- Email: [email protected]
Bifurcations and chaotic behavior on the Lanford system
Chaos, Solitons and Fractals 21 (2004) 803–808 www.elsevier.com/locate/chaos
Bifurcations and chaotic behavior on the Lanford system Svetoslav Nikolov b
a,*
, Bozhan Bozhkov
b
a Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72 Blvd., 1784 Sofia, Bulgaria
Accepted 8 December 2003
Abstract The aim of this article is to investigate in details the bifurcation behavior and show existence of chaotic solutions in the Lanford system. The regular behavior of the model was thoroughly studied in [Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981; Theory of Chaos, Bulgarian Acad. Press, 2001]. Using Lyapunov– Andronov theory, we define the analytical formulas for the first Lyapunov value (this is not Lyapunov exponent) at the boundaries of stability. Here, for specific parametric choice, we obtain chaotic behavior of the Lanford system for the first time to our knowledge. We also calculate the maximal Lyapunov exponent in the parameters space where chaotic motion of this system exists. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In the past 30 years (since the emergence of chaos in the middle of 1970s), autonomous three-dimensional dynamical systems play an outstanding role in modern nonlinear dynamics [1] (and references there). These systems can display a rich diversity of periodic and chaotic solutions dependent upon the specific values of one or more bifurcation (control) parameters [4,5]. Here, we investigate regular and chaotic behavior of a third-order system of nonlinear ordinary differential equations x_ ¼ ðv 1Þx y þ xz; y_ ¼ x þ ðv 1Þy þ yz; 2
2
ð1Þ
2
z_ ¼ vz ðx þ y þ z Þ; which is obtained for the first time in [2] by Hopf. Following [3], the system (1) (in this form) was suggested by Lanford in a private report. That is why, following [3,4], the system (1) is called the Lanford system (LS). For different choice of dimensionless parameter v only the regular motion of LS has been investigated by [3,4]. Also, Hassard [3] gave a good qualitative analysis and proofed that LS have a very rich bifurcation behavior. The system (1) has two equilibrium (steady state) points with coordinates x1 ¼ y1 ¼ z1 ¼ 0;
ð2Þ
x2 ¼ y2 ¼ 0; z2 ¼ v:
ð3Þ
*
Corresponding author. Tel.: +35-92-979-6428; fax: +35-92-870-7498. E-mail address: [email protected] (S. Nikolov).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.040
804
S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808
Now, we continue the investigation (qualitatively and numerically) of the dynamics of LS and we show for the first time to our knowledge that it is a rich dynamical system possessing a vast number of regular and chaotic solutions. Following [6], we calculate the first Lyapunov value (this is not Lyapunov exponent––see appendix in [7] or for a detailed discussion [8]) at the boundary of stability regions R ¼ r ¼ 0 of the system (1). In accordance with Lyapunov– Andronov theory we have: (i) the sign of Lyalunov’s value determines the character (stable or unstable) of equilibrium state at R ¼ 0; (ii) at the boundary of stability r ¼ 0, two cases occur––(a) if the first Lyapunov’s value is different from zero, then the equilibrium state is unstable, (b) if the first Lyapunov’s value is zero, then the equilibrium state is stable; (iii) the character of equilibrium state, at R ¼ r ¼ 0, qualitatively determines the reconstruction of phase portrait (including stability or instability of limit cycle) at transition from R < 0 (r < 0) to R > 0 (r > 0) [6,8]. We also calculate the maximal Lyapunov exponent where in phase space LS is chaotic. The scheme of the present paper is as follows: In Sections 2 and 3 we present analytical and numerical results concerning the system (1) for different values of parameter v. In Section 4 we discuss and summarize our results.
