Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here we analyse one such recurrence map defined through a 2-dimensional difference equation in (X, Y ) called Gumowski-Mira transformation [18]. This system involves a non linear function f(x) that is advanced in time by one iteration, in the y equation. It has been reported that computer simulations of this iterative scheme [19] give rise to a variety of 2-dimensional images in the phase plane (x, y) called GM patterns that resemble natural objects like star fish, jelly fish, wings of a butterfly, sections of fruits, flowers of varying shapes etc. Apart from a mere fascination in creating these patterns, one can implement a scheme like Iterative function scheme for producing 2-dimensional fractals or fractal objects of nature using this map for different values of the parameters involved. Moreover, the final pattern is found to depend very sensitively on the parameters, a feature which can be exploited in decision making algorithms and control techniques in computing and communication. 61
Our motivation in this work is to analyse the underlying dynamics behind the formation of these patterns and their possible bifurcation sequences. In this context, we would like to mention that the period doubling sequences and chaos doubling phenomenon in this system has been reported earlier for a chosen set of parameter values [11]. We are concentrating on another set that was used in [19] for generating GM patterns. Here the dynamics involved is different in the sense that the intermittency route and odd period cycles predominate and the route to chaos is via formation of quasiperiodic bands and band mergings. The detailed evolution of the basin structure during the scenario until onset of chaos and final escape forms part of our work. The scaling behaviour near intermittency, stability of the lowest periodic cycles are also analysed. Moreover our work provides evidence for periodic window and repeated substructures in the bifurcation scenario which show self similarity with respect to scaling in the parameter. This can therefore account for the sensitive dependence of the asymptotic dynamical states or phase portraits on the relevant parameter of the system. We also analyse a variant of the GM map called modified GM map in which the nonlinear function f(x) itself is used in the second relation. The system is found to have a prominent period doubling sequence in the bifurcation scenario. 3.2 Gumowski-Mira map The transformation called Gumowski-Mira map can be written as a two dimensional recurrence relation defined in the x y plane as x n+1 = y n + a(1 by 2 n )y n + f(x n ) y n+1 = x n + f(x n+1 ) (3.1) with f(x n ) = µx n + 2(1 µ)x2 n 1+x 2 n and µ as the control parameter. 62
(a) (b) Figure 3.1: Variation of a) the function f(x) b) its first iterate for three different values of µ.3,.16 and.39 It should be noted that the function f(x) in the y equation is advanced in time by one iteration which provides richer and complex dynamical behaviour. The nature of variation of the non linear function f(x ) as well as f(x 1 ) for different µ values in the range.3 to +.39 with a =.8 and b =.5 is shown in Fig 3.1a and 3.1b. Here x is the starting seed value and x 1 is the first iterate. These figures give an indication of how different the functions in the x and y equations are and how fast and differently they change as iterations proceed. The iterates of this map give rise to a variety of interesting patterns in the (X, Y ) plane and hence it is useful as a two dimensional iterative scheme for producing fractal objects. The phase space plots known as GM patterns in Fig 3.2 illustrates this. In these calculations µ is varied in steps of.1. Fig 3.3 gives another set of GM patterns which resemble marine living creatures, slice of a tomato, a plankton etc. [19]. It is widely recognised that in nature fractal geometry plays a crucial role in forming various self similar object like fern leaf. Analogously one is tempted to assume that the shapes of living marine creatures appear to be a realization of the GM transformation, and can argue that the mathematics of the complexity of systems such as chaos and fractals underlies the living world. 63
