DE GRUYTER
DOI 10.1515/ijnsns-2011-0032
Int. J. Nonlinear Sci. Numer. Simul. 2013; 14(1): 77–85
Marcos Roberto de Lima, Fábio Claro, Wallace Ribeiro, Sérgio Xavier and A. López-Castillo*
The Numerical Connection between Map and its Differential Equation: Logistic and Other Systems Abstract: The logistic map, in this article, is used to show the propagation of the numerical error for numerically solved logistic ODE’s. When this equation is solved numerically (discretization procedure) the solution shows a chaotic behavior by varying the control parameter (integration step-size). However, the logistic ODE dx=dt ¼ ðk 1Þx kx 2 also has an analytical solution, i.e., not presenting the signature of chaos. The connection between the two limits, analytical solution and chaos, allows us to establish a region of acceptability for the numerical error. Therefore the limits of the numerical process can be found in between the limits of the analytical and chaotic solutions. This study can be extended to any other ODE or map. For example, there is a similar numerical connection between the chaotic bi-dimensional Cat Map with its respective integrable ODE, which has an analytical solution. Our discussion applied to those particular systems is generic, that is, the solutions obtained from the numerical discretization can be more instable than those obtained from the continuous procedure. The important point is to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution since the instabilities can be unavoidable. Keywords: chaotic maps, numerical discretization procedure, differential equations PACS= (2010). 37M99; 65P20
ics [2], which can also be modeled, e.g., by the Lotka nonlinear differential equation [3]. In this study, we mainly consider the logistic map to show the propagation of the numerical error when the logistic ODE is solved numerically. This study shows that the limits of a numerical process can be found in between the analytical and chaotic solutions. The area between the two limits of the analytical solution and chaos allows us to establish the region of acceptability for the numerical method. The analytical solution does not show chaos, but the numerical one (iterative map) can show it due to the discretization procedure. The importance of the numerical error becomes clear when we compare the analytical solution of an ODE (ordinary differential equation) with, e.g., the iterative Euler method with the step-size T for the same equation [4, 5]. Chaos can emerge from a finite propagation length for any finite step-size and for any numerical method. The solutions obtained from the numerical discretization are equal to or more instable than those obtained from a continuous procedure. It is important to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution. The numerical connection between ODEs and maps was studied, for example: the Bernoulli shift map and the bi-dimensional cat map [6]. The Brusselator system [7] was also introduced to show a possible application to a bi-dimensional non-linear ODE, which does not have a generic analytic solution. This paper is structured as: a) a short revision on the logistic system with a discussion for the complete range 2 < k < 4 and its stability; b) studies of the general logistic map and its numerical simulation; c) other examples of map and its respective ODE, mainly the cat map.
*Corresponding author: A. López-Castillo: Departamento de Química, Universidade Federal de São Carlos (UFSCar), São Carlos, São Paolo, Brazil. E-mail:
[email protected] Marcos Roberto de Lima, Fábio Claro, Wallace Ribeiro, Sérgio Xavier: Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC), Santo André, São Paolo, Brazil
2 Theory
1 Introduction
2.1 The logistic map and its ODE
The chaotic systems appear in several fields of knowledge such as Mathematics, Computation, Physics, Chemistry, Biology, among others [1]. Particularly, the chaotic logistic map can be used to study the population dynam-
The logistic ODE [8] is defined as dx=dt ¼ ðk 1Þx kx 2 ; with asymptotic solutions x0 ¼ 0 and x0 ¼ ðk 1Þ=k.