2. Analytical and numerical analysis In this section, we consider the system (1), which presents an autonomous dynamical model. The constant v of this model is real and can be negative, zero or positive. In order to determine the character of first and second fixed points (Eqs. (2) and (3)) we make the following substitutions into (1) x ¼ x1;2 þ x1 ¼ x1 ;
y ¼ y1;2 þ x2 ¼ x2 ;
z ¼ z þ x3 ;
ð4Þ
where z can be z1 or z2 . Hence, after accomplishing some transformations the system (1) (in local coordinates) has the form x_ 1 ¼ ðv 1 þ zÞx1 x2 þ x1 x3 ; x_ 2 ¼ x1 þ ðv 1 þ zÞx2 þ x2 x3 ;
ð5Þ
x_ 3 ¼ ðv 2zÞx3 ðx21 þ x22 þ x23 Þ: The divergence of the flow (5) is D3 ¼
o_x1 o_x2 o_x3 þ þ ¼ 3ðv zÞ 2: ox1 ox2 ox3
ð6Þ
The system (5) is dissipative and has attractor when D3 < 0. According to [6], the Routh–Hurwitz conditions for stability of (2) and (3) can be written in the form p ¼ 2 3v > 0;
ð7Þ
q ¼ 1 þ ðv 1 þ zÞð3v 3z 1Þ > 0; 2
r ¼ ðv 2zÞ½ðv 1 þ zÞ þ 1 > 0; R ¼ pq r > 0:
ð8Þ ð9Þ ð10Þ
Here the notations p, q, r and R are taken from [6]. When conditions (9) or (10) are not valid, the steady states (2) and (3) become unstable. In order to define the type of stability loss of steady states (2) and (3) it is to calculate the so called first Lyapunov value [6,7]. In case of three first order differential equations, this value can be determined analytically (at the boundary of stability R ¼ 0) by the formula in [6] p h ð2Þ ð3Þ ð2Þ ð3Þ ð2Þ ð2Þ ð2Þ 2 A33 A33 A22 A22 þ 2A23 A22 þ A33 L1 ðk0 Þ ¼ 4q i pffiffiffi ð2Þ ð3Þ ð3Þ ð3Þ ð3Þ ð2Þ ð3Þ 2A23 A22 þ A33 þ 3 q A222 þ A333 þ A233 þ A223 ð11Þ n h i p ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ þ pffiffiffi 2 p2 2A22 3A12 þ A13 þ 2A33 A12 þ 3A13 þ 4A23 A13 þ A12 4p qðp þ 4qÞ i o pffiffiffih ð1Þ ð1Þ ð2Þ ð3Þ ð1Þ ð3Þ ð2Þ ð1Þ ð1Þ ð2Þ ð3Þ þ 16q A22 þ A33 A12 þ A13 ; þ 4p q A22 A33 A13 þ A12 þ 2A23 A13 A12
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805
where k0 is defined as a value of v for which the relation R ¼ 0 takes place. Here we note that for first fixed point (Eq. (2)) R ¼ 0 when 3 1 v3 2v2 þ v ¼ 0; 2 2
ð12Þ
i.e. when v1 ¼ 1 and v2;3 ¼ 0:5 0:5i, and for second fixed point (Eq. (3)) R ¼ 0 when 5 v3 v2 þ 3v 1 ¼ 0; 2
ð13Þ
i.e. when v1 ¼ 0:5 and v2;3 ¼ 1 i. For the system (5) ð2Þ
ð3Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
A22 ¼ A22 ¼ A33 ¼ A33 ¼ A23 ¼ A23 ¼ 0:
ð14Þ
After substitution of (14) into (11) and accomplishing some transformations and algebraic operations for the first Lyapunov value L1 ðk0 Þ (for the system (5)) we obtain L1 ðk0 Þ ¼
p ½p2 ð2B1 þ 2B2 þ B3 Þ þ 8qðB1 þ B2 Þ; pffiffiffi 2p qðp2 þ 4qÞ
ð15Þ
where B1 ¼ ðv 2zÞ2 þ ½1 þ ðv 1 þ zÞ2 þ ðv 2zÞðv 1 þ zÞ2 ;
ð16Þ
B2 ¼ q½ð2v 1 zÞ2 þ 1; o pffiffiffin B3 ¼ 2 q 2z v þ ½1 þ ðv 1 þ zÞ2 þ ðv 2zÞðv 1 þ zÞ :
ð17Þ ð18Þ
The first Lyapunov value (in (15)) can be negative or positive. If L1 ðk0 Þ is negative, then in case of transition through the boundary R ¼ 0 from positive values to negative ones, a stable limit cycle (self-oscillations) emerges. Inversely, in case of a transition from negative values to positive ones the stable limit cycle disappears, i.e. the self-oscillations cease [6]. In theory of dynamic system the type of bifurcation behavior near the boundary R ¼ 0 is often called soft loss of stability, i.e. when the bifurcation parameter k0 changes, the system has reversible behavior. If L1 ðk0 Þ is positive, then in case of transition through the boundary R ¼ 0 from positive values to negative ones, an unstable limit cycle emerges. Inversely, in case of transition from negative values to positive ones, the unstable limit cycle disappears. This type of bifurcation behavior near the boundary R ¼ 0 is often called hard loss of stability, i.e. the system has irreversible behavior and the boundary R ¼ 0 is dangerous. 