
6
E
Fig. 8. Effects of the parameter b on the superharmonic amplitude–response curves for a = 0.5 and r0 = 0.05. (a) b = 0.03, (b) b = 0.5, (c) b = 50.
409 4.2. Subharmonic resonances 410 412
To analyze the subharmonic resonances, we introduce another detuning parameter r according to X ¼ 3 þ lr.
ð51Þ
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413 From Eq. (43), the condition under which the secular terms are now cancelled is given by dA 2AN2 þ að12A2 AN 2 þ 2A3 A 2 4XA2 AN þ 6AN4 Þ b½20AN6 þ 2ð43 23XÞA2 AN 4 þ A A2 A dT 1 2 N a½6ð2 XÞA 2 N3 þ 4ð1 XÞAA 3 N þ b½30ð2 XÞA 2 N5 2 þ 5A4 A 3 þ fð2 XÞA þ 60A3 AN
2
415
3
4
N3 þ ð26 15XÞA2 A NÞg expðjrT 1 Þ ¼ 0. þ 2ð56 27XÞAA
ð52Þ
418
OO F
416 Once more, we introduce the polar notation (46) and after some algebraic calculations, it comes that dq ¼ 11 q þ 12 q3 þ 13 q5 þ 14 q7 þ ð 1b5 q2 þ 1b6 q4 þ 1b7 q6 Þ cos v; dT 1 dv ð 1b q2 þ 1b6 q4 þ 1b7 q6 Þ sin v; ¼r3 5 dT 1 q
ð2 XÞð1 6aN2 ÞN 30bðX þ 2ÞN5 ; 4 ð26 15XÞbN ; v ¼ rT 1 3h. 1b7 ¼ 64 1b5 ¼
421
1b6 ¼
PR
419 where
ð53Þ
ð56 27XÞbN3 2að1 XÞ ; 8
D
422 For steady-state motions, we obtain after eliminating v via some algebraic manipulations the following equation: " 2
2
# ¼ 0.
CO
RR
EC
r2 9
UN
424
2 þ ð211 12 1b5 Þq2 þ 121 þ
TE
q2 14 q12 þ ð213 14 1b7 Þq10 þ ð123 þ 212 14 2 1b6 1b7 Þq8 þ ð211 14 þ 212 13 1b6 2 1b5 1b7 Þq6 þ ð122 þ 211 13 2 1b5 1b6 Þq4
Fig. 9. Subharmonic frequency–response curves for the parameters a = 1.0, b = 0.25, E = 5.5 and l = 0.05.
ð54Þ
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425 In this case, the motion of the subharmonic oscillatory states is described by the following equation: X t þ v þ N cos Xt þ OðlÞ. xðtÞ ¼ q cos 427 3
CO
RR
EC
TE
D
PR
OO F
The resolution of Eq. (44) using the Newton–Raphson algorithm enables us to plot in Fig. 9 the behavior of the amplitude q when the detuning parameter r varies for some fixed values of the parameters a and b. Apart the harmonic, superharmonic and subharmonic oscillatory states display by the model, it can also bifurcate from the regular to the chaotic regime. Therefore, it seems very interesting to find the range of parameters for which the model switches from a regular to a chaotic oscillatory states and from a chaotic to a regular oscillation.
UN
428 429 430 431 432
ð55Þ
Fig. 10. Bifurcation diagram (a) and its corresponding Lyapunov exponent (b) when l is varied with the parameters a = 2.55, b = 1.70, X = 3.465 and E = 8.27.
