BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible dynmics is very rich. In prticulr, we will consider dynmicl systems depending on prmeter. When the vlue of the prmeter chnges continuously, the behviour of the system my chnge in discontinuous wy. One sys tht bifurction occurs for n isolte vlue of the prmeter t which the type of dynmic chnges.. Discrete one-dimensionl dynmicl systems A discrete one-dimensionl dynmicl system is system subjected to single eqution of this type () (t + ) = f() where I R nd f is function of. The vrible t is in generl considered s the time, but in discrete systems the time tkes only discrete vlues, so tht it is possible to tke t Z. A trjectory is set {} t= of points stisfying the bove eqution. It is evident tht the initil point = () determines the entire trjectory. The behviour of the dynmicl system is therefore given by ll the trjectories { : () = } for ll initil vlues I. A dynmicl system depending on prmeter is described