A scheme for designing extreme multistable discrete dynamical systems

Transcription

1 Pramana J. Phys. (7) 89:4 DOI.7/s y Indian Academy of Sciences A scheme for designing etreme multistable discrete dynamical systems PRIYANKA CHAKRABORTY Department of Applied Mathematics, University of Calcutta, 9 APC Road, Kolkata 7 9, India priya.chakraborty8@gmail.com MS received August 6; revised 6 February 7; accepted 6 April 7; published online August 7 Abstract. In this paper, we propose a scheme for designing discrete etreme multistable systems coupling two identical dynamical systems. Eistence of infinitely many attractors in the system is obtained via partial synchronization between two systems for a given set of parameters. We give a conjecture that etreme multistable systems can be designed by coupling two m-dimensional dynamical systems in such a way that i ( i m ) number of state variables of the two systems synchronize completely and (m i) number of state variables keep constant difference. We demonstrate the applicability of our scheme in two-dimensional (D) as well as threedimensional (3D) discrete dynamical systems. In particular, we discuss our scheme taking coupled D Hénon maps, coupled D Duffing maps and coupled 3D Hénon maps. We have analytically shown the eistence of fied points and period- orbits in the coupled system with the variation of initial conditions. These analytically derived conditions matched very well with the numerical simulation results. Variation of the largest Lyapunov eponent with the initial conditions is shown to confirm the eistence of etreme multistability in the model. Our scheme may be useful for designing physically, chemically and biologically useful multistable discrete dynamical systems. Keywords. PACS No. Multistability; Hénon map; Duffing map; Lyapunov eponent a. Introduction The phenomenon of multistability has been found in almost all areas of science and nature, e.g. in visual perceptions [,], in biological systems [3,4], in hydrodynamics [5], in optical systems [6,7], in semiconductor materials [8], in chemical reactions [9 ], in ecosystems [,3], in neuron dynamics [4], in climate dynamics [5 8], in social systems [9,] etc.a multistable dynamical system is one that possesses a large number of asymptotic stable states for a fied set of parameters depending on initial conditions. Trivial multistability of a system can be considered as the coeistence of several stable fied points, i.e. the coeistence of multiple point attractors in the phase space. On the other hand, generalized multistability occurs when there is coeistence of many non-trivial attractors such as limit cycles, chaotic attractors etc. Therefore, multistable systems are sensitive to initial conditions. Generalized multistability was first discovered in an eperimental nonlinear dynamical system by Arecchi et al [] in 98 and in a computational set-up by Arecchi et al [6] in 985. The phenomenon of multistability has been identified in different classes of systems, such as weakly dissipative systems, coupled systems, delayed feedback systems, parametrically ecited systems and stochastic systems []. The appearance of a multitude of attractors depends in general on the most important parameters characterizing a particular system class, such as, the strength of dissipation, the kind and strength of coupling, the value of the time delay, amplitude and frequency of the parameter perturbation and the noise intensity [3]. Here our basic motivation to study multistable systems is to identify the mechanisms that lead to multistability and to prove rigorously under what circumstances the phenomenon may occur. In etreme multistability, the number of coeisting attractors is infinite and this kind of system has been reported in a coupled continuous system by Sun et al [4]. It has been shown that the reason for the emergence of infinitely many attractors lies in the appearance of a conserved quantity in the long-term limit. Recently, a coupling scheme to obtain multistable continuous dynamical systems has been developed by