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
ð1Þ
78
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
DE GRUYTER
Fig. 1: Bifurcation diagram
Eq. (1) can be reduced to the logistic map [2, 8, 9, 10] by finite differentiation, adopting the step-size as unity ðΔT ¼ 1Þ, as dx=dt A Δx=ΔT ¼ Δx=1 ¼ xnþ1 xn ¼ ðk 1Þxn kxn 2 , and finally xnþ1 ¼ kxn ð1 xn Þ:
ð2Þ
The logistic map diagram for allowed values of k is shown in the Figure 1. Usually, the logistic map is studied only for k > 0 [8], the negative region of k is not discussed in literature since k must be bigger than zero due to biological arguments (biotic potential) [2, 8], however these arguments are mathematically irrelevant. As an example, the correspondent web diagram [8] for the negative k value is shown in Figure 2. For a negative k, the doubling periods accumulate at k A 1:545 and the solution x ¼ 0 is stable until the first doubling period at k ¼ 1 (Figure 1). The solution of the logistic ODE is given by: xðtÞ ¼ A=½1 þ ðA=x0 Þeð1kÞt ;
ð3Þ
where x0 is the initial condition and A ¼ ðk 1Þ=k. The chaotic regime does not belong to the solution of the logistic ODE. The solution of the map for 1 a k a 3 coincides with the asymptotic one of the ODE. The stability of the system when near to the critical point x0 A 0, for the whole range of k, can be obtained by neglecting the quadratic term of the map as xnþ1 A kxn or xn A k n x0, the equation is given below:
Fig. 2: Web diagram, k ¼ 1:8 and x0 ¼ 0:1
8 < 0 ) jkj < 1 ) 1 < k < 1 lim xn A x0 ) jkj ¼ 1 ) k ¼ 1; k ¼ 1 n!y : y ) jkj > 1 ) k < 1; k > 1
ð4Þ
The system is stable near the critical point of x0 A 0 for 1 < k < 1 and becomes unstable if k < 1 and k > 1. The stability around x0 A ðk 1Þ=k can be found described in the literature [8, 9, 10]. In this case, the system is stable for 1 < k < 3 and unstable for k > 3 and k < 1. The logistic map does not show bounded solutions if k is bigger than kmax ¼ 4 or less than kmin ¼ 2.
3 Results 3.1 A general logistic map A general logistic map can be obtained from the logistic ODE considering the Euler method with non-unitary stepsize ΔT 1 T > 0 as dx=dt ¼ ðk 1Þx kx 2 A Δx=ΔT ¼ Δx=T ) Δx ¼ ½ðk 1Þx kx 2 T; with Δx ¼ xnþ1 xn and assuming x 1 xn , we obtain xnþ1 xn ¼ ½ðk 1Þxn kxn2 T and finally we have xnþ1 ¼ ðkT T þ 1Þxn kTxn2
ð5Þ
or xnþ1 ¼ Tkxn ð1 xn Þ þ ð1 TÞxn ; if T 1 1 the usual logistic map is recovered. In particular for the logistic map, a scaling relation can be used in order to reduce the general logistic map to an ordinary one. For example,
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
DE GRUYTER
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
if q 1 ðkT T þ 1Þ and zn ¼ ½kT=ðkT T þ 1Þxn the map xnþ1 ¼ ðkT T þ 1Þxn kTxn2 can be rewritten as znþ1 ¼ qzn ð1 zn Þ. Considering the scaling relation, xn can be given by xn ¼ cðT; kÞzn, where cðT; kÞ ¼ ðkT T þ 1Þ=ðkTÞ. The numerical connection between ODEs and maps was studied, for example: a) Bernoulli shift map and its ODE; b) bi-dimensional cat map and its ODE; c) Brusselator bi-dimensional non-linear ODE [7] and its non-linear map. These examples are given in section 3.3.