2.1. Investigation of the first fixed point (Eq. (2)) In this case, the equilibrium (steady state) value (2) of the system (1) is the zero one. After substitution of z ¼ 0 and v ¼ 1 into (7)–(9) we obtain that p and r are negative. Therefore, the first Lyapunov value at the boundary R ¼ 0 cannot be calculated. But for v ¼ 0 the boundary r is zero. Following [6], the first Lyapunov value lðk0 Þ at the boundary r ¼ 0 has the form lðk0 Þ ¼ aAð1Þ þ bAð2Þ þ cAð3Þ ;
ð19Þ ðiÞ
where the coefficients a, b, c and A (i ¼ 1–3) are also defined by corresponding formulas presented by [6]. For the system (5) (when z ¼ 0) a ¼ b ¼ 0. Hence, after accomplishing some transformations, we obtain for lðk0 Þ lðk0 Þ ¼ 0:5 6¼ 0;
ð20Þ
where c ¼ 2 and Að3Þ ¼ 0:25. According to [6], if lðk0 Þ is different from zero, then in case of a transition from negative values to positive ones, the equilibrium state becomes unstable double point, the system has irreversible behavior and the boundary r ¼ 0 is dangerous. Also, from sign of the added condition D ¼ p2 q2 þ 4q3 ;
ð21Þ
we can have two cases: (i) if D < 0, then the equilibrium state becomes saddle-knot; (ii) if D > 0, then the equilibrium state becomes saddle-focus [6]. Here, D ¼ 16 > 0, therefore (2) is from saddle-focus type.
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2.2. Investigation of the second fixed point (Eq. (3)) After substitution of z ¼ v ¼ 0:5 and accomplishing some algebraic operations for the first Lyapunov value L1 ðk0 Þ (for the system (5)) we have L1 ðk0 Þ ¼ 13:6682 < 0:
ð22Þ
Therefore, following [6,7] the boundary R ¼ 0 is undangerous and the system has reversible behavior, i.e. a stable limit cycle (self-oscillations) emerges or ceases. This result is in accordance with these obtained by [3,4]. In the following section, we demonstrate numerically these types of behavior. Also, we obtain that in some intervals of variation of the parameter v the system (1) has chaotic behavior.
3. Numerical results In previous section we introduced the analytical tools that we will use in our numerical analysis of the system (1). In Fig. 1(a) the case when v ¼ 0:49 is shown. It is seen that here the system is stable. After v ¼ 0:5 the stable limit cycle occur (see Fig. 1(b)) and the system (1) has periodic solution. These results are in accordance with results obtained in our study in previous section. Fig. 2 shows a bifurcation diagram for the system (1): values of xn are plotted against v regarded as a continuously varying control parameter. What could one observe in the figure? When v 2 ½0:635; 0:655, the system has regular solution with period 1. As v increased further, there is a period-doubling bifurcation that results in a double-loop attractor (see Fig. 3(a)). We see that chaos occurs there after v P 0:6666. A confirmation of our conclusions is the
Fig. 1. Stable (a) and periodic (b) solution of the system (1) at v ¼ 0:49 and v ¼ 0:51.