2 4 Page of 5 Pramana J. Phys. (7) 89:4 Hens et al [5]. The coupling scheme is developed in a systematic way by using the principle of partial synchronization. Hens et al [5] formulated one precondition for the emergence of etreme multistability and reported that the coeistence of infinitely many attractors in two coupled m-dimensional continuous systems will be possible if m of the variables of the two systems are completely synchronized and one of them obeys a constant difference between them. We propose a scheme for designing multistable discrete dynamical systems and we also generalize the conditions of Hens et al [5] and reformulate that the coeistence of infinitely many attractors in two m-dimensional coupled systems will be possible if i of the variables of the two systems are completely synchronized and m i (where i m ) of them obey a constant difference between them. Many works have been done for generating multistable system coupling continuous dynamical systems [6 8] but there is no simple coupling scheme for generating multistable discrete dynamical systems. In this paper, we propose a scheme for generating generalized multistable systems coupling discrete dynamical systems. We discuss the effectiveness of our scheme taking D Hénon map, D Duffing map and 3D Hénon map. The paper is organized as follows: in, a generalized scheme for designing multistable discrete system is proposed. Section 3 illustrates our scheme with the help of two-dimensional maps, e.g., Hénon map, Duffing map and three-dimensional Hénon map and the corresponding bifurcation analysis of each modified coupled map. The numerical simulation results are analysed in 4. Finally, a conclusion is drawn in 5.. Scheme for designing multistable systems In this section we propose a scheme for generating discrete multistable systems coupling two identical discrete dynamical systems. We consider an m-dimensional map of the following type: (n + ) = f ((n)), () where f is a vector function with m components. Now, we couple two identical m-dimensional maps of the above type in the following way: p (n + ) = f p ( (n), (n),..., m (n)) + u p ( (n), (n),..., m (n); y (n), y (n),...,y m (n)) () and y p (n + ) = f p (y (n), y (n),...,y m (n)) + v p ( (n), (n),..., m (n); y (n), y (n),...,y m (n)), (3) where u p and v p (p =,, 3,...,m) are the controllers. We have to choose suitable controllers to obtain multistable systems. We define the synchronization error between systems ()and(3)ase p (n) = y p (n) p (n), p =,,...,m and obtain dynamical equations for time evolution of error as follows: e p (n + ) = f p (y (n), y (n),...,y m (n)) f p ( (n), (n),..., m (n)) + v p u p (4) p =,, 3,...,m. We conjectured that multistable systems can be designed by choosing u, u, u 3,...,u m and v,v,v 3,...,v m in such a way that i ( i m ) number of state variables synchronize completely and (m i) number of state variables keep constant difference. Therefore, according to our scheme, we choose u p and v p (p =,,...,m)insuchawaythat, e r (n) =, r =,,...,i (5) and e i+s (n) = c s, s =,,...,m i, (6) where i m. Then, for such choice the system formed by coupling ()and(3) may show multistability. Therefore, the errors e r (n) must tend to zero, i.e., y r (n) = r (n) (r =,,...,i) ande i+s remain constant in time, y i+s (n) = i+s (n)+c s (s =,,...,m i)asn. Hence, c s s are initial condition-dependent constants. Now the dynamics of the coupled systems ()and(3) is equivalent to the following system: p (n) = f ( (n), (n),..., m (n)) + u p ( (n), (n),..., m (n); (n), (n),..., i (n), i+ (n) + c,..., m (n) + c m i ). (7) p =,,...,m and c, c, c 3,...,c m i s are initial condition-dependent constants. System (7) showsmultistable behaviour if its dynamics changes qualitatively with variation of c s s (s =,,...,m i). 3. Theoretical results In this section, we illustrate our scheme for constructing multistable systems by considering coupled D Hénon