3.2 Numerical results for logistic system The importance of the numerical error becomes clear when we compare the analytical solution of the logistic
79
ODE with the iterative Euler method with step-size T (general logistic map) for the same equation. The analytical solution does not show chaos, but the numerical one (iterative map) can show it due to the discretization procedure. The diagrams of the general logistic map are presented in Figures 3a to 3d, having several values for T: 1; 0:01; 2, and 100. The diagram for T ¼ 0:01 and 100 are similar to T ¼ 1. In particular for the logistic map, some singularities appear on the bifurcation diagram when k A 0 for T ¼ 1 and 2. If k A 0 xn ¼ ½ðkT T þ 1Þ=ðkTÞzn diverges for q ¼ ðkT T þ 1Þ A 1 T, i.e., 2 a q a 1 for 2 a T a 3. Singularities also appear for a negative value of T. The map for T ¼ 1
Fig. 3: Bifurcation diagrams
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
80
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
Fig. 4: kðT Þ as function of T > 0: I) k ¼ 1 þ 3=T (chaos limit for k positive), II) k ¼ 1 þ 2=T (bifurcation limit for k positive), III) k ¼ 1 2=T (bifurcation limit for k negative), and IV) k ¼ 1 3=T (chaos limit for k negative). Regions: A – the map solution coincides with the asymptotic one of the ODE; B – unstable regions
can be considered a backward logistic map. Independent of the logistic map scaling relation the numerical problem remains, for practical purposes, for all systems which require a numerical procedure to solve it. The solution of the general logistic map is defined in the interval ð1 3=TÞ < k < ð1 þ 3=TÞ (2 < q < 4 for T ¼ 1). The bifurcations do not appear in the interval ð1 2=TÞ < k < ð1 þ 2=TÞ (1 < q < 3 for T ¼ 1). The map solutions coincide with the asymptotic solutions of the ODE in this non-bifurcation interval and the range of this interval is given by ð1 þ 2=TÞ ð1 2=TÞ ¼ 4=T, see Figure 1 and Figure 3. Figure 4 shows kðTÞ, the curve k ¼ 1 þ 3=T describes the limit of the chaos ending and k ¼ 1 þ 2=T describes the limit of the bifurcation beginning for positive k. The curve k ¼ 1 2=T is analogous to k ¼ 1 þ 2=T, and k ¼ 1 3=T can be compared to k ¼ 1 þ 3=T for negative k. The A region on Figure 4 represents the place where the map solutions coincides with the asymptotic solutions of the ordinary differential equation, the B regions are connected to bifurcations and chaos. The solutions of the logistic map above must be given by an asymptotic solution (xðtÞ ¼ A=½1 þ ðA=x0 Þ expð1 kÞt, in the t ! Gy limits) of the logistic EDO (Eq. (1)). xðk; t; x0 ¼ 0:5Þ, i.e., x as a function of k for some fixed values of t and x0 ¼ 0:5, is given in Figure 5. The map solutions are near to t ¼ G10 3 , i.e., the asymp-
DE GRUYTER
Fig. 5: Some solutions of logistic differential equation
totic solutions (t ! Gy limits) for logistic differential equation coincide with the map solution for T ! 0G limits. It is possible to appreciate the divergence region 0 a k < 1 and the distortion region of k < 0 for the map with T < 0 in Figure 5. These divergences appear when the general logistic map is solved for k ¼ 0, 2 a T a 3 and T a 0.
3.3 Other systems Several other systems can be considered to study the numerical connection between a map and its ODE. As an example, we describe below the Bernoulli shift map, the bi-dimensional cat map, and the non-linear Brusselator ODE. The Bernoulli shift and cat maps consider the mod 1 procedure to maintain finite solution since the ODE solutions diverge exponentially in time. For comparison, the logistic map is self-limited (for 2 < k < 4) since its ODE solution is also limited. We can classify the systems in accordance to the asymptotic behavior of the correspondent ODE analytical solution (without mod 1 procedure): a) Convergent – as logistic system or generalized cat EDO; b) Oscillatory – as generalized cat EDO; c) Divergent – as usual cat and Bernoulli EDOs, as described below. Systems with exponential divergent behavior, e.g., the bi-dimensional cat map, are extreme cases where the numerical connection between ODEs and maps can be considered.