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433 5. Strange attractors and transitions to chaos
Lya ¼ lim
448
t!1
OO F
As nonlinear systems, biological systems can show up regular or chaotic motions depending on some perturbation of the initial states in their environment. Thus to find the ways under which strange attractors arise in biological system is useful. Indeed, chaotic motions are of interest in executing activity adaptation and state transitions in response to environmental changes, and consequently creates a rich repertoire of responses [26]. The quenching of chaos in biological systems is important in medical science because, chaos control techniques are expected to bring about new diagnostic tools and therapies for certain types of diseases, including cardiac arrhythmias [27,28] and epilepsy [29]. On the other hand, the existence of chaos is sometimes needed. Thus, the interests have been devoted in the idea that the brain may utilize transition in and out of chaos [30–32] in its processing to form a complex system [33] displaying self-organization [32], capable of generating new types of structure through bifurcation, a sudden qualitative change in structure occurring at a critical value of a continuously varying parameter. In this section, we analyze the way chaos arises in the MLCVdPo described by Eq. (9). For this purpose, we solve it numerically using the Runge–Kutta algorithm and drawn the resulting bifurcation diagram and the variation of the corresponding Lyapunov exponent as the amplitude E and the coefficient l are varied. The Lyapunov exponent is defined as ln½dðtÞ ; t
PR
434 435 436 437 438 439 440 441 442 443 444 445 446
449 with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 þ dv2x ;
RR
EC
TE
D
where dx and dvx are the variations of x and x_ respectively. The time period of the periodic stroboscopic bifurcation diagram used to map the transition is T ¼ 2p . For the set of parameters a = 2.55, b = 1.70, X = 3.465 and E = 8.27, X it is found that chaos appears in the system within the range l 2 [1.572, 1.577] [ [1.586, 1.588] [ [1.937, 1.944] [ [1.973, 1.985] [ [1.999, 2.001] [ [3.706, 3.709] as one can observe in the bifurcation diagram and its corresponding Lyapunov exponent shown in Fig. 10. Thus the phase portrait showing the chaotic behavior of the oscillator is plotted in Fig. 11 for l = 2.0. Fig. 12 presents the bifurcation diagram and the corresponding variation of the Lyapunov exponent respectively when the amplitude E is varied and the following transitions are observed. When the amplitude of the
CO
452 453 454 455 456 457 458
dðtÞ ¼
UN
451
ð56Þ
Fig. 11. Chaotic phase portrait of the model with the parameters a = 2.55, b = 1.70, X = 3.465, l = 2.0, E = 8.27 and the initial conditions ðxð0Þ; x_ ð0ÞÞ ¼ ð0; 1Þ.
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EC
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external excitation E increases from the value E = 0, the system moves from a quasiperiodic states to a period-5 orbit at E = 1.34. That period-5 orbit remains until E = 8.224 where chaos occurs with some sporadic windows of quasiperiodic orbit alternately and this persists until E = 9.302 where only quasiperiodic oscillations continue to be displayed. For E = 9.50, there is a transition from quasiperiodic behavior to a period-8 orbit. The period-8 orbit exist until E = 9.55 where a period-5 orbit occur and lead to a period-4 orbit. At E = 9.86, once also have a transition from period-4 orbit to a small range of chaos with some sporadic windows of quasiperiodic orbit alternately and this continues to be in place until E = 9.96 where a period-3 orbit takes place before leading to a period-1 orbit (harmonic oscillations) at E = 17.056. For another set of parameter a = 0.144, b = 0.05, l = 3.5, X = 3.465, E = 11.40 with two different sets of initial conditions, the model exhibits two particular types of chaotic attractors (see Fig. 13) which are complementary since one of the attractor can evolves toward another one through the phenomenon of degenerescence or symmetry inversion. Such
UN
459 460 461 462 463 464 465 466 467 468 469
CO
Fig. 12. Bifurcation diagram (a) and the corresponding Lyapunov exponent (b) when E varies with the parameters a = 2.55, b = 1.70, X = 3.465 and l = 2.0.
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Fig. 13. Degenerated chaotic trajectories obtained for two different sets of initial conditions. (a) ðxð0Þ; x_ ð0ÞÞ ¼ ð2:5; 2:5Þ (upward _ attractor), (b) ðuð0Þ; uð0ÞÞ ¼ ð3:0; 3:0Þ (downward attractor).
UN
470 type of attractors have also been reported by Leung recently in the study of the synchronization of two classical Van der 471 Pol oscillator [19].
472 6. Conclusion 473 474 475 476 477 478
In this paper, we have studied the nonlinear dynamics of the biological model. A specific example of brain waves model has been used to establish the equation that governs the MLC-VdPo. Oscillatory states have been derived both in the non-autonomous and autonomous cases, using respectively the averaging and the harmonic balance methods. The phenomenon of birhythmicity has been exhibited by the model in the autonomous regime. In the non-autonomous case, the stability boundaries of the harmonic oscillations have been derived through the Floquet theory and it has been established that the model exhibited Neimark instability. Superharmonic and subharmonic oscillatory states have been
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investigated also. Lyapunov exponents and bifurcation diagrams showing transitions from regular to chaotic and from chaotic to regular motions have been drawn. For a particular set of initial conditions, two degenerate chaotic attractors have been obtained. As the MLC-VdPo is concerned, some insights have been given for biological systems. A good comprehension of the dynamics of such a model is of importance. In biochemistry for example, the stable limit cycles correspond to two enzymes oscillatory states. Therefore, to investigate the synchronization of such model is of interest since it possess many applications.