3 Pramana J. Phys. (7) 89:4 Page 3 of 5 4 maps, coupled D Duffing maps and coupled 3D Hénon maps. Consider the following two-dimensional discrete dynamical system: n+ = f ( n, y n ) y n+ = g( n, y n ). (8) The Jacobian matri of this system is given by ( ) f / n f / y J = n. g/ n g/ y n Let (, y ) be a fied point of system (8). Then the Jacobian matri at the fied point (, y ) is J (,y ), denoted as J. Assume that and are two roots of the characteristic equation of the Jacobian matri J (,y ), then we have the following well-known Definition and Lemma [9,3]. DEFINITION A fied point (, y ) is called (i) sink or locally asymptotically stable if < and <, (ii) source or locally unstable if > and >, (iii) saddle if > and < (or < and > ) and (iv) non-hyperbolic if either one of the eigenvalue is of unit modulus, i.e., =or =. Lemma. Let H() = + B+C. Suppose that H () > ; and are roots of equation H() =. Then we have:. < and < if and only if H( ) > and C < ;. < and > (or > and < ) if and only if H( ) <; 3. > and > if and only if H( ) > and C >; 4. = and = if and only if H( ) = and B =, ; 5. and are comple and = = if and only if B 4C < and C =. Lemma. Let J is a Jacobian matri at a fied point, then we have. All eigenvalues of J satisfy, < if and only if tr J < det J <.. Assume that tr J = det J. If tr J >, then the eigenvalues of J are = and = det J. If tr J <, then the eigenvalues of J are = and = det J. 3. Assume that tr J det J =. Then the eigenvalues of J are, = e ±iω where ω = cos (tr J /). 3. Coupled D Hénon maps The Hénon map [3] is a classical eample of twodimensional discrete dynamical system described by (n + ) = a (n) + y(n), y(n + ) = b(n), (9) where a, b are parameters. For a =.4andb =.3the map shows chaotic behaviour. Two Hénon maps describe by the state variables i and y i (i =, ) are coupled in a highly nonlinear way through controllers u ij (i, j =, ) in the following manner: (n + ) = a (n) + y (n) + u y (n + ) = b (n) + u (n + ) = a (n) + y (n) + u y (n + ) = b (n) + u. () The error dynamics of the coupled Hénon map is given by e (n + ) = a( (n) + (n))e (n) + e (n) + u u e (n + ) = be (n) + u u. () To construct multistability in the coupled system, we choose u ij (i, j =, ) in such a way that e (n), i.e. (n) (n) and e (n) c, i.e., y (n) y (n) + c (c is a constant which depends on initial conditions of the system). 3.. Case I. We can easily choose u = a( (n) (n)) ( (n) (n)) + y (n) y (n) u = b( (n) (n)) u = u = y (n) y (n). () Then () becomes e (n + ) = e (n), i.e., e (n + ) = n e () and e (n + ) = e (n), n =,, 3,... As time increases e (n+) ande (n+) e (n), then () takes the following form: (n + ) = a (n) + y (n) ( (n) (n)) y (n + ) = b (n)

4 4 Page 4 of 5 Pramana J. Phys. (7) 89:4 (n + ) = a (n) + y (n) y (n + ) = b (n) + y (n) y (n). (3) Then the dynamics of coupled system can be determined completely by the system given by the equations (n + ) = a (n) + y (n) y (n + ) = b (n) + c. (4) 3.. Case II. Again, we choose different sets of controllers as u = a( (n) (n)) ( (n) (n)) + y (n) y (n) u = b( (n) (n)) + y (n) y (n) c u = u = y (n) y (n). (5) Then () becomes e (n + ) = e (n), i.e. e (n + ) = n e () and e (n + ) = be (n) + c, n =,,, 3... As time increases e (n + ) which implies that e (n +) e (n)+c (c is any constant). So, as e (n), e (n+) c.again,c = e (n+) which implies c = y (n+) y (n+). If we decrease n, thenc = y (n) y (n),,c = y () y (). Hence, () takes the following form: (n + ) = a (n) + y (n) ( (n) (n)) y (n + ) = b (n) b (n) + y (n) y (n) y () + y () (n + ) = a (n) + y (n) y (n + ) = b (n) + y (n) y (n). (6) Again the dynamics of the above coupled system can be determined completely by the system given by the equations (n + ) = a (n) + y (n) y (n + ) = b (n) + y () y (). (7) 3..3 Bifurcation analysis of modified D Hénon map. The fied points of system (4) are obtained from = a + y y = b + c. (8) The fied points are ( ) b + β b(b + β) P, + c a a and ( b β b(b β) P, + c ), a a where β = (b ) + 4a( + c )(>). The Jacobian matri of (4) at the fied point (, y) can be written as ( ) a J =. b Now, we analyse the local stability of the fied point P as well as P. The characteristic equation of the Jacobian matri J of system (4) evaluated at the fied point P can be written as + B + C =, where B = a and C = b.leth() = + B + C,then H() = + a b (>) and H( ) = a b. With Lemma, we have the following results: Theorem... When c <(3(b ) /4a) and b >, the fied point P is sink and locally asymptotically stable.. When c >(3(b ) /4a), the fied point P is saddle. 3. When c <(3(b ) /4a) and b <, the fied point P is source and locally unstable. 4. When c = (3(b ) /4a), c =, a b, the fied point P is non-hyperbolic, the system may undergo flip bifurcation (period-doubling bifurcation). For a =., b =.3, the fied point P ( c, c + c )issink and asymptotically stable for the range of c along c <.675 shown in figure a. For the special case that c =.675, eigenvalues of the Jacobian matri J are,.3. So one eigenvalue is of unit modulus and hence system (4) possesses period-doubling bifurcation, i.e., the fied point (3.5, 3.75) is non-hyperbolic. By Lemma, for the fied point P ( b β a, b(b β) a + c ), we have the following result: Theorem... If c <(3(b ) /4a), the fied point P is asymptotically stable.. Assume b β =.3, which implies β = b.3, b.7.now, if (i)β>b, then the eigenvalues of J are = and =.3 and if (ii) β <b, then the eigenvalues of J are = and =.3. So, P is non-hyperbolic. Now, at a =., b =.3, P ( c, c + c ) is saddle as modulus of one