(a) (b) (c) (d) (e) (f) Figure 3.2: GM patterns generated in the phase space ie. (X, Y ) plane by the discrete system in equation (3.1). Here a =.8, b =.5, X =.1, Y =...6 µ.1, µ =.1. It is clear that the asymptotic dynamic states depends very sensitively on the control parameter µ. We observe that the final attractor of the GM map depends very sensitively on the value of µ. Even very small changes in µ 1 6 results in drastic changes in the nature and shape of the attractor and this can be exploited in decision making algorithms and control techniques. 64
(a) µ =.15 (b) µ =.2 (c) µ =.22 (d) µ =.31 (e) µ =.55 Figure 3.3: GM patterns for different µ values which resemble marine living creatures, slice of a tomato, a plankton etc. 65
3.3 Dynamical states and bifurcation scenario It is found that the non linear function f(x) is the y equation is advanced in time by one iteration which provides richer and complex dynamical behaviour. The one cycle fixed points of the system is determined by solving the equations x = y + a(1 by 2 2(1 µ)x2 )y + µx + 1 + x 2 2(1 µ)x2 y = (µ 1)x + (3.2) 1 + x 2 using MAXIMA and the only real and bounded solutions of interest are found to be (, ) and (1, ), which are unstable in the relevant parameter regions of our study. The most prominent elementary stable cycle here is a four cycle born by saddle node bifurcation. The stability of this four cycle can be established numerically by calculating the eigen values of the Jacobian M for chosen values of µ and the corresponding elements of the four cycle. Thus if (δx n, δy n ) are the small perturbations to the cycle elements, we have ( ) ( ) δxn+1 δxn = M. (3.3) δy n+1 δy n where M 11 = 4x3 (1 µ) 4x(1 µ) + + µ (1 + x 2 ) 2 1 + x 2 M 12 = 1 2aby 2 + a(1 by 2 ) 66
( M 21 = 1 + µ 4x3 (1 µ) + (1 + x 2 ) 2 ( 4(1 µ) ) 4x(1 µ) + µ 1 + x 2 ) ( ) 3 + 4x(1 µ) 1+x + µ y + ay(1 by 2 ) + 2x2 (1 µ) 2 1+x + xµ 2 ( ) 2 + 1 + (y + ay(1 by 2 ) + 2x2 (1 µ) 1+x + xµ) 2 ) ( 2 ) y + ay(1 by 2 ) + 2x2 (1 µ) 1+x + xµ 2 4x3 (1 µ) (1+x 2 ) 2 ( 4(1 µ) 4x3 (1 µ) (1+x 2 ) + 4x(1 µ) 2 1+x + µ 2 1 + (y + ay(1 by 2 ) + 2x2 (1 µ) 1+x 2 + xµ) 2 M 22 = (1 2aby 2 + a(1 by 2 ))µ 4(1 2aby 2 + a(1 by 2 ))(1 µ)(y + ay(1 by 2 ) + 2x2 (1 µ) 1 x + xµ) 3 ( ) 2 2 + 1 + (y + ay(1 by 2 ) + 2x2 (1 µ) 1 x + xµ) 3 2 4(1 2aby 2 + a(1 by 2 ))(1 µ)(y + ay(1 by 2 ) + 2x2 (1 µ) 1+x 2 + xµ) 1 + (y + ay(1 by 2 ) + 2x2 (1 µ) 1+x 2 + xµ) 2 The eigen values of M evaluated for a few typical µ values with a =.8; b =.5 in the four cycle window are given in 3.1. Table 3.1: The Eigen values of the stability matrix for different µ values in the four cycle window E < 1 in all the above cases indicating stability of the corresponding four cycle. µ E 1 E 2.964.63.63.95.64.64.5.93.93.2.81.81.3.78.78 Bifurcation scenario of the Gumowski-Mira map is studied for a particular value of parameters a = 1.1, b =.2 [11]. The bifurcation diagram for of GM map for these parameter values shows that as the control parameter increases a kind of chaos doubling phenomenon is observed. 67
We numerically investigate in detail, the possible dynamical states and their bifurcation patterns as µ is varied. We find that the bounded interval for the map lies in the interval [ 1, 1] for a =.8 and b =.5. Moreover the most prominent elementary cycle of periodic behaviour is four which occurs in many intermittent windows of µ in the bifurcation diagram. The full scenario is given in Fig 3.4a which is mostly dominated by broad windows of odd cycles like 7, 11 etc. Here out of 1, iterate 9 are discarded as initial transients and the next 1 are plotted. The specific regions of the windows of such cycles are zoomed and reproduced in Fig 3.4b, 3.4c and 3.4d. x 8 6 4 2 2 4 6 8 1.9.8.7.6.5.4.3.2.1 µ (a) (b).3 < µ <.25 (c).2 < µ <.55 (d).1 < µ <.95 Figure 3.4: a) Bifurcation scenario in the range µ = 1 to, for a =.8, b =.5. The windows of periodic cycles are zoomed for details in b, c and d as indicated. 68
