Flip and Neimark-Sacker Bifurcations in a Coupled Logistic Map System
FF LIP AND N EIMARK -S ACKER B IFURCATIONS IN A C OUPLED L OGISTIC M AP S YSTEM
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A. Mareno and L.Q. English Department of Mathematics and Computer Science, Pennsylvania State University, Capital College, Middletown PA, 17057 email: [email protected] Department of Physics, Dickinson College, Carlisle, PA 17013May 19, 2020 A BSTRACT
In this paper we consider a system of strongly coupled logistic maps involving two parameters. Weclassify and investigate the stability of its fixed points. A local bifurcation analysis of the systemusing Center Manifold is undertaken and then supported by numerical computations.This reveals theexistence of reverse flip and Neimark-Sacker bifurcations. K eywords logistic map · flip bifurcation · Neimark-Sacker bifurcation · center manifold theory Coupled logistic maps originally gained attention in the mathematical biology literature via their utility in modelsof, for instance, populations of migrating species and environmental heterogeneity [9, 11]. Recent years, however,have seen a renewed interest in the dynamics of coupled logistic maps. At least two developments have spurredthis re-examination: (a) the realization that discrete coupled maps could be usefully exploited in digital encryptionschemes [16, 1, 6], and (b) success with their experimental implementation using electronic circuits [10, 12, 7]. Bothof these recent threads have revealed intricate and non-intuitive behavior of these coupled maps. One such behavior –spontaneous symmetry breaking - was recently highlighted and explored [7]. That reference, however, did not attemptto analyze the chaotic regime in this coupled system (i.e., for large values of r ), focusing primarily on symmetrybreaking and its basin of attraction pertaining to n-cycles. Here we revisit the problem in a mathematically rigorousway and thus shed light on the origins of some of the unusual bifurcations seen experimentally [12] in this system.In particular, we start by systematically classifying all the fixed points and their bifurcation properties that manifest inthis coupled system, taking the coupling strength, (cid:15) , to be our bifurcation parameter and not the growth rate, r , whichis typically chosen. We then focus on the symmetry-broken 1-cycle – a fixed point unique to the coupled system -and proceed to apply the center-manifold-theoretic framework to prove that it becomes stable via a flip bifurcationas the coupling strength parameter is increased. This transition is an interesting phenomenon also seen previously inexperiments [12]. In this paper, we explore this flip bifurcation also numerically and see excellent agreement with thepredictions derived from theorem established herein.For even higher values of the coupling strength, the symmetry-broken 1-cycle loses stability again (something alsoseen experimentally). In this context, we prove that the origin of this instability is a Neimark-Sacker bifurcation. Wethen explore this bifurcation numerically, and again demonstrate excellent agreement with our theoretical results.Throughout this work we consider the following discrete system x n +1 = (1 − (cid:15) ) f ( x n ) + (cid:15)f ( y n ) y n +1 = (cid:15)f ( x n ) + (1 − (cid:15) ) f ( y n ) , (1)where f ( z ) = rz (1 − z ) . (2) a r X i v : . [ n li n . C D ] M a y PREPRINT - M AY
19, 2020For convenience the system can be rewritten in the form: F ( x, y ) = ((1 − (cid:15) ) f ( x ) + (cid:15)f ( y ) , (cid:15)f ( x ) + (1 − (cid:15) ) f ( y )) (3)where the parameter (cid:15) ∈ [0 , and r ∈ (0 , . The organizational structure of this paper is as follows: In section 2 we discuss the basic framework and relevantterminology for this work. In section 3, we classify and determine the stability of the fixed points of the systemthe r(cid:15) plane. Section 4 is devoted to a rigorous mathematical treatment of the flip and Neimark-Sacker bifurcationsmanifested by this system. Finally, in section 5 we give numerical evidence to support our theoretical results fromsection 4.