5 Pramana J. Phys. (7) 89:4 Page 5 of 5 4 of the eigenvalues is greater than. Again, for the second case, β =,.4, but β cannot be negative. So, when β =, the fied points are identical as ((b )/a, (b(b )/a) + c ), i.e., ( 3.5,.5 + c ) and the corresponding eigenvalues of Jacobian matri J are,.3 for any value of c. Then the fied point is nonhyperbolic. 3. Coupled D Duffing maps The Duffing map takes a point ( n, y n ) in the plane and maps it to a new point given by (n + ) = y(n) y(n + ) = b(n) + ay(n) y 3 (n). (9) It has chaotic dynamics for a =.75 and b =.. Couple the Duffing map through the controllers as follows: (n + ) = y (n) + u y (n + ) = b (n) + ay (n) y 3 (n) + u (n + ) = y (n) + u y (n + ) = b (n) + ay (n) y 3 (n) + u. () The synchronization error dynamics of the two coupled Duffing maps are given by e (n + ) = e (n) + u u e (n + ) = be (n) + ae (n) (y 3 (n) y3 (n)) + u u. () We design the controllers in such a way that, e (n), i.e., (n) (n) and e (n) c, i.e., y (n) y (n) + c. We choose, u = y (n) y (n) ( (n) (n)) u = a(y (n) y (n)) (y 3 (n) y3 (n)) b( (n) (n)) u = u = y (n) y (n). () Now system () takes the following form: (n + ) = y (n) ( (n) (n)) y (n + ) = b (n) + ay (n) y 3 (n) (n + ) = y (n) y (n + ) = b (n) + ay (n) y 3 (n) + y (n) y (n). (3) Since (n) = (n) and y (n) = y (n) c we determine the dynamics of the above system with the help of the following system only: (n + ) = y (n) y (n + ) = b (n) + ay (n) y 3 (n) + c. (4) Again, if we choose the controllers as, u = (y (n) y (n)) ( (n) (n)) u = (a )(y (n) y (n)) (y 3 (n) y3 (n)) b( (n) (n)) u = y (n) y (n) u =. (5) System () takes the following form: (n + ) = y (n) + (y (n) y (n)) ( (n) (n)) y (n + ) = b (n) + ay (n) y 3 (n) y (n) + y (n) (n + ) = y (n) + (y (n) y (n)) y (n + ) = b (n) + ay (n) y 3 (n). (6) Hence, by error dynamics the generalized system is given by (n + ) = y (n) + c y (n + ) = b (n) + ay (n) y 3 (n). (7) 3.. Bifurcation analysis of the modified D Duffing map. Here we shall perform bifurcation analysis of (4). The fied points of the system are obtained from = y y = b + ay y 3 + c. (8) The above system reduced to 3 +( a+b) c =. By solving we get the fied point as (, y) where y =, ( c = + β and β = ) /3 ( c + β ) /3 = κ + κ /3 c + 4 ( a + b 3 The fied points are ( κ + κ P /3, κ ) + κ /3, ) 3. ( κ ω + κ ω P /3, κ ω + κ ω /3 ( κ ω + κ ω P 3 /3, κ ω + κ ω /3 ), )