Fig. 2. Bifurcation diagram xn versus v generated by computer solution of system (1). Note that v 2 ½0:635; 0:66669.
S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808
807
Fig. 3. Phase portrait of the system (1) at (a) v ¼ 0:657; (b) v ¼ 0:66668.
Fig. 4. Trajectories of the system (1) with two different initial conditions at v ¼ 0:6666. For dotted line x0 ¼ 0:1 and for solid line x0 ¼ 0:1.
strange attractor shown in Fig. 3(b), solutions with exponential sensitivity to initial conditions and obtained for this case maximal Lyapunov exponent. The maximal Lyapunov exponent kmax shows the kind of motion on the phase space: (i) if kmax < 0 the motion is a stable fixed point; (ii) if kmax ¼ 0 the motion is a stable limit cycle; (iii) if 0 < kmax < 1 the motion is chaotic and (iv) if kmax ¼ 1 the motion is noise [9]. Following [9], the maximal Lyapunov exponent for a given data set can be calculated by means of the sum 1 0 N X 1 X 1 ð23Þ SðDnÞ ¼ ln @ jsn þDn snþDn jA; N n0 jWðbn0 Þj b 2W 0 n0
where reference points bn0 are embedding vectors, Wðbn0 Þ is the neighborhood of bn0 with diameter e, sn0 is the last element of bn0 and sn0 þDn is outside the time span covered by the delay vector bn0 . For the numerical calculation of kmax we use the TISEAN software package [10]. The obtained maximal Lyapunov exponent (per unit time) is: kmax ¼ 0:1139 0:0076, when v ¼ 0:66668. In Fig. 4 we show that after v ¼ 0:6666 the chaotic behavior of system takes place. When we have different initial conditions (which are very close) for x, the system (1) has different trajectories. Here we note that strange attractor for these two cases is topological one and the same.
4. Summary and conclusions The paper presents a study of the behavior of Lanford system. Using Lyapunov–Andronov’s theory, we find new analytical formulas for the first Lyapunov’s value at the stability limit. It enables one to study in detail the bifurcation behavior of the dynamic system (1). The approach (for first Lyapunov value) proposed here has a basic advantage, which consist in the following: we can answer the question for structural stability (respectively unstability) of the Lanford system. Here we note that for all simulations the initial conditions were x0 ¼ y0 ¼ 0:1, z0 ¼ 0:07.
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S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808
Generalizing the results obtained in Sections 2 and 3, we can conclude that: 1. The emergence of a stable limit cycle with period 1 takes place for a value of the bifurcation parameter v ¼ 0:5 under a soft stability loss. 2. The boundary of stability r ¼ 0 (for first fixed point Eq. (2)) is dangerous and the Lanford system (1) has irreversible behavior for v > 0. 3. The first fixed point (Eq. (2)) after v > 0 becomes unstable double point from saddle-focus type. 4. The first Lyapunov value at the boundary R ¼ 0 (for first fixed point Eq. (2)) cannot be calculated. 5. For values of the bifurcation parameter v, larger than v ¼ 0:6666, the system (1) is in a chaotic state (till v ¼ 0:6667).
Acknowledgements This work was supported by the National Science Fund of the Ministry of Education and Science (Bulgaria), project MM 1302/2003.