We begin by stating important terminology and concepts relevant to this work (see for instance [4, 17]). Generallywe can say that a set S is an invariant set if iterates of the map for any element of S stay in S for all integers. Wewill loosely think of an invariant manifold as a set which locally has the structure of Euclidean space, typically assurfaces imbedded in R n , for which the function representing the surface has maximal rank and can therefore, belocally represented as a graph, by way of applying the Implicit Function Theorem.We now define three important linear subspaces, relevant to the study of dynamical systems, spanned by the(generalized) eigenvectors of the Jacobian matrix DF ( x, y ) at a fixed point ( x, y ) : E s (the stable subspace), E u (theunstable subspace) and E c (the center subspace). The associated eigenvalues of each subspace have modulus lessthan one, greater than one or equal to one respectively. When DF ( x, y ) has no eigenvalues of unit modulus ( x, y ) iscalled a hyperbolic point and so its stability is determined entirely by the eigenvalues themselves. Furthermore, forhyperbolic points E c does not exist.A hyperbolic fixed point is called a sink if the eigenvalues of the Jacobian matrix evaluated at the fixed point havemagnitude less than one. Such a fixed point is locally asymptotically stable. If the magnitudes of both eigenvaluesare greater than one, the hyperbolic fixed point is called a source and is locally asymptotically unstable. Moreover, ahyperbolic fixed point is called a saddle point if only one of the eigenvalues has magnitude greater than one.The Stable Manifold Theorem [8] guarantees the existence of local stable and unstable invariant manifolds W sloc and W uloc which can be viewed as nonlinear analogues of the linear subspaces E s and E u respectively. These invariantmanifolds are tangent to these the two linear subspaces, have the same dimensions as these subspaces and are assmooth as the underlying map.The Center Manifold Theorem ( see chapter 1 in [8] or [4] )asserts the existence of an invariant manifold tangentto the center eigenspace E c which can be non-unique and ‘non-smooth’ (in a certain sense) (see-chapter 3 in [8]or[4]) where the dynamics of the nonlinear system (at say the trivial fixed point) restricted to the center manifold isdetermined by a c-dimensional map, a map whose dimension is the same as that of the center subspace E c , where say ( x, y ) ∈ R c × R s and both R s , R c are subsets of R n . So for a two-dimensional system such as the system studied inthis paper the dynamics of our nonlinear map are determined by a one-dimensional map.Herein lies the significance of Center Manifold Theorem - rather than studying the map on the entire domain ofthe map to determine its dynamics we can restrict this analysis to the center manifold, an invariant manifold withdimension equal to the dimension of the center subspace, which is less than the dimension of the maps’ domain. Inaddition, using the invariance of the center manifold one can derive a quasi-linear partial differential equation thatthe c-dimensional map characterizing the center manifold must satisfy in order for its graph to be an invariant centermanifold. To find this map, one must solve this partial differential equation. Thus this theorem can be viewed as typeof reduction principle that one can apply to ascertain the stability of non-hyperbolic fixed points, when say E u istrivial.Therefore, in this paper we restrict our use of Center Manifold Theory to the case where the Jacobian matrix has itsspectrum inside the unit circle apart from one or two eigenvalues. For an additional reference on Center ManifoldTheory see [5]. 2 PREPRINT - M AY
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We begin our analysis of the system (3) by solving the equations (1 − (cid:15) ) rx (1 − x ) + (cid:15)ry (1 − y ) = x (4) (cid:15)rx (1 − x ) + (1 − (cid:15) ) ry (1 − y ) = y. (5)and obtaining the fixed points of our system: (0 , , ( rr − , rr − , ( x ∗ , y ∗ ) , ( y ∗ , x ∗ ) (6)where x ∗ = r (2 (cid:15) −
1) + 1 − (cid:112) ( r (1 − (cid:15) ) − r (1 − (cid:15) ) + 4 (cid:15) − r (2 (cid:15) − (7) y ∗ = r (2 (cid:15) −