6 4 Page 6 of 5 Pramana J. Phys. (7) 89:4 where κ 3 = c + β and κ 3 = c β. The Jacobian at the fied points can be written as ( ) J = b a 3y. The characteristic equation for the matri is H() = + B + C = whereb = 3y a and C = b.now H() = +3y a+b(>). H( ) = 3y +a+b. With Lemma, we have the following theorem: Theorem... When (κ +κ ) > /3 3 (+a +b) and b <, the fied point P is the sink and locally asymptotically stable.. When (κ +κ ) < /3 3 (+a +b), the fied point P is the saddle. 3. When (κ + κ ) > /3 3 ( + a + b) and b >, the fied point P is the source and unstable. 4. When (κ + κ ) = /3 3 ( +a + b), the fied point P is non-hyperbolic and the system may undergo flip bifurcation (period-doubling bifurcation). For a =.9andb =. (<), β c. Then the real fied point P (c /3, c /3 ). The fied point P is (i) sink and asymptotically stable for c (, ) (, ). P is (ii) saddle for the range < c <. (iii) For c = ±, P is nonhyperbolic. When c =±, the eigenvalues of J at fied point P (±.8, ±.8) are,.. So one eigenvalue is and hence flip bifurcation occurred for system (4) as shown in figure 7a. Again a =.9 andb =. (<), and other two comple fied points are P (c /3 ω, c /3 ω) and P 3 (c /3 ω, c /3 ω ). 3.3 Coupled 3D Hénon maps Consider the three-dimensional Hénon map [3], (n + ) = a (n) + by(n) y(n + ) = (n) + z(n) z(n + ) = (n) (9) and the map is chaotic for a =. andb =.8. We coupled two systems of the type (9) using the controllers u ij (i, j =,, 3) in the following way: (n + ) = a (n) + by (n) + u y (n + ) = (n) + z (n) + u z (n + ) = (n) + u 3 (n + ) = a (n) + by (n) + u y (n + ) = (n) + z (n) + u z (n + ) = (n) + u 3. (3) In this case, the synchronization error dynamics obey the following system: e (n + ) = ( (n) (n)) + be (n) + u u e (n + ) = e (n) + e 3 (n) + u u e 3 (n + ) = e (n) + u 3 u 3. (3) Here we choose the controllers in such a way that one variable of the three-dimensional coupled system is completely synchronized and two of them maintain a constant difference between them. Then, e (n), i.e. (n) (n), e (n) c 4,i.e.y (n) y (n) + c 4 and e 3 (n) c 5,i.e.z (n) z (n) + c 5 (where c 4 and c 5 are initial condition-dependent constants) can be done by the proper choice of controllers. Notice that in this case there are two initial condition-dependent parameters. Now we choose the controllers as u = ( (n) (n)) ( (n) (n)) + b(y (n) y (n)) u = z (n) z (n) + (n) (n) u 3 = ( (n) (n)) u = u = y (n) y (n) u 3 = z (n) z (n). (3) Then system (3) is transformed to (n + ) = a (n) + by (n) ( (n) (n)) y (n + ) = (n) + z (n) z (n + ) = (n) (n + ) = a (n) + by (n) y (n + ) = (n) + z (n) + y (n) y (n) z (n + ) = (n) + z (n) z (n) (33) and the error dynamics (3) changes as e (n + ) = e (n), e (n + ) = e (n) and e 3 (n + ) = e 3 (n). So the dynamics of the coupled system is completely identified by the following system: (n + ) = a (n) + by (n) y (n + ) = (n) + z (n) + c 4 z (n + ) = (n) + c 5. (34) 3.3. Bifurcation analysis of the modified 3D Hénon map. The fied points of system (34) are obtained from = a + by