References [1] Eichhorn R, Linz S, Hanggi P. Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flow. Phys Rev E 1998;58(6):7151–64. [2] Hopf E. A mathematical example displaying the features of turbulence. Comm Pure Appl Math 1948;1:303–22. [3] Hassard B, Kazarinoff ND, Wan YH. Theory and applications of Hopf bifurcation. Cambridge University Press; 1981. [4] Panchev S. Theory of Chaos. Bulgarian Acad. Press; 2001. [5] Phillipson P, Schuster P. Bifurcation dynamics of three-dimensional systems. Int J Bifurcat Chaos 2000;10(8):1787–804. [6] Bautin N. Behavior of dynamical systems near the boundary of stability. Moscow: Nauka; 1984. [7] Nikolov S, Petrov V. New results about route to chaos in Rossler system. Int J Bifurcat Chaos 2004;14(1):1–16. [8] Andronov A, Vitt A, Khaikin S. Theory of oscillations. Reading, MA: Addison-Wesley; 1966. [9] Kantz H, Schreiber T, editors. Nonlinear time series analysis. Cambridge Univ. Press; 1997. [10] Hegger R, Kantz H, Schreiber T. Practical implementation of nonlinear time series methods: the TISEAN package. Chaos 1999;9:413–35.
Bifurcations and chaotic behavior on the Lanford system Svetoslav Nikolov b
a,*
, Bozhan Bozhkov
b
a Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72 Blvd., 1784 Sofia, Bulgaria
Accepted 8 December 2003
Abstract The aim of this article is to investigate in details the bifurcation behavior and show existence of chaotic solutions in the Lanford system. The regular behavior of the model was thoroughly studied in [Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981; Theory of Chaos, Bulgarian Acad. Press, 2001]. Using Lyapunov– Andronov theory, we define the analytical formulas for the first Lyapunov value (this is not Lyapunov exponent) at the boundaries of stability. Here, for specific parametric choice, we obtain chaotic behavior of the Lanford system for the first time to our knowledge. We also calculate the maximal Lyapunov exponent in the parameters space where chaotic motion of this system exists. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In the past 30 years (since the emergence of chaos in the middle of 1970s), autonomous three-dimensional dynamical systems play an outstanding role in modern nonlinear dynamics [1] (and references there). These systems can display a rich diversity of periodic and chaotic solutions dependent upon the specific values of one or more bifurcation (control) parameters [4,5]. Here, we investigate regular and chaotic behavior of a third-order system of nonlinear ordinary differential equations x_ ¼ ðv 1Þx y þ xz; y_ ¼ x þ ðv 1Þy þ yz; 2
2
ð1Þ
2
z_ ¼ vz ðx þ y þ z Þ; which is obtained for the first time in [2] by Hopf. Following [3], the system (1) (in this form) was suggested by Lanford in a private report. That is why, following [3,4], the system (1) is called the Lanford system (LS). For different choice of dimensionless parameter v only the regular motion of LS has been investigated by [3,4]. Also, Hassard [3] gave a good qualitative analysis and proofed that LS have a very rich bifurcation behavior. The system (1) has two equilibrium (steady state) points with coordinates x1 ¼ y1 ¼ z1 ¼ 0;
ð2Þ
x2 ¼ y2 ¼ 0; z2 ¼ v:
ð3Þ
*
Corresponding author. Tel.: +35-92-979-6428; fax: +35-92-870-7498. E-mail address: [email protected] (S. Nikolov).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.040
804
S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808
Now, we continue the investigation (qualitatively and numerically) of the dynamics of LS and we show for the first time to our knowledge that it is a rich dynamical system possessing a vast number of regular and chaotic solutions. Following [6], we calculate the first Lyapunov value (this is not Lyapunov exponent––see appendix in [7] or for a detailed discussion [8]) at the boundary of stability regions R ¼ r ¼ 0 of the system (1). In accordance with Lyapunov– Andronov theory we have: (i) the sign of Lyalunov’s value determines the character (stable or unstable) of equilibrium state at R ¼ 0; (ii) at the boundary of stability r ¼ 0, two cases occur––(a) if the first Lyapunov’s value is different from zero, then the equilibrium state is unstable, (b) if the first Lyapunov’s value is zero, then the equilibrium state is stable; (iii) the character of equilibrium state, at R ¼ r ¼ 0, qualitatively determines the reconstruction of phase portrait (including stability or instability of limit cycle) at transition from R < 0 (r < 0) to R > 0 (r > 0) [6,8]. We also calculate the maximal Lyapunov exponent where in phase space LS is chaotic. The scheme of the present paper is as follows: In Sections 2 and 3 we present analytical and numerical results concerning the system (1) for different values of parameter v. In Section 4 we discuss and summarize our results.