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
DE GRUYTER
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
3.3.1 Bernoulli shift map
3.3.2 Cat map
The Bernoulli shift map is defined as:
The (bi-dimensional) cat map can be written as
Xnþ1 ¼ 2Xn ðmod 1Þ:
ð6Þ
That map can have periodic ðX0 A QÞ or chaotic ðX0 A IÞ behaviors. We can rewrite the Bernoulli shift map as ΔX ¼ Xnþ1 Xn ¼ Xn :
ð7Þ
Considering that the step-size ðΔtÞ of this map is Δt ¼ 1; it is possible to obtain the ODE (smoothed version) of the Bernoulli shift map as lim ΔX=Δt ¼ dX=dt ¼ Xðmod 1Þ:
Δt!0
ð8Þ
Xnþ1 ¼ Xn þ Yn ðmod 1Þ Ynþ1 ¼ Xn þ 2Yn ðmod 1Þ:
XðtÞ ¼ X0 expðtÞðmod 1Þ;
ð9Þ
where X0 is the initial condition. The Bernoulli shift map can be chaotic, but the correspondent ODE must not be chaotic. The choice of a finite step-size ðΔt 0 0; in particular Δt ¼ 1Þ, is the source of the chaos on the Bernoulli map. Note that, XðtÞ is XðtÞ ¼ 0 after t1 ¼ lnðX0 Þ for any t > t1 if the mod 1 procedure is applied exactly ðT ¼ 0Þ on the analytic ODE solution, i.e., Xðt1 Þ ¼ 1 ! ðmod 1Þ ! Xðt > t1 Þ ¼ 0. The map, for any value of Δt 1 T, i.e., Euler approximation of the Eq. (8), is Xnþ1 ¼ ð1 þ TÞXn :
ð10Þ
Note that if 2 < T < 0 the map converges to zero, similarly to its ODE solution for t < 0. If the following Bernoulli map is defined as: Xnþ1 ¼ kXn ðmod 1Þ:
ð11Þ
then its general map is: Xnþ1 ¼ ð1 þ ðk 1ÞTÞXn :
ð12Þ
The original Bernoulli map has k ¼ 2 and T ¼ 1. The Bernoulli map is a particular case of the generalized cat map studied below.
ð13Þ
The respective ODEs are dX=dt ¼ Y ) d 2 X=dt 2 dX=dt X ¼ 0 dY=dt ¼ X þ Y ) d 2 Y=dt 2 dY=dt Y ¼ 0;
ð14Þ
with following analytic solutions: XðtÞ ¼ C1 exp½γt þ C2 exp½ðγ þ 1Þtðmod 1Þ; YðtÞ ¼ γC1 exp½γt þ ðγ þ 1ÞC2 exp½ðγ þ 1Þtðmod 1Þ;
with the following analytical solution
81
ð15Þ
pffiffiffi where γ ¼ ð 5 1Þ=2 is the golden mean with limt!y YðtÞ=XðtÞ ¼ ðγ þ 1Þ: Xð0Þ ¼ C1 þ C2 and Yð0Þ ¼ γC1 þ ðγ þ 1ÞC2 . One possible general cat map is: Xnþ1 ¼ ½1 þ ðk1 1ÞTXn þ k2 TYn ðmod 1Þ Ynþ1 ¼ k3 TXn þ ½1 þ ðk4 1ÞTYn ðmod 1Þ:
ð16Þ
The original cat map has k1 ¼ k2 ¼ k3 ¼ 1, k4 ¼ 2, and T ¼ 1. The original Bernoulli map can be obtained from Eq. (16) if k1 ¼ 2, k2 ¼ k3 ¼ k4 ¼ 0, and T ¼ 1. The respective general cat ODEs are dX=dt ¼ ðk1 1ÞX þ k2 Y dY=dt ¼ k3 X þ ðk4 1ÞY;
ð17Þ
where the analytic solution ðX; YÞ can have different behaviors: as constant (const), convergent exponential (conv-exp), linear (lin), oscillatory (osc), and divergent exponential (div-exp), which depend on the k’s values as shown in Table 1. The solution of the Eq. (17) is given by: XðtÞ ¼ ð2AÞ1 expðBt=2Þ fexpðAt=2Þ½ðk1 k4 þ AÞC10 þ 2k2 C20 expðAt=2Þ½ðk1 k4 AÞC10 þ 2k2 C20 g ðmod 1Þ; ð18Þ
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
82
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
Table 1: Some analytic solution ðX; YÞ behaviors of the general cat ODEs ðT ¼ 0Þ for different k’s. The symbol (*) indicates explicit examples in Figure 6. X /Y behavior
k1
k2
k3
k4
const/const const/conv-exp* const/conv-exp conv-exp conv-exp conv-exp osc osc* const/div-lin const/div-exp div-osc-exp* div-exp div-exp div-exp div-exp div-exp (Cat)*
1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 1 1 1 2 2 1 1 1 2 2
DE GRUYTER
and YðtÞ ¼ ð2AÞ1 expðBt=2Þ fexpðAt=2Þ½ðk1 þ k4 þ AÞC20 þ 2k3 C10 expðAt=2Þ½ðk1 þ k4 AÞC20 þ 2k3 C10 g ðmod 1Þ; ð19Þ where A ¼ ½ðk1 k4 Þ 2 þ 4k2 k3 1=2 , B ¼ k1 þ k4 2, Xð0Þ ¼ C10 , and Yð0Þ ¼ C20 . Figure 6 and Table 1 show the behavior of the some solutions for integer values of k ’s. Note that, the procedure mod 1 was not applied. The asymptotic behavior of the convergent exponential solution (Figure 6a) is T independent for 0 < T < 2. That is, XðtÞ ¼ Xn ¼ Xð0Þ and limt!y YðtÞ ¼ limn!y Yn P j ¼ Xð0Þ y j¼0 Tð1 TÞ ¼ Xð0Þ for j1 Tj < 1. Similar behavior appears for different (real number) k’s if B < 0ðk1 þ k4 < 2Þ and jAj a jBjð½ðk1 k4 Þ 2 þ 4k2 k3 1=2 a jk1 þ k4 2jÞ. As an example of k variation, the oscilla-
Fig. 6: Y ðtÞ solution of general Cat Map without mod 1 procedure for different k’s and T ’s with Xð0Þ ¼ C10 ¼ 1 and Yð0Þ ¼ C20 ¼ 0.