7 Pramana J. Phys. (7) 89:4 Page 7 of 5 4 y = + z + c 4 z = + c 5. (35) The fied points are ( + + 4(a + b(c4 + c 5 )) P, c 4 + c 5, ) + 4(a + b(c 4 + c 5 )) and ( + 4(a + b(c4 + c 5 )) P, c 4 + c 5, + ) + 4(a + b(c 4 + c 5 )). The fied points are real if c 4 + c 5 a b 4b. The Jacobian matri (J ) of the system for the fied points P and P can be written as b. Let i (i =,, 3) are three eigenvalues of the above Jacobian. The eigenvalues of J are, ± + b. It is clear that all the eigenvalues are real for real fied points. Now for a =., b =.3 andc 5 =.,thefied points are ( c4 P, c 4 +, ).6 +.c 4 and (.6 +.c4 P, c 4 +, + ).6 +.c 4. Now eigenvalues at P are, p ± ( + p) +.3, where p = c 4.So, one eigenvalue is of unit modulus and the remaining two will lie within the unit circle ( < ) if c 4 lies in 4 < c 4 <.4. Similar condition will be obtained for fied point P. For both the fied points, isaneigenvalueofthe Jacobian matri, i.e. dynamics became oscillatory [33], flip bifurcation occur for system (34) and the corresponding figure is plotted in 3a. 4. Numerical results 4. Coupled D Hénon maps In figure,weplottheerrortermse (n) and e (n) with respect to time for system (6)fora =.andb =.3. The figure demonstrate that the term e (n) gradually tends to a constant term and e (n) tends to zero as time increases. In system (4), c = y () y () (for n = ) is initial condition-dependent constant. If we consider y () =, then c = y (). Hence, c is the initial condition of system (4). Then the bifurcation diagram.5.5 e (n) Error e (n) Time (n) Figure. Error dynamics of system (6) fora =. and b =.3.

8 4 Page 8 of 5 Pramana J. Phys. (7) 89: y () y () y () y () Figure. Bifurcation diagram of of system (4) with respect to initial condition y () (control parameter c ) when a =., b =.3, a =, b =.3, a =.9, b =.3, a =., b = y () y () y () y () Figure 3. Bifurcation diagram of of system (4) with respect to control parameter y () when a =.4, b =.39, a =.4, b = 6, a =.4, b =.5, a =.4, b =.

9 Pramana J. Phys. (7) 89:4 Page 9 of y () y () y () y () Figure 4. Maimum Lyapunov eponent of system (4) with respect to initial condition y () (control parameter c ) when a =., b =.3, a =, b =.3, a =.9, b =.3, a =., b = y () y () y () y () Figure 5. Maimum Lyapunov eponent of system (4) with respect to control parameter c when a =.4, b =.39, a =.4, b = 6, a =.4, b =.5, a =.4, b =.

10 4 Page of 5 Pramana J. Phys. (7) 89: y () Figure 6. Bifurcation diagram of of system (6) with respect to initial condition y () (control parameter c ) when a =.4, b =.3. with respect to y () of system (4) for different values of a keeping the parameter b fied are plotted in figure. In figure a we have chosen the values of a and b such that the original map has stable fied point. From figure a, it is observed that numerical simulation results are consistent with the theoretically derived period-doubling bifurcation point of system (4). We have chosen the values of a and b such that the original map has period- orbit and draw the corresponding bifurcation diagram in figure b. The multistable nature of the system is clear from figure b. In figures c and d the parameter values are taken in such a way that the original map has period-4 orbit and chaotic orbit respectively. The bifurcation diagram with respect to y () of system (4) for different b keeping the parameter a fied are plotted in figure 3. The parameter values for figures 3a 3d are chosen in such a way that the original map has fied point, period- orbit, period-4 orbit and chaotic orbit respectively. The bifurcation diagrams confirm the multistable nature of the system. For system (6), c = y () y () as e (n), and if we choose y () = then the corresponding bifurcation diagram is given in figure 6. Lyapunov eponents are drawn in figures 4 and 5 as parameters of a or b with respect to initial condition y (). The oscillation of the largest Lyapunov eponent from negative to positive values guarantees the etreme multistable behaviour of the designed system c c c c Figure 7. Bifurcation diagram of of system (4) with respect to control parameter c when a =.9, b =., a =.5, b =., a =., b =., a =.5, b =..