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485 Appendix A 486
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The coefficients Cl of Eq. (43) are the following:
dA 2N2 A þ að12A2 AN 2 þ 2A3 A2 4XA2 AN 2 þ 6AN4 Þ b½20AN6 þ ð86 46XÞA2 AN 4 þ 60A3 AN 2 þ 5A4 A3 ; þ A A2 A dT 1 3 4A2 A2 N þ 6XA2 A2 b½5XN7 þ 60XAAN 5 þ ð186X 4ÞA2 A2 N3 þ 20XA3 A3 ; C2 ¼ ð1 2AAÞXN XN3 þ að2XN5 þ 14XAAN 2 3 3 5 3 7 5 C3 ¼ XN þ aXð3N þ 10AAN Þ b½9xN þ 88XAAN þ ð78X AÞA AN ;
C1 ¼ 2
PR
b½30ð2 XÞA2 N5 þ ð112 54XÞA3 AN 3 þ ð26 15XÞA4 A2 N; C4 ¼ ðX 2ÞA2 N þ a½6ð2 XÞA2 N3 þ 4ð1 XÞA3 AN 3 3 3 4 3 4 4 2 5 2 C ¼ A þ a½4ð3 þ XÞA N þ 3A A þ b½ð90 þ 16XÞA N þ 4ð2a þ XÞA AN þ 9A A ; 5
b½4ð15 þ 38XÞA2 N5 þ 4ð53 37XÞA3 AN 3 þ ð74 þ 15XÞA4 A2 N; C6 ¼ ð2 þ XÞA2 N þ a½6ð2 þ XÞA2 N3 þ 4ð3 þ XÞA3 AN 2 b½15ð1 þ 2XÞAN6 þ 2ð85 þ 58XÞA2 AN 4 þ 2ð61 þ 26XÞA3 A2 N2 ; C ¼ Að1 þ 2XÞN2 þ 4a½ð1 þ 2XÞAN4 þ ð1 þ 3XÞA2 AN 7
TE
C10 ¼ aXA4 N b½ð40 þ 21XÞA4 N3 þ 2ð10 þ 3XÞA5 AN;
D
2 Þ b½15ð1 2XÞAN6 60XA2 N5 þ 8ð22 21XÞA2 AN 4 þ 2ð64 25XÞA3 AN 3 ; C8 ¼ Að1 2XÞN2 þ 4að1 2XÞðAN4 þ A2 AN 5 5 2 6 C9 ¼ aA b½2ð11 þ 2XÞA N þ 5A A; C11 ¼ að4 XÞA4 N b½ð56 7XÞA4 N3 þ 2ð10 3XÞA5 AN; 4 2 4 6 C12 ¼ að1 þ 4XÞA N b½6ð1 þ 4XÞAN þ ð19 þ 56XÞA AN ; 5 ; C13 ¼ aXN5 bX½5N7 þ 28AAN
EC
4 A2 A; C14 ¼ að1 4XÞA4 N b½6ð1 4XÞAN6 þ 7ð2 26XÞA2 AN 3 ; C15 ¼ 2að2 þ 3XÞA2 N3 b½15ð2 þ 3XÞA2 N5 þ 4ð13 þ 11XÞA3 AN 2 ; C16 ¼ 2að3 2XÞA3 N2 b½12ð5 3XÞA3 N4 þ 2ð25 13XÞA4 AN 2 3 2 5 2 3 C17 ¼ 2að2 3XÞA N b½2ð15 þ 7XÞA N þ ðXA þ 32A AÞN3 ; 5
C19 ¼ bð9 þ 4XÞA3 N4 ;
RR
C18 ¼ b½XA6 N þ ð1 þ 6XÞAN6 ;
C20 ¼ bð15 6XÞA N ; C21 ¼ bð6 XÞA6 N; A3 ÞN2 2ð7 10XÞ; C23 ¼ 15XbA4 N3 ; C22 ¼ b½ð15A4 A 2
C24 ¼ bð6 þ 9XÞA2 N5 ;
C25 ¼ bð4 þ XÞA4 N3 ; 2 ; C26 ¼ b½2ð27 þ 18XÞA N þ ð43 þ 22XÞA4 AN 4
CO
3
5
2
C28 ¼ bð1 þ 2XÞA N ; 2
C29 ¼ 2bA N;
C30 ¼ b½2ð3 14XÞA N XA2 N3 ; 7
C32 ¼ bA ;
489 References 490 491 492 493 494 495 496
5
C31 ¼ 6XbAN6 ;
7
C33 ¼ XbN .
UN
488
C27 ¼ 20bA4 N3 ;
6
[1] Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley; 1979. [2] Chedjou JC, Fotsin HB, Woafo P. Behavior of the Van der Pol oscillator with two external periodic forces. Phys Scr 1997;390:393–455. [3] Van der Pol B. Philos Mag 1922;700:43, 1926;978:7–2, 1927;65:7–3, Proc. IRE 1934;1051:22. [4] Hayashi C. Nonlinear oscillations in physical systems. New York: McGraw-Hill; 1964. [5] Parlitz U, Lauterborn W. Period doubling cascades and devil’s staircases of the driven Van der Pol oscillator. Phys Rev A 1987;1428:1434–6.
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CO
RR
EC
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