(a) (b) (c) (d) Figure 3.5: The intermittency behaviour before the stabilisation of a 7 cycle is shown in the x n -n plot in (a) and corresponding phase portrait in (b) for µ =.2734. Note that this is the typical behaviour near the onset of each periodic window. The self similar and repeating substructures inside this stability window is shown in Figs 3.5c and 3.5d where the 7 and 22 cycles are seen to recurr. (The rectangle shown in 3.5c is zoomed in 3.5d). In general these windows of periodic cycles born by tangent bifurcations or saddle node bifurcations at their left ends exhibit intermittency behaviour in their iterates. The transition to chaos takes place for small increase of values of µ when the periodic cycle becomes unstable giving rise to quasiperiodic bands which become chaotic and merge together. We illustrate this for the seven cycle window. Near the left end of a periodic seven cycle window Fig 3.5a gives the x n n plot with µ =.2734 for 6 iterations showing intermittently laminar and chaotic behaviour. Fig 3.5b gives the corresponding x y plot. It is interesting to note that the periodic window structure shows self similarity in their substructures with respect to 69
x 15 1 5 5 1 15 2 4 6 8 1 12 14 16 18 2 n (a) log< l > 2.72 2.7 2.68 2.66 2.64 2.62 2.6 2.58 2.56 2.54 2.52 2.5 6 5.95 5.9 5.85 5.8 5.75 5.7 5.65 5.6 (b) log µ µ c Figure 3.6: Type I intermittency near the onset of a 1 cycle a) the laminar and irregular behaviour for µ =.31251 b) the scaling of average laminar region l as function of µ µ c where µ c =.312498. The scaling index in this case is.49. scaling in µ. This is evident from the plots in Fig 3.5c and Fig 3.5d where a 7 cycle and a 22 cycle patterns repeat at two levels of zooming in the parameter range. This recurring self similarity with the consequent intermittency in the beginning of each periodic cycle and quasi-periodicity at its end, accounts for the variety and richness of the GM patterns in the x y plane with its highly sensitive dependence on µ. The nature of intermittency before the birth of a periodic cycle is analysed by calculating the average life time of the laminar region l in the x n n plots. For one such region near µ =.31251 before a ten cycle window Fig 3.6a shows the laminar and irregular behaviour for µ =.31251, the variation of l as a function of µ µ c is studied in Fig 3.6b. Here µ c is the critical value when it becomes chaotic. We can write l µ µ c ν. ν then defines the scaling index that helps to identify the type of intermittency. Here µ c =.312498 and ν =.49 so that the intermittency is Type I [111] which is usually associated with saddle node bifurcation. The basin of attraction for different dynamical states inside the four cycle window is shown in Fig 3.7a 3.7c. Here black region corresponds to 7
(a) (b) (c) (d) Figure 3.7: The basin of attraction for the different dynamical states inside the 4 cycle window. Here black region corresponds to the basins of the stable 4 cycle as the µ value increases from intermittency towards periodic cycle. a) µ =.815 b) µ =.81 c) µ =.8. In (d), the black region corresponds to bounded chaos while white region that of escape for µ = 1.1. basin of the stable four cycle as the µ value increases from intermittency toward periodic cycle. Fig 3.7d gives the various boundary between bounded and escape regions, where escape taken place beyond µ = +1. Black region corresponds to escape at µ = 1.1. To calculate the spectrum of Lyapunov exponents for a two dimensional map, the eigen values of the Jacobian matrix are calculated [112]. For a two dimensional map, the two eigen values are j 1 and j 2 such that 71