11 Pramana J. Phys. (7) 89:4 Page of 5 4 Figure 8. Bifurcation diagram of of system (4) with respect to control parameter c when a =.7, b =.3, a =.7, b =, a =.7, b =, a =.7, b = c c Figure 9. Bifurcation diagram of of system (7) with respect to control parameter c when a =.6, b =.8, a =.6, b = Coupled D Duffing maps The bifurcation diagram with respect to c of system (4) for different a keeping the parameter b fied are plotted in figure 7. The parameter values for figures 7a 7d are chosen in such a way that the original map has fied point, period- orbit, period-4 orbit and chaotic orbit respectively. The bifurcation diagrams confirm

12 4 Page of 5 Pramana J. Phys. (7) 89: c c c c Figure. Maimum Lyapunov eponent of system (4) with respect to control parameter c when a =.9, b =., a =.5, b =., a =., b =., a =.5, b = c c c c Figure. Maimum Lyapunov eponent of system (4) with respect to control parameter c when a =.7, b =.3, a =.7, b =, a =.7, b =, a =.7, b =.9.

13 Pramana J. Phys. (7) 89:4 Page 3 of c c Figure. Maimum Lyapunov eponent of system (7) with respect to control parameter c when a =.6, b =.8, a =.6, b =.7. Figure 3. Bifurcation diagram of of system (34). a =., b =.3, c 5 =., a =., b =.6, c 5 =., a =., b =, c 4 =., a =.5, b =.8, c 4 =.. the multistable nature of the system. We theoretically derived period-doubling bifurcation for particular a =.9 andb =., which is consistent with our finding in figure 7a. The bifurcation diagram with respect to c of system (4) for different b keeping the parameter a fied are plotted in figure 8. Also, bifurcation diagram with respect to c of system (7) for different b and fied a =.6areplottedinfigure 9. This diagram confirms the multistability nature of system (7). Lyapunov eponents are drawn in figures, and as parameters of a or b with respect to parameter c. It is obvious from the figures that the largest Lyapunov eponent fluctuates between negative and positive values which is an indication of the etreme multistable behaviour of the model. 4.3 Coupled 3D Hénon maps Figure 3 represents the bifurcation diagram of system (34) with respect to one constant keeping the other one fied for different values of parameters a and b. Infig-

14 4 Page 4 of 5 Pramana J. Phys. (7) 89: c c c c 5 Figure 4. Maimum Lyapunov eponent of system (34). a =., b =.3, c 5 =., a =., b =.6, c 5 =., a =., b =, c 4 =., a =.5, b =.8, c 4 =.. ure 3a we have plotted bifurcation diagram of of (34) with respect to c 4 taking c 5 =., a =. and b =.3. Also, we observed that numerical simulation results for c 5 =., a =. andb =.3 are consistent with the theoretically derived period-doubling bifurcation point of system (34). In figure 3b, c 5 =., a =., b =.6 and the corresponding bifurcation diagram of is presented with respect to c 4. The bifurcation diagram with respect to c 5 for a =., b = and c 4 =. is shown in figure 3c. In figure 3d the bifurcation diagram with respect to c 5 is depicted for a =.5, b =.8, c 4 =.. From these figures we observe the eistence of many attractors (e.g., steady state, period- orbit, period-4 orbit, chaos etc.) with the variation of initial condition dependent parameters. Hence the system is sensitive to initial conditions and therefore the coupled system (33) is a multistable system. Lyapunov eponents are plotted in figure 4 as parameters of a or b with respect to parameters c 4 (figures 4a, 4b) and c 5 (figures 4c, 4d). It is clear from the figure that the largest Lyapunov eponent fluctuates between positive and negative values with the variation of initial conditions which confirms the etreme multistable behaviour of the model. Hence, we can conclude from these eamples that our scheme can successfully design multistable discrete dynamical systems. 5. Conclusion We propose a scheme for designing discrete multistable systems by coupling two identical discrete dynamical systems. We propose a conjecture that the coeistence of infinitely many attractors in two m-dimensional coupled systems will be possible if m i of the variables of the two systems are completely synchronized and i (where i m ) of them obey a constant difference between them. We illustrate our theory by considering the eamples of coupled D and 3D Hénon maps and coupled D Duffing maps. The conditions for period-doubling bifurcations are derived analytically and verified numerically. In all the cases, bifurcation diagrams and variation of largest Lyapunov eponents with respect to the initial condition-dependent parameter are presented to show the effectiveness of our scheme. This scheme will also be useful to design multistable system coupling two non-identical discrete dynamical systems. Our scheme of designing multistable systems will be helpful to identify mechanisms that lead to multistabil-