2. Analytical and numerical analysis In this section, we consider the system (1), which presents an autonomous dynamical model. The constant v of this model is real and can be negative, zero or positive. In order to determine the character of first and second fixed points (Eqs. (2) and (3)) we make the following substitutions into (1) x ¼ x1;2 þ x1 ¼ x1 ;
y ¼ y1;2 þ x2 ¼ x2 ;
z ¼ z þ x3 ;
ð4Þ
where z can be z1 or z2 . Hence, after accomplishing some transformations the system (1) (in local coordinates) has the form x_ 1 ¼ ðv 1 þ zÞx1 x2 þ x1 x3 ; x_ 2 ¼ x1 þ ðv 1 þ zÞx2 þ x2 x3 ;
ð5Þ
x_ 3 ¼ ðv 2zÞx3 ðx21 þ x22 þ x23 Þ: The divergence of the flow (5) is D3 ¼
o_x1 o_x2 o_x3 þ þ ¼ 3ðv zÞ 2: ox1 ox2 ox3
ð6Þ
The system (5) is dissipative and has attractor when D3 < 0. According to [6], the Routh–Hurwitz conditions for stability of (2) and (3) can be written in the form p ¼ 2 3v > 0;
ð7Þ
q ¼ 1 þ ðv 1 þ zÞð3v 3z 1Þ > 0; 2
r ¼ ðv 2zÞ½ðv 1 þ zÞ þ 1 > 0; R ¼ pq r > 0:
ð8Þ ð9Þ ð10Þ
Here the notations p, q, r and R are taken from [6]. When conditions (9) or (10) are not valid, the steady states (2) and (3) become unstable. In order to define the type of stability loss of steady states (2) and (3) it is to calculate the so called first Lyapunov value [6,7]. In case of three first order differential equations, this value can be determined analytically (at the boundary of stability R ¼ 0) by the formula in [6] p h ð2Þ ð3Þ ð2Þ ð3Þ ð2Þ ð2Þ ð2Þ 2 A33 A33 A22 A22 þ 2A23 A22 þ A33 L1 ðk0 Þ ¼ 4q i pffiffiffi ð2Þ ð3Þ ð3Þ ð3Þ ð3Þ ð2Þ ð3Þ 2A23 A22 þ A33 þ 3 q A222 þ A333 þ A233 þ A223 ð11Þ n h i p ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ þ pffiffiffi 2 p2 2A22 3A12 þ A13 þ 2A33 A12 þ 3A13 þ 4A23 A13 þ A12 4p qðp þ 4qÞ i o pffiffiffih ð1Þ ð1Þ ð2Þ ð3Þ ð1Þ ð3Þ ð2Þ ð1Þ ð1Þ ð2Þ ð3Þ þ 16q A22 þ A33 A12 þ A13 ; þ 4p q A22 A33 A13 þ A12 þ 2A23 A13 A12
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805
where k0 is defined as a value of v for which the relation R ¼ 0 takes place. Here we note that for first fixed point (Eq. (2)) R ¼ 0 when 3 1 v3 2v2 þ v ¼ 0; 2 2
ð12Þ
i.e. when v1 ¼ 1 and v2;3 ¼ 0:5 0:5i, and for second fixed point (Eq. (3)) R ¼ 0 when 5 v3 v2 þ 3v 1 ¼ 0; 2
ð13Þ
i.e. when v1 ¼ 0:5 and v2;3 ¼ 1 i. For the system (5) ð2Þ
ð3Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
A22 ¼ A22 ¼ A33 ¼ A33 ¼ A23 ¼ A23 ¼ 0:
ð14Þ
After substitution of (14) into (11) and accomplishing some transformations and algebraic operations for the first Lyapunov value L1 ðk0 Þ (for the system (5)) we obtain L1 ðk0 Þ ¼
p ½p2 ð2B1 þ 2B2 þ B3 Þ þ 8qðB1 þ B2 Þ; pffiffiffi 2p qðp2 þ 4qÞ
ð15Þ
where B1 ¼ ðv 2zÞ2 þ ½1 þ ðv 1 þ zÞ2 þ ðv 2zÞðv 1 þ zÞ2 ;
ð16Þ
B2 ¼ q½ð2v 1 zÞ2 þ 1; o pffiffiffin B3 ¼ 2 q 2z v þ ½1 þ ðv 1 þ zÞ2 þ ðv 2zÞðv 1 þ zÞ :
ð17Þ ð18Þ