1) + 1 + (cid:112) ( r (1 − (cid:15) ) − r (1 − (cid:15) ) + 4 (cid:15) − r (2 (cid:15) − . (8)We note that x ∗ , y ∗ are real valued if and only if ∆ = (1 − (cid:15) )( r − + 4 (cid:15) r ( r − ≥ . This occurs when r ∈ (3 , and (cid:15) ∈ (cid:20) , r − r (cid:21) or (cid:15) ∈ (cid:20) r − r − , (cid:21) (9)In addition x ∗ = y ∗ if and only if ∆ = 0 which occurs when (cid:15) = r − r or (cid:15) = r − r − , and so for these values of (cid:15) the fixed point ( x ∗ , y ∗ ) coincides with one of the two symmetric fixed points: (0 , or ( r − r , r − r ) , respectively.Throughout this work we consider only ( x ∗ , y ∗ ) and not ( y ∗ , x ∗ ) -its flipped counterpart. To determine conditionsfor a fixed point to be classified as a hyperbolic/non-hyperbolic fixed point and to determine the stability type ofhyperbolic fixed points we compute the Jacobian of our map F : DF ( x, y ) = (cid:18) (1 − (cid:15) ) r (1 − x ) (cid:15)r (1 − y ) (cid:15)r (1 − x ) (1 − (cid:15) )(1 − y ) (cid:19) (10)By solving the characteristic equation det ( DF ( x, y ) − λI ) = 0 , (11)the eigenvalues of the Jacobian evaluated at a fixed point ( x, y ) are computed as follows: λ , = r (1 − (cid:15) ) + r ( (cid:15) − x + y ) ± (cid:112) r (1 − (cid:15) )( x − y ) + (cid:15) ( x + y − (12)Although the characteristic equation is characterized by the three principle invariants, where each is in turn is a functionof the eigenvalues of the Jacobian and therefore one can use say, the Jury conditions to determine the stability of thefixed points, we take a more straight forward approach and analyze the eigenvalues and their magnitudes directly; thisdirect approach yields more ‘directional’ information about the magnitudes of both eigenvalues.Using these definitions and the eigenvalues associated with each fixed point we determine the parameter dependentregions where each of the fixed points is asymptotically stable, unstable, a saddle point and a non-hyperbolic point,which are stated in the following theorem: Theorem 1-Fixed Point Classification and Stability
A. (i)The fixed point (0 , is sink if r ∈ (0 , and (cid:15) ∈ [0 , .(ii) (0,0) is a source if r ∈ (0 , and (cid:15) ∈ [0 , r − r ) or (cid:15) ∈ ( r +12 r , .(iii) (0,0) is a saddle point r ∈ (1 , and (cid:15) ∈ (cid:0) r − r , r +12 r (cid:1) . (Here, | λ | > and | λ | < ).(iv) (0,0) is a non-hyperbolic point (specifically here, λ = − and | λ | > if r ∈ (1 , and (cid:15) = r + 12 rλ = 1 , | λ | > for r ∈ (1 , , (cid:15) = r − rλ = 1 , λ = − for (cid:15) = 1 , r = 1 PREPRINT - M AY
19, 2020 λ = 1 , | λ | < for (cid:15) ∈ (0 , , r = 1 λ = λ = 1 for (cid:15) = 0 , r = 1 (1:1 resonance) . B. (i) The symmetric fixed point ( r − r , r − r ) is a sink if r ∈ (1 , for all (cid:15) in [0 , . (ii) ( r − r , r − r ) is a source if r ∈ (0 , and (cid:15) ∈ (cid:20) , r − r − (cid:19) , or r ∈ (0 , and (cid:15) ∈ (cid:18) r − r − , (cid:21) or r ∈ (3 , and (cid:15) ∈ (cid:20) , r − r − (cid:19) , or r ∈ (3 , and (cid:15) ∈ (cid:18) r − r − , (cid:21) . (iii) ( r − r , r − r ) is a saddle point (in this case it means | λ | < and | λ | > ) if r ∈ (0 , and (cid:15) ∈ (cid:18) r − r − , r − r − (cid:19) or r ∈ (3 , and (cid:15) ∈ (cid:18) r − r − , r − r − (cid:19) . (iv) ( r − r , r − r ) is a non-hyperbolic point if r ∈ (0 , or r ∈ (3 , and (cid:15) = r − r − (here λ = − , | λ | > or (cid:15) ∈ (0 , and r = 3 (here λ = − , | λ | < or (cid:15) ∈ (0 , and r = 1 (here λ = 1 , | λ | < or (cid:15) = r − r − and r ∈ (0 , or r ∈ (3 , and λ = 1 , | λ | > . Furthermore, λ = λ = − , (cid:15) = 0 , r = 3 (1:2 resonance ) λ = 1 , λ = − , (cid:15) = 1 , r = 3 . C. (i) The non-symmetric fixed point ( x ∗ , y ∗ ) is a sink if r ∈ (3 , √ and (cid:15) ∈ (cid:32)
12 + √ (cid:115) r ( r − , (cid:35) or r ∈ (1 + √ , and (cid:15) ∈ (cid:32)
12 + √ (cid:115) r ( r − , f ( r ) (cid:33) where f ( r ) = 14 (cid:34) − r + 2 r r ( r −
2) + √ − r + 8 r r ( r − (cid:35) . (ii) ( x ∗ , y ∗ ) is a source if r ∈ (3 , and (cid:15) ∈ (cid:34) , − √ (cid:115) r ( r − (cid:33) or r ∈ (1 + √ , and (cid:15) ∈ ( f ( r ) , . PREPRINT - M AY
19, 2020(iii) ( x ∗ , y ∗ ) is a saddle point if r ∈ (3 , and (cid:15) ∈ (cid:32) r − r − ,
12 + √ (cid:115) r ( r − (cid:35) , (specifically | λ | < , | λ | > or r ∈ [3 , and (cid:15) ∈ (cid:32) − √ (cid:115) r ( r − , r − r (cid:33) , (specifically | λ | < , | λ | > . (iv) ( x ∗ , y ∗ ) is a non-hyperbolic point if r ∈ [1 + √ , and (cid:15) = f ( r ) (here | λ | = | λ | = 1 , λ i ∈ C , i = 1 , ) or r ∈ (3 , and (cid:15) = 12 + √ (cid:115) r ( r − (specifically , λ = − , | λ | < or r ∈ (3 , and (cid:15) = 12 − √ (cid:115) r ( r − (specifically , λ = − , | λ | > or r = 3 and (cid:15) = 0 ,where our system now corresponds to an uncoupled pair of logistic maps. Proof . For the trivial fixed point (0 , , λ = r, λ = r (1 − (cid:15) ) . By inspection we see | λ i | < , for i=1,2 if and only if r ∈ (0 , for any epsilon in [0 , . The remaining parts of A can easily be deduced.For the symmetric fixed point ( r − r , r − r ) , λ = r (1 − (cid:15) )( r − and λ = r ( r − . Again a straightforwardcalculation shows that parts (i)-(iv) of B hold.For the anti-symmetric fixed point ( x ∗ , y ∗ ) a direct calculation shows that the eigenvalues are λ = (cid:15) − (cid:15) − (cid:113) (cid:15) +(1 − (cid:15) )∆(1 − (cid:15) ) ) (cid:15) − , λ = (cid:15) − − (cid:15) ) (cid:113) (cid:15) +(1 − (cid:15) )∆(1 − (cid:15) ) ) (cid:15) − from which one can establish (i)-(iv).In Figure 1(a) we illustrate the stable, unstable and saddle regions for the fixed point (0 , . Figures 1 (b) and (c) showthese three regions for the fixed points ( r − r , r − r ) and ( x ∗ , y ∗ ) .In 1(a) above the upper curve r +12 r is the flip curve, and r − r , r = 1 are fold curves. In 1(b) the two upper dashedcurves denote flip and fold curves respectively as well as the lines r = 3 and r = 1 respectively. In 1(c) we define h = r − r , h = r − r − , g = + √ (cid:113) r ( r − , g = − √ (cid:113) r ( r − and f = f ( r ) which was defined earlier.Here g , g are flip curves, f ( r ) is a Neimark-Sacker curve and h , h are the curves bounding the saddle regions. Wealso note that for the two symmetric fixed points we have symmetric regions of stability/instability whose boundingcurves the translation symmetry (cid:15) (cid:55)→ − (cid:15) inherent in the system’s defining equations. For the anti-symmetric fixedpoint ( x ∗ , y ∗ ) this translation symmetry manifests in the equations for the bounding curves g , g but not in theregions bounded by these curves. Now we determine the stability of the non-hyperbolic fixed point ( x ∗ , y ∗ ) via center manifold theory. In particularwe demonstrate that system (3) undergoes a flip bifurcation at ( x ∗ , y ∗ ) where λ = − and λ = (cid:15) − (cid:15) − and wherewe choose (cid:15) as our bifurcation parameter and allow it to vary in a small neighborhood of ( x ∗ , y ∗ ) . Generically, a5 PREPRINT - M AY
19, 2020Figure 1: Diagrams of the regions of stability for three of the four fixed points of system (3) in the ( r, (cid:15) ) planeflip bifurcations is characterized by a the loss of stability of a periodic orbit as a parameter crosses a critical value(from above or below) and at which point locally, either there exists stable periodic orbits with double the period forparameter values near the critical parameter forming a new branch that emerges at the critical parameter value (super-critical period doubling) or unstable periodic orbits with double the period coalescing with and destroyed by stableperiodic orbits (sub-critical period doubling). Moreover, a flip bifurcation occurs at an eigenvalue of -1 of the Jacobianof the map.In order to apply Center Manifold theory we assume that our discrete system has the form: x n +1 = Ax n + F ( x n , y n ) y n +1 = By n + F ( x n , y n ) (13)where all of the eigenvalues of the matrix A (an n × n matrix) are on the unit circle and the eigenvalues of the matrix B (an mxm matrix) are within the unit circle, and the Jacobian matrix for the system has the form (cid:20) A B (cid:21) We assume without loss of generality that the system has the origin as a fixed point. We use a slight modification ofthe following version of the Center Manifold Theorem in [5]:
Theorem 5 . There exists a C r -center manifold for system (13) that can be represented locally as W cloc (0 ,
0) = { ( x, y, µ ) ∈ R | y = h ( x, µ ) , | x | < δ , | µ | < δ , , h (0 ,
0) = Dh (0 , , | x | < (cid:15), | µ | < δ } Furthermore, the dynamics of the system restricted to W cloc (0) are given locally by the map x (cid:55)−→ Ax + f ( x, µ, h ( x, µ )) , for x ∈ R . In addition we state the following theorem from [8] which gives criteria for the existence of a flip bifurcation:
Theorem 3.5.1
Let f µ : R → R be a one parameter family of mappings such that f µ has a fixed point x with aneigenvalue of value − . Assume ∂f∂µ ∂ f∂x + 2 ∂ f∂x∂µ (cid:54) = 0 at ( x , µ ); PREPRINT - M AY
19, 2020 (cid:18) ∂ f∂x (cid:19) + 13 (cid:18) ∂ f∂x (cid:19) (cid:54) = 0 at ( x , µ ) . Then there is a smooth curve of fixed points of f µ passing through ( x , µ ) , the stability of which changes at ( x , µ ) .There is also a smooth curve γ passing through ( x , µ ) so that γ − ( x , µ ) is a union of hyperbolic period 2 orbits.The curve γ has quadratic tangency with the line R × { µ } at ( x , µ ) . We begin the establishment of a flip bifurcation at ( x ∗ , y ∗ ) by first defining H F P = (cid:40) ( r, (cid:15) ) : r ∈ [3 , , (cid:15) = 12 + √ (cid:115) r ( r − (cid:41) (14)the set containing the parameters that satisfy the second condition for a hyperbolic point in C (iv) from Theorem 1.For arbitrary parameters ( r s , (cid:15) s ) ∈ H F P and by the change of variables u n = x n − x ∗ , v n = y n − y ∗ where we alsoset ¯ (cid:15) = (cid:15) − (cid:15) s ( and so (cid:15) s = + √ (cid:113) r ( r − ) be a new independent variable, we transform the fixed point ( x ∗ , y ∗ ) into (0 , . System (3) now has the form (cid:18) u n +1 v n +1 (cid:19) = (cid:18) a u n + a v n + a u n + a v n + b ∗ ¯ (cid:15) + b ¯ (cid:15)u n − b ¯ (cid:15)v n a u n + a v n + a u n + a v n − b ∗ ¯ (cid:15) − b ¯ (cid:15)u n + b ¯ (cid:15)v n (cid:19) (15)where a = r s (1 − (cid:15) s )(1 − x ∗ ) a = r s ¯ (cid:15) (1 − x ∗ ) a = r s (cid:15) s (1 − y ∗ ) a = r s (1 − e s )(1 − y ∗ ) a = r s ( (cid:15) s − a = − r s (cid:15) s (16)and b ∗ = r s ¯ (cid:15) (cid:0) ( x ∗ ) − x ∗ + y ∗ − ( y ∗ ) − (1 − x ∗ ) u n + (1 − y ∗ ) v n (cid:1) , b = r s We begin the process of putting the system into the format of the equations in (13) by first defining an invertiblematrix T = (cid:18) − a − a a + 1 a − λ (cid:19) determined by the eigenvectors associated with the linearization of the system at (0 , . Using the transformation (cid:18) u n v n (cid:19) = T (cid:18) X n Y n (cid:19) (17)and letting µ = ¯ (cid:15) the system now takes the desired form: (cid:18) X n +1 Y n +1 (cid:19) = (cid:18) − λ − a (cid:19) (cid:18) X n Y n (cid:19) + (cid:18) F ( X n , Y n , µ ) G ( X n , Y n , µ ) (cid:19) (18)where F ( X n , Y n , µ ) = b a (1 + λ ) (cid:2) ( a a − a b ) X n + (( a a − a b ) Y n + (2 a ( a − b b ) X n Y n + µb ∗ (cid:3) + b a (1 + λ ) (cid:2) b µ ([ a − b ] X n + [ a − b ] Y n + 2[ a − b b ] X n Y n ) (cid:3) + 11 + λ (cid:2) ( a ( b − a )) X n + ( a ( b − a )) Y n + (2 a ( b b − a ) X n Y n − µb ∗ (cid:3) PREPRINT - M AY
19, 2020 + 11 + λ (cid:2) − b µ ([ a − b ] X n + [ a − b ] Y n + 2[ a − b b ] X n Y n (cid:3) and G ( X n , Y n , µ ) = − b a (1 + λ ) (cid:2) a ( a − b ) X n + a ( a − b ) Y n + 2( a ( a − b b )) X n Y n + µb ∗ (cid:3) − b a (1 + λ ) (cid:2) b µ (cid:2) ( a − b ) X n + ( a − b ) Y n + 2( a − b b ) X n Y n ) X n Y n (cid:3)(cid:3) −
11 + λ (cid:2) a ( b − a ) X n + a ( b − a ) Y n + 2 a ( b b − a ) X n Y n − µb ∗ (cid:3) −
11 + λ (cid:2) − b µ (cid:2) ( a − b ) X n + ( a − b ) Y n + 2( a − b b ) X n Y n (cid:3)(cid:3) where b = a +1 , b = a − λ . By applying the center manifold theorem we see that there exists a center manifoldfor system (3) defined as W cloc (0 ,
0) = { ( x, y, µ ) ∈ R (cid:51) | y = h ( x, µ ) , | x | < δ , | µ | < δ , , h (0 ,
0) = Dh (0 , , | x | < (cid:15), | µ | < δ } for sufficiently small (cid:15) and δ . To actually find the center manifold as the graph of y = h ( x, µ ) we consider a powerseries representation for this map: y = h ( x, µ ) = A X + A Xµ + A µ + O (( | X | + | µ | ) ) which we then substitute into (13). Hence, the center manifold must satisfy the equation h ( − x + F ( x, h ( x, µ ) , µ ) , µ ) = λ h ( x, µ ) + G ( x, h ( x, µ ) , µ ) . (19)By writing F in the form F ( X, Y, µ ) = ( f − g ) (cid:2) e X + e Y + e XY + µe + µe ( e X + e Y + e XY ) (cid:3) and G in the form G ( X, Y, µ ) = ( f − g ) (cid:2) e X + e Y + e XY + µe + µe ( e X + e Y + e XY ) (cid:3) where e = a ( a − b ) e = a ( a − b ) e = 2 a ( a − b b ) e = b ∗ e = a − b e = a − b e = 2( a − b b ) e = b (20) f = b a (1 + λ ) , g = 11 + λ , g = − g , f = − b a (1 + λ ) By substituting the equations for F , G and h into the center manifold equation (19) and equating the coefficients oflike terms on either side of the equation, we determine the coefficients A , A , A : A = ( f − g ) e − λ A = 2 A ( g − f ) µe λ A = ( f − g )[ A e + A e ] λ − PREPRINT - M AY
19, 2020The restriction of our map to the center manifold is defined as the map K ( X, µ ) := − X + ( f − g ) (cid:2) ( e + µe e ) X + ( e + µe e )( X + A X µ + A µ X ) (cid:3) +( f − g ) (cid:2) ( e + µe e )( A X + A X µ + A µ + 2 A A X µ + 2 A X A µ + 2 A A µ X (cid:3) . Straightforward but detailed calculations shows that α = ∂K∂µ ∂ K∂X + 2 ∂ K∂X∂µ = 2 e e ( f − g ) (cid:12)(cid:12)(cid:12) (0 , (cid:54) = 0 and α = 12 (cid:18) ∂ K∂X (cid:19) + 13 (cid:18) ∂ K∂X (cid:19) = 2( f − g )[( f − g ) e + A e ] (cid:12)(cid:12)(cid:12) (0 , (cid:54) = 0 By Theorem 5.1 and Theorem 3.5.1 above, the following result is now established:
Theorem 2 If α , α (cid:54) = 0 then the map undergoes a flip bifurcation at the fixed point ( x ∗ , y ∗ ) when the parameter (cid:15) varies in a small neighborhood of (cid:15) s . Moreover if α > ( respectively α < the period-2 orbits that bifurcatefrom ( x ∗ , y ∗ ) are stable (unstable). A Neimark-Sacker bifurcation is characterized by a stable fixed point becoming unstable at a certain critical valueof the bifurcation parameter of the system in which an an attracting closed invariant curve manifests or a repellingclosed invariant curve emerges as the values of the parameter cross this critical value.In the former case, we say thebifurcation is a supercritical Neimark -Sacker bifurcation; in the latter case a subcritical Neimark-Sacker bifurcation.In either case such a a bifurcation is associated with discrete systems whose eigenvalues are complex conjugates ofmodulus one.Here we state a slight modification of a theorem from [5], (Chapter 5), which outlines the criteria for the emergenceof such a bifurcation.