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
DE GRUYTER
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
tory analytic trajectory YðtÞ (Figure 6b) is shown in Figure 7 as a function of the k3 and t for k1 ¼ 1, k2 ¼ 1, and k4 ¼ 1. The oscillatory (Figure 6b), oscillatory-divergent exponential (Figure 6c), and divergent exponential (Figure 6d) behaviors have crescent dependence of T. The divergent exponential behavior is most sensible to parameter T. For example, the k1 ¼ 1, k2 ¼ 1, k3 ¼ 1, k4 ¼ 2 (Figure 6c) and k1 ¼ 1, k2 ¼ 1, k3 ¼ 1, k4 ¼ 2 (Figure 6d) maps with mod 1 procedure are shown in Figures 8a and b.
83
Note that, the result of the map with T ¼ 0:01 is similar to that with T ¼ 106 for a small t. A more complete study is shown in Figure 9. In general, the region of acceptability of T depend on the values of k ’s as a logistic system and the time propagation ðtÞ for divergent behavior. The Figure 9 shows the agreement (jYT ðtÞ YT¼106 ðtÞj < 0:1) between the map solution YT>106 ðtÞ for any T > 106 and YT¼106 ðtÞ for T ¼ 106 for the examples given in Figure 8. Potentially, the solutions obtained with those T-maps can coincide with EDO (for analytic or with step-size f T) solution until time t delimited by curves shown in Figure 9, indicated by region “A’’. For example, the solution of the map for T A 0:01 is acceptable for t < 10. Of course, more curves TðtÞ can be added to Figure 9 for a larger range of k ’s and T’s in order to obtain a better estimative of the acceptable region of T: “A’’. It is important to address that, any numerical procedure with variable step-sizes (e.g., Runge-Kutta-Fehlberg) for solving the EDO can not give correct solutions if the mod 1 procedure is added since the solution strongly depends on the step-size.
3.3.3 Brusselator ODE The (bi-dimensional and non-linear) Brusselator [7] ODE is given by:
Fig. 7: YðtÞ analytic solution of general cat map as a function of k3 and t for k1 ¼ 1, k2 ¼ 1, and k4 ¼ 1 and Xð0Þ ¼ 1 and Yð0Þ ¼ 0.
dX=dt ¼ A ðB þ 1ÞX þ X 2 Y dY=dt ¼ BX X 2 Y;
Fig. 8: Y ðtÞ solution of general cat map with mod 1 procedure for different k’s and T ’s with Xð0Þ ¼ C10 ¼ 0:999 and Y ð0Þ ¼ C20 ¼ 0.
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
ð20Þ
84
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
Fig. 9: T as a function of t. Acceptable region of T , A for: k1 ¼ 1, k2 ¼ 1, k3 ¼ 1, k4 ¼ 2 ð1; 1; 1; 2Þ and k1 ¼ 1, k2 ¼ 1, k3 ¼ 1, k4 ¼ 2 ð1; 1; 1; 2Þ. The curves delimit the agreement: jYT ðtÞ YT ¼106 ðtÞj < 0:1.
where A and B are parameters. Its discrete general map is: Xnþ1 ¼ TA þ ½1 TðB þ 1ÞXn þ TXn2 Yn Ynþ1 ¼ TBXn þ Yn TXn2 Yn :
ð21Þ
There are no analytical solutions for the Brusselator ODE. This solution can show that regular and chaotic behavior depends on the initial conditions (Xðt0 Þ and Yðt0 Þ) and on parameters A and B. Of course, the solution of the Eq. (21) for T ! 0 coincides to that one of the Eq. (20). However, for finite values of T the solution of the Eq. (21) can be more unstable than one of the Brusselator ODE. In particular, if Xðt0 Þ ¼ A and Yðt0 Þ ¼ B=A the trajectories of Eq. (20) and Eq. (21) are reduced to a constant (fixed point).
DE GRUYTER
solvable (non-chaotic) system with an analytical solution that becomes chaotic due to the “variation’’ of a control parameter (step-size of the Euler method or, e.g., other Runge-Kutta methods). The bi-dimensional general cat map and ODE were studied for different sets of parameters (k’s), which lead to different asymptotic behaviors, such as convergent, oscillatory, and divergent. In this system there are two sources of instabilities: a) mod 1 procedure; b) the stepsize of integration ðTÞ. The mod 1 procedure depends strongly on the step-size of integration ðTÞ. Generically, for any finite step-size (numerical discretization procedure) and for any numerical method, chaos can emerge from a finite propagation length. The connection between these two limits, analytical solution and chaos, allows us to establish the region of acceptability for the numerical method. The limits of a numerical process can be found in between the limits of the analytical and chaotic solution. In general, the rounding and truncation errors and the finite number of interactions can disturb the process and invalidate the numerical results. These studies can be made for any ODE with an analytical solution (or not) and its correspondent chaotic (or not) map. The solutions obtained from the numerical discretization are equal to or more instable than those obtained from a continuous procedure. Since more instabilities can be unavoidable, it is important to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution. Acknowledgments: ALC acknowledged to Prof. Marcus M. A. Aguiar (Unicamp) and Dr. Marcio F. da Silva (UFABC) for fruitful discussions and to the Universidade Federal do ABC (UFABC) for the Visiting Professor position during the period from February/2008 to August/ 2009. Support by FAPESP (Brazil) is also acknowledged. Received: June 30, 2011. Accepted: December 19, 2012.
4 Discussions and conclusion
References
The extended logistic map obtained from the logistic ODE, considering the Euler method with non-unitary step-size, was studied. The numerical (asymptotic) solution coincides with the ODEs analytical one with the Euler method if ð1 2=TÞ < k < ð1 þ 2=TÞ. However, if k > 1 þ 2=T or k < 1 2=T, the map reaches the bifurcation region and, therefore, the numerical solution cannot agree with analytical one. This case is an example of a
[1] [2] [3] [4]
Gleick, J., Chaos: Making a New Science, Penguin (Non-Classics) (1988). May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261(1976), 459–467. Lotka, A.J., Undamped oscillations derived from the law of mass action, J. Am. Chem. Soc., 42(1920), 1595–1599. Lorenz, E.N., Computational Chaos – A prelude to computational instability, Physica D, 35(1989), 299–317.
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
DE GRUYTER
[5]
M.R. de Lima et al., Numerical Connection between Map and its Differential Equation
Yamaguti, M.; Ushiki, S., Chaos in Numerical Analysis of Ordinary Differential Equations, Physica D, 3(1981), 618–626. [6] Gutzwiller, C., Chaos in Classical and Quantum Mechanics, Springer, New York (1990). [7] Nicolis, G.; Prigogine, I., Self-Organization in Nonequilibrium Systems, John Wiley & Sons, New York (1976).
85
[8] Stewart, J., Calculus, Brooks/Cole Publishing Company, 4 th edition (1999). [9] Boyce, W.E., DiPrima, R.C., Elementary Differential Equations and Boundary Value Problems, Wiley, 7 th edition (2000). [10] Zill, D.G., First Course in Differential Equations, Brooks Cole, 5 th edition (2000).
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
Authenticated |
[email protected] author's copy Download Date | 3/7/13 1:51 AM