The first Lyapunov value (in (15)) can be negative or positive. If L1 ðk0 Þ is negative, then in case of transition through the boundary R ¼ 0 from positive values to negative ones, a stable limit cycle (self-oscillations) emerges. Inversely, in case of a transition from negative values to positive ones the stable limit cycle disappears, i.e. the self-oscillations cease [6]. In theory of dynamic system the type of bifurcation behavior near the boundary R ¼ 0 is often called soft loss of stability, i.e. when the bifurcation parameter k0 changes, the system has reversible behavior. If L1 ðk0 Þ is positive, then in case of transition through the boundary R ¼ 0 from positive values to negative ones, an unstable limit cycle emerges. Inversely, in case of transition from negative values to positive ones, the unstable limit cycle disappears. This type of bifurcation behavior near the boundary R ¼ 0 is often called hard loss of stability, i.e. the system has irreversible behavior and the boundary R ¼ 0 is dangerous. 2.1. Investigation of the first fixed point (Eq. (2)) In this case, the equilibrium (steady state) value (2) of the system (1) is the zero one. After substitution of z ¼ 0 and v ¼ 1 into (7)–(9) we obtain that p and r are negative. Therefore, the first Lyapunov value at the boundary R ¼ 0 cannot be calculated. But for v ¼ 0 the boundary r is zero. Following [6], the first Lyapunov value lðk0 Þ at the boundary r ¼ 0 has the form lðk0 Þ ¼ aAð1Þ þ bAð2Þ þ cAð3Þ ;
ð19Þ ðiÞ
where the coefficients a, b, c and A (i ¼ 1–3) are also defined by corresponding formulas presented by [6]. For the system (5) (when z ¼ 0) a ¼ b ¼ 0. Hence, after accomplishing some transformations, we obtain for lðk0 Þ lðk0 Þ ¼ 0:5 6¼ 0;
ð20Þ
where c ¼ 2 and Að3Þ ¼ 0:25. According to [6], if lðk0 Þ is different from zero, then in case of a transition from negative values to positive ones, the equilibrium state becomes unstable double point, the system has irreversible behavior and the boundary r ¼ 0 is dangerous. Also, from sign of the added condition D ¼ p2 q2 þ 4q3 ;
ð21Þ
we can have two cases: (i) if D < 0, then the equilibrium state becomes saddle-knot; (ii) if D > 0, then the equilibrium state becomes saddle-focus [6]. Here, D ¼ 16 > 0, therefore (2) is from saddle-focus type.
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S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808
2.2. Investigation of the second fixed point (Eq. (3)) After substitution of z ¼ v ¼ 0:5 and accomplishing some algebraic operations for the first Lyapunov value L1 ðk0 Þ (for the system (5)) we have L1 ðk0 Þ ¼ 13:6682 < 0:
ð22Þ
Therefore, following [6,7] the boundary R ¼ 0 is undangerous and the system has reversible behavior, i.e. a stable limit cycle (self-oscillations) emerges or ceases. This result is in accordance with these obtained by [3,4]. In the following section, we demonstrate numerically these types of behavior. Also, we obtain that in some intervals of variation of the parameter v the system (1) has chaotic behavior.