15 Pramana J. Phys. (7) 89:4 Page 5 of 5 4 ity in many natural systems and it will also be useful for designing physically or biologically useful multistable systems. Acknowledgements The author is grateful to the Editor and reviewers of Pramana for their critical comments and suggestions which helped a lot to improve the manuscript. She would like to epress her sincere thanks to Dr Swarup Poria, Department of Applied Mathematics, University of Calcutta for his valuable suggestions to prepare the manuscript. This research is supported by the Centre with Potential for Ecellence in Particular Area (CPEPA), University of Calcutta under University Grants Commission (UGC) scheme Ref. F.R.8-/8 (NS/PE), dated 4th December,. References [] F Atteneave, Sci. Am.5, 63 (97) [] J L Schwartz, N Grimault, J M Hupé, B C J Moore and D Pressnitzer, Philos. Trans. R. Soc. B367, 896 () [3] W Knorre, F Bergter and Z Simon, Stud. Biophys. 49, 8 (975) [4] D Angeli, J E Ferrell Jr and E D Sontag, Proc. Natl Acad. Sci. USA (7), 8 (4) [5] F Ravelet, L Marié, A Chiffaudel and F Daviaud, Phys. Rev. Lett. 93, 645 (4) [6] F T Arecchi, R Badii and A Politi, Phys. Rev. A 3(), 4 (985) [7] H M Gibbs, S L McCall and T N C Venkatesan, Phys. Rev. Lett. 36, 35 (976) [8] J Kastrup, H T Grahn, K Ploog, F Prengel, A Wacker and E Schöll, Appl. Phys. Lett. 65, 88 (994) [9] I Hudson and J Mankin, J. Chem. Phys. 74, 67 (98) [] R Simonyi, A Wolf and H Swinney, Phys. Rev. Lett. 49, 45 (98) [] J Wang, P Sorensen and F Hymne, J. Chem. Phys. 98, 75 (994) [] R May, Nature 69, 47 (977) [3] P M Groffman et al, Ecosystems 9, (6) [4] J Hertz, A Krogh and R G Palmer (Addison-Wesley, New York, 99) [5] D Paillard, Nature 39, 378 (998) [6] R Calov and A Ganopolski, Res. Lett. 3, L77 (5), DOI:.9/5GL458 [7] S Bathiany, M Claussen and K Fraedrich, Clim. Dynam. 38, 775 () [8] T N Palmer, J. Clim., 575 (999) [9] G C M A Ehrhardt, M Marsili and F Vega-Redondo, Phys. Rev. E 74, 366 (6) [] A Naimzada and M Pireddu, Appl. Math. Comp. 39, 375 (4) [] F T Arecchi, R Meucci, G Puccioni and J Tredicce, Phys. Rev. Lett. 49, 7 (98) [] U Feudel, Int. J. Bifurcat. Chaos 8, 67 (8) [3] U Feudel, Phys. Rep. 54, 67 (4) [4] H Sun, S Scott and K Showalter, Phys. Rev. E 6, 3876 (999) [5] C R Hens, R Banerjee, U Feudel and S K Dana, Phys. Rev. E 85, 35(R) () [6] S Pal, B Sahoo and S Poria, Phys. Scr. 89, 94 (4) [7] S Pal, B Sahoo and S Poria, Pramana J. Phys. 86(6), 83 (6) [8] S Pal and S Poria, Phys. Scr. 9(3), 353 (5) [9] X L Liu and D M Xiao, Chaos, Solitons and Fractals 3, 8 (7) [3] K Murakami, J. Differ. Equ. Appl. 3(), 9 (7) [3] M Hénon, Commun. Math. Phys. 5, 69 (976) [3] P Pal, S Debroy, M K Mandal and R Banerjee, Nonlinear Dyn. 79, 79 (5) [33] M Sonis, Disc. Dynam. Nature Soc. 4, 333 (999)