Theorem 5.4 (Neimark-Sacker)
Consider the family of C r maps ( r ≥ , F µ : R (cid:50) × R → R (cid:50) such that thefollowing conditions hold: . F µ (0) = 0 , i.e., the origin is a fixed point of F µ . . DF µ (0) has two complex conjugate eigenvalues λ , ( µ ) = r ( µ ) e ± iθ ( µ ) , where r (0) = 1 , r (cid:48) (0) (cid:54) = 0 , θ (0) = θ . . e ikθ (cid:54) = 1 for k = 1 , , , (absence of strong resonances condition) . If in addition, a (cid:54) = 0 where a = − Re (cid:20) (1 − λ )¯ λ ζ ζ − λ (cid:21) − | ζ | − | ζ | + Re (¯ λζ ) , ( a is called the first Lyapunov coefficient ) , then for sufficiently small µ , F µ there exists a unique invariant closed curve enclosing that bifurcates from the originas a passes through 0. If a¿0 we have a supercritical Neimark-Sacker bifurcation. If a < we have a subcriticalNeimark-Sacker bifurcation. The complex conjugate eigenvalues of our system are given by the following formulas: λ i = (cid:15) − ± i (2 (cid:15) − (cid:113) − (cid:15) +(2 (cid:15) − − (cid:15) ) ) (cid:15) − , for i = 1 , . (21)9 PREPRINT - M AY
19, 2020A simple calculation shows that | λ i | = 1 , for i=1,2 if and only if (cid:114) ∆ − (cid:15) − , or ∆ = 2 (cid:15). (22)Thus the range of parameters for which the eigenvalues associated with the fixed point ( x ∗ , y ∗ ) are complex conjugatesand have magnitude 1 can be described by the set H NS = { ( r, (cid:15) ) : (cid:15) + (1 − (cid:15) )∆ < , ∆ = 2 (cid:15) } ≡ { ( r, (cid:15) ) : r ∈ [1 + √ , , (cid:15) = f ( r ) } (23)We now show that a Neimark-Sacker bifurcation occurs at ( x ∗ , y ∗ ) for arbitrary parameters ( e h , r h ) ∈ H NS , taking (cid:15) as our bifurcation parameter and allowing it to vary in a small neighborhood of e h . So we consider a small perturbationof the parameter (cid:15) as follows: ¯¯ (cid:15) = (cid:15) − (cid:15) h and transform the fixed point ( x ∗ , y ∗ ) to the origin (0 , as before to producethe system (where we are essentially replacing e s by e h in an earlier statement of our system) with coefficients thatwere defined in Section 3: (cid:18) u n +1 v n +1 (cid:19) = (cid:18) a u n + a v n + a u n + a v n + b ∗ ¯¯ (cid:15) + b ¯¯ (cid:15)u n − b ¯¯ (cid:15)v n a u n + a v n + a u n + a v n − b ∗ ¯¯ (cid:15) − b ¯¯ (cid:15)u n + b ¯¯ (cid:15)v n (cid:19) (24)Now the characteristic equation at ( u n , v n ) = (0 , is as follows: λ − λ ( r h (1 − (cid:15) h − ¯¯ (cid:15) )(1 − x ∗ ) + r h ( (cid:15) h + ¯¯ (cid:15) )(1 − y ∗ )) + r h (1 − (cid:15) h − ¯¯ (cid:15) ))(1 − x ∗ )(1 − y ∗ ) (25)where λ , = ( (cid:15) h + ¯¯ (cid:15) ) − ± i (2( (cid:15) h + ¯¯ (cid:15) ) − (cid:113) − ( (cid:15) h +¯¯ (cid:15) ) +(2( (cid:15) h +¯¯ (cid:15) ) − − (cid:15) h +¯¯ (cid:15) )) ) (cid:15) h + ¯¯ (cid:15) ) − . (26)A straightforward calculation shows that dd ¯¯ (cid:15) ( | λ , | ) = dd ¯¯ (cid:15) ( (cid:115) ˜∆ − (cid:15) + (cid:15) h ) −
1) ) (cid:12)(cid:12)(cid:12)(cid:12) ¯¯ (cid:15) =0 = 2(1 − (cid:15) h ) + ( r h ) − r h > for ( r h , (cid:15) h ) ∈ H NS , where (27) ˜∆ = (1 − (cid:15) + (cid:15) h ))( r − + 4(¯¯ (cid:15) + (cid:15) h ) r h ( r h − (cid:15) + (cid:15) h ) . Now we state conditions for the absence of strong resonances, i.e. λ m , ( (cid:15) h ) (cid:54) = 1 , m = 1 , , , for ¯¯ (cid:15) = 0 . Here wenote that the condition that the eigenvalues are a pair of complex conjugates leads to the following condition deduciblefrom equation (17) using ∆ = 2 (cid:15) : We can write λ , = (cid:15) − ± i (2 (cid:15) − (cid:113) (cid:15) − (cid:15) (1 − (cid:15) ) (cid:15) − (28)An examination of the condition λ m ( e h ) (cid:54) = 1 for m = 1 , , , , leads to the constraints (cid:15) (cid:54) = 0 , , , . For r ∈ [1 + √ , these (cid:15) constraints, again for (cid:15) ∈ H NS , are equivalent to r (cid:54) = 1 + √ which we now require. Now we studythe normal form of our system when ¯¯ (cid:15) = 0 by first computing the following Taylor expansion at ( u n , v n ) = (0 , : (cid:18) u n +1 v n +1 (cid:19) = (cid:18) a u n + a v n + a u n + a v n a u n + a v n + a u n + a v n (cid:19) (29)where the coefficients a , a , a , a , a , a were defined earlier. Next we define A = (cid:15) − (cid:15) − and A = (cid:113) (cid:15) − (cid:15) (1 − (cid:15) ) ; these coefficients represent the real and imaginary parts of λ , . Upon finding the eigenvectors associ-ated with these eigenvalues we construct the following invertible matrix T = (cid:18) − a a − A A (cid:19) Using the transformation (cid:18) u n v n (cid:19) = T (cid:18) X n Y n (cid:19) (30)10 PREPRINT - M AY
19, 2020the system can be rendered in the form X n +1 = A X n − A Y n + F ( X n , Y n ) (31) Y n +1 = A X n + A Y n + G ( X n , Y n ) (32)where F ( X n , Y n ) = c X n + c X n Y n + c Y n (33)and G ( X n , Y n ) = c X n + c X n Y n + c Y n (34)Here, the coefficients are defined as A = A − A ( a + a ) + a a − a a A , (35) A = a + a − A (36) c = A a − A a a + ( a ) a − A a a + a a a + a a A (37) + a a + 3 A a a − A a − A a a a A (38) c = 2 a a − A a + 2 A a − A a a + 2 a a a (39) c = A a + − A A a + a A a a (40) c = 2 A a a − A a − a a a (41) c = 2 A A a − a A a a (42) c = A a a (43)In addition we have F x n x n (cid:12)(cid:12)(cid:12) (0 , = 2 c F x n y n (cid:12)(cid:12)(cid:12) (0 , = c F y n y n (cid:12)(cid:12)(cid:12) (0 , = 2 c F x n x n x n (cid:12)(cid:12)(cid:12) (0 , = F x n x n y n (cid:12)(cid:12)(cid:12) (0 , = F x n y n y n (cid:12)(cid:12)(cid:12) (0 , = F y n y n y n (cid:12)(cid:12)(cid:12) (0 , = 0 and G x n x n (cid:12)(cid:12)(cid:12) (0 , = 2 c G x n y n (cid:12)(cid:12)(cid:12) (0 , = c G y n y n (cid:12)(cid:12)(cid:12) (0 , = 2 c G x n x n x n (cid:12)(cid:12)(cid:12) (0 , = G x n x n y n (cid:12)(cid:12)(cid:12) (0 , = G x n y n y n (cid:12)(cid:12)(cid:12) (0 , = G y n y n y n (cid:12)(cid:12)(cid:12) (0 , = 0 Now we must show that a (cid:54) = 0 where λ, ¯ λ = e ± iθ and a = − Re (cid:20) (1 − λ )¯ λ ζ ζ − λ (cid:21) − | ζ | − | ζ | + Re (¯ λζ ) (44)11 PREPRINT - M AY
19, 2020where ζ = 18 [( F x n x n − F y n y n + 2 G x n y n ) + i ( G x n x n − G y n y n − F x n y n ] (cid:12)(cid:12)(cid:12) (0 , = 14 [( c − c + c ) + i ( c − c − c )] ζ = 14 [( F x n x n + F y n y n ) + i ( G x n x n + G y n y n ] (cid:12)(cid:12)(cid:12) (0 , = 12 [( c + c + i ( c + c )] ζ = 18 [( F x n x n − F y n y n − G x n y n ) + i ( G x n x n − G y n y n + 2 F x n y n ] (cid:12)(cid:12)(cid:12) (0 , = 14 [( c − c − c ) + i ( c − c + c )] ζ = 116 [( F x n x n x n + F x n y n y n + G x n x n y n + G y n y n y n ) + i ( G x n x n x n + G x n y n y n − F x n x n y n − F y n y n y n ] (cid:12)(cid:12)(cid:12) (0 , = 0 We summarize our work now as a theorem indicating that a Neimark-Sacker bifurcation occurs at ( x ∗ , y ∗ ) and thenature of the resulting bifurcation curve: Theorem 3 If r (cid:54) = 1 + √ and a (cid:54) = 0 then the map undergoes a Neimark-Sacker bifurcation at the fixed point ( x ∗ , y ∗ ) when the parameter (cid:15) varies in a small neighborhood of (cid:15) h . Moreover if a < (respectively a > ) then an attracting(respectively repelling) invariant closed curve bifurcates from the fixed point for (cid:15) > (cid:15) h (respectively (cid:15) < (cid:15) h ). In this section we use Mathematica to numerically verify and illustrate the conclusions of Theorems 1, 2 and 3 withrespect to the fixed point ( x ∗ , y ∗ ) .Figure 2: Bifurcation diagram for r = 3 . , (cid:15) = 0 . .Using the flip equation (cid:15) = + √ (cid:113) r ( r − for r = 3 . we have (cid:15) = 0 . and ( x ∗ , y ∗ ) = ( . , . and α = − . . Since the corresponding value α ¡0 the period-2 orbits that bifurcate from ( x ∗ , y ∗ ) are unstableand they are succeeded by a stable period-1 orbit. In figure 2, we observe the emergence of the period-1 orbit atthe bifurcation point. The flip bifurcation occurs at (cid:15) = 0 . . Here we include a vertical line at (cid:15) = 0 . to show at least numerically that there is another flip bifurcation for (cid:15) = − √ (cid:113) r ( r − . Figure 3 shows that theunstable flip occurs in the chaotic region and the subsequent stable one cycle thereafter.In Figures 4, 5, and 6 we show further numerical evidence of a flip bifurcation at several other values of r . Next weconsider r = 3 . which corresponds to (cid:15) = 0 . . Here the corresponding fixed point is (0 . , . and the value of α = − . . The bifurcation diagram in Figure 4 shows the onset of flip bifurcations at the twomarked off vertical lines (cid:15) = 0 . , (cid:15) = 0 . . 12 PREPRINT - M AY