3. Numerical results In previous section we introduced the analytical tools that we will use in our numerical analysis of the system (1). In Fig. 1(a) the case when v ¼ 0:49 is shown. It is seen that here the system is stable. After v ¼ 0:5 the stable limit cycle occur (see Fig. 1(b)) and the system (1) has periodic solution. These results are in accordance with results obtained in our study in previous section. Fig. 2 shows a bifurcation diagram for the system (1): values of xn are plotted against v regarded as a continuously varying control parameter. What could one observe in the figure? When v 2 ½0:635; 0:655, the system has regular solution with period 1. As v increased further, there is a period-doubling bifurcation that results in a double-loop attractor (see Fig. 3(a)). We see that chaos occurs there after v P 0:6666. A confirmation of our conclusions is the
Fig. 1. Stable (a) and periodic (b) solution of the system (1) at v ¼ 0:49 and v ¼ 0:51.
Fig. 2. Bifurcation diagram xn versus v generated by computer solution of system (1). Note that v 2 ½0:635; 0:66669.
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Fig. 3. Phase portrait of the system (1) at (a) v ¼ 0:657; (b) v ¼ 0:66668.
Fig. 4. Trajectories of the system (1) with two different initial conditions at v ¼ 0:6666. For dotted line x0 ¼ 0:1 and for solid line x0 ¼ 0:1.
strange attractor shown in Fig. 3(b), solutions with exponential sensitivity to initial conditions and obtained for this case maximal Lyapunov exponent. The maximal Lyapunov exponent kmax shows the kind of motion on the phase space: (i) if kmax < 0 the motion is a stable fixed point; (ii) if kmax ¼ 0 the motion is a stable limit cycle; (iii) if 0 < kmax < 1 the motion is chaotic and (iv) if kmax ¼ 1 the motion is noise [9]. Following [9], the maximal Lyapunov exponent for a given data set can be calculated by means of the sum 1 0 N X 1 X 1 ð23Þ SðDnÞ ¼ ln @ jsn þDn snþDn jA; N n0 jWðbn0 Þj b 2W 0 n0
where reference points bn0 are embedding vectors, Wðbn0 Þ is the neighborhood of bn0 with diameter e, sn0 is the last element of bn0 and sn0 þDn is outside the time span covered by the delay vector bn0 . For the numerical calculation of kmax we use the TISEAN software package [10]. The obtained maximal Lyapunov exponent (per unit time) is: kmax ¼ 0:1139 0:0076, when v ¼ 0:66668. In Fig. 4 we show that after v ¼ 0:6666 the chaotic behavior of system takes place. When we have different initial conditions (which are very close) for x, the system (1) has different trajectories. Here we note that strange attractor for these two cases is topological one and the same.
4. Summary and conclusions The paper presents a study of the behavior of Lanford system. Using Lyapunov–Andronov’s theory, we find new analytical formulas for the first Lyapunov’s value at the stability limit. It enables one to study in detail the bifurcation behavior of the dynamic system (1). The approach (for first Lyapunov value) proposed here has a basic advantage, which consist in the following: we can answer the question for structural stability (respectively unstability) of the Lanford system. Here we note that for all simulations the initial conditions were x0 ¼ y0 ¼ 0:1, z0 ¼ 0:07.
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Generalizing the results obtained in Sections 2 and 3, we can conclude that: 1. The emergence of a stable limit cycle with period 1 takes place for a value of the bifurcation parameter v ¼ 0:5 under a soft stability loss. 2. The boundary of stability r ¼ 0 (for first fixed point Eq. (2)) is dangerous and the Lanford system (1) has irreversible behavior for v > 0. 3. The first fixed point (Eq. (2)) after v > 0 becomes unstable double point from saddle-focus type. 4. The first Lyapunov value at the boundary R ¼ 0 (for first fixed point Eq. (2)) cannot be calculated. 5. For values of the bifurcation parameter v, larger than v ¼ 0:6666, the system (1) is in a chaotic state (till v ¼ 0:6667).
Acknowledgements This work was supported by the National Science Fund of the Ministry of Education and Science (Bulgaria), project MM 1302/2003.
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