.4.3.2 λ.1.1.2.3 1.9.8.7.6.5.4.3.2.1 µ Figure 3.8: Lyapunov exponent for the GM map as a function of control parameter µ, µ is varied from to 1. j 1 > j 2. Lyapunov exponents are then defined by 1 λ i = lim n n log 2 j i (n). where j i are the absolute values of eigen values. Fig 3.8 shows the Lyapunov exponent for the GM map as a function of control parameter µ and µ is varied from to 1. The regions showing chaos where λ > and periodic orbits where λ < are clearly visible. 3.4 Modified Gumowski-Mira map Modified Gumowski-Mira map is a variant of Gumowski-Mira map defined by the following set of equations. x n+1 = y n + a(1 by 2 n)y n + f(x n ) y n+1 = x n + f(x n ) (3.4) with f(x n ) = µx n + 2(1 µ)x2 n 1 + x 2 n 72
x 1.2 1.8.6.4.2.2.4.9.8.7.6.5.4.3.2.1 µ (a).145.15.155.16 x.165.17.175.18.185.6495.649.6485.648.6475.647.6465.646.6455.645 µ (b) Figure 3.9: a) Bifurcation scenario of the MGM map as µ is varied from.9 to. b) A small region in the bifurcation diagram is zoomed and self similar structure is observed. In this map the nonlinear function f(x n ) in the iteration in both x and y are in step and does not differ by one iteration. This leads to consequent changes in the bifurcation scenario. Here we fix the values of a and b as a =.8 and b =.5. Bifurcation scenario of the MGM map can be explained using Fig 3.9a as µ is varied slowly. The various periodic orbits are clearly observed in the bifurcation diagram. A small region in the bifurcation diagram is zoomed so that self similarity can be observed. This shown in Fig 3.9b. For µ >, the x y plots spirals into a stable fixed point at (, ). As µ is reduced slowly, a limit cycle is developed. Fig 3.1a shows the limit cycle behaviour of the MGM map for µ =.1. As µ is slowly reduced, there is breaking of symmetry and it split into a stable four cycle. Fig 3.1b shows the stable four cycle of the MGM map for µ =.4. Then the period doubling phenomenon takes place and it subsequently lead to chaos for µ =.7. Chaotic attractor of the MGM map is shown in Fig 3.1c for µ =.7. MGM map undergoes Hopf bifurcation and period doubling bifurcation. The dynamical states of the MGM map is very much different from the GM map. Since intermittency is not observed in this map, the interesting patterns are not observed in the phase plots as in GM maps. 73
y.4.3.2.1.1.2.2.1.1.2.3.4 x (a) y.8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8 (b) x y.7.6.5.4.3.2.1.1.2.3.3.2.1.1.2.3.4.5.6.7 x (c).8 Figure 3.1: a. The limit cycle behaviour of the MGM map for µ =.1; b. The stable four cycle of the MGM map for µ =.4; c. Chaotic attractor for µ =.7. λ.5.5 1 1.5 2 2.5 3 3.5 1.9.8.7.6.5.4.3.2.1 Figure 3.11: The variation of the largest Lyapunov exponent as a function of the parameter µ. µ 74
We have also calculated the Lyapunov Exponents using standard algorithms [113]. Fig 3.11 depicts the variation of the largest Lyapunov exponent as a function of the parameter µ. In Fig 3.11 the regions showing λ > correspond to chaos and λ < correspond to periodic orbits. 3.5 Conclusion In this chapter we analyse the large variety of interesting and lively patterns exhibited by the GM map and find that they are attractors of the system in the phase plane in the neighbourhood of higher order cycles. Most prominent and frequently occuring periodic cycle is a four cycle and higher order periodic cycles are also common in GM map. The intermittency observed is of Type I with an exponent.5. This is usually associated with saddle node bifurcation. The dependence of the phase space structure of the patterns on minute changes in the parameter makes it a useful tool in decision making algorithms. This sensitive dependence is due to the recurring periodic and self similar substructures in the bifurcation scenario, each with its own intermittency, periodicity, quasi periodic band and merging of bands leading to chaos. The bifurcation scenario and the corresponding Lyapunov exponent are also studied. A variant of GM, modified GM map is also studied. The non linear function occuring in the x and y equation are in step. MGM map follows a Hopf bifurcation sequence in the bifurcation scenario. The route to chaos is then through period doubling bifurcation. This bifurcation structure is analysed in detail. In the following chapter, we will be using these two maps as model systems for studying synchronisation in two dimensional discrete systems. 75