19, 2020Figure 3: Maximum Lyapunov Exponent Plot for r = 3 . .Figure 4: Bifurcation diagram for r = 3 . , (cid:15) = 0 . ,with initial conditions (0 . , . .Figure 5 gives a sequence of time series plots revealing a stable symmetric two cycle before the critical value of (cid:15) isreached and a weak two cycle at the critical value of (cid:15) . In the last plot we see the emergence of a one cycle for a valueof (cid:15) nearby but larger than our critical value. Here the chosen values of (cid:15) are . , . , . respectively.For contrast, we consider a fairly high value of r = 3 . , deep into the chaotic regime of the system. Here (cid:15) =0 . and the initial conditions are (0 . , . . The fixed point is (0 . , . and α = − . .(The corresponding lower value of (cid:15) where a flip may occur is (cid:15) = 0 . ). The accompanying sequence of timeseries plots shows a chaotic cycle colliding with a two cycle at our critical value and the birth of a one cycle for a valueof (cid:15) > . close to our critical value. Additional time series plots (not included here) in fact show a pattern ofintermittency-periods of stability and instability of a symmetric and anti-symmetric two cycles- before the one cycleis reached. In the panel the chosen values of (cid:15) are . , . , . , . respectively.Using the relation (cid:15) = f ( r ) and substituting . for r we get that (cid:15) = 0 . and the fixed point ( x ∗ , y ∗ ) =(0 . , . . Figures 7 and 8 below show the formation of a Neimark-Sacker bifurcation and chaotic regionsin the phase plane for the initial conditions (0 . , . . In Figure 7(a) where (cid:15) = 0 . < . the fixed pointis stable. Figures 7(b) illustrates the loss of stability of the fixed point at (cid:15) = 0 . . In figures 7(c),(d),(e) and (f) (cid:15) = 0 . , . , . , . , respectively.Here we see that for increasing (cid:15) > (cid:15) = 0 . the gradual development of a closedinvariant curve, in other words, a subcritical Neimark-Sacker bifurcation occurs. In addition, A detailed computationof a yields a negative value. Furthermore, in figures, 8(a),8(b), 8(c) and 8(d) (here (cid:15) = 0 . , . , . , . showthe transition to a chaotic state with the appearance of 11 coexisting chaotic attractors in figure 8(b) and a chaoticattracting set in figures 8(c) and 8(d) , for values of (cid:15) further away from . .13 PREPRINT - M AY
19, 2020Figure 5: Time Series plots for r = 3 . , (cid:15) = 0 . , . , . respectively.Figure 6: Time Series plots for r = 3 . , (cid:15) = 0 . , . , . , . ,respectively.14 PREPRINT - M AY
19, 2020Figure 7: Formation of a Neimark-Sacker Bifurcation15
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19, 2020Figure 8: Emergence of chaosThe vertical line in the accompanying bifurcation diagram shows the birth of Neimark-Sacker bifurcation. A plot ofthe maximum Lyapunov exponent for r = 3 . for (cid:15) in the range [0 . , is also included. Negative exponents indicateFigure 9: Bifurcation diagram and plot of the maximum Lyapunov exponent for r = 3 . .stable regions within the otherwise chaotic regime and positive exponents are indicative of the chaotic regions. In this work we investigated the dynamics of a discrete coupled system of logistic maps. We determined the stability ofthe systems’ fixed points and used center manifold and bifurcation theory to prove the existence of a flip and Neimark-Sacker bifurcation for the non-symmetric fixed point ( x ∗ , y ∗ ) . Using (cid:15) as our bifurcation parameter our numericalresults revealed that the flip bifurcation is a reverse flip bifurcation (or period halving bifurcation) in that at the criticalvalue of the parameter a newly unstable period 2 cycles bifurcates to a stable period 1 cycle (rather than a 1 cyclebecoming unstable and giving rise to a stable period 2 cycle). This result contrasts the usual ’period doubling cascade’observed in logistic map systems where typically r (not (cid:15) ) is chosen to be the bifurcation parameter. A generalexamination of the constant a in Theorem 3 and our numerical evidence show that the Neimark- Sacker bifurcation is16 PREPRINT - M AY
19, 2020subcritical. The rich dynamics of the system also includes interesting chaotic sets which will be analyzed further in aforthcoming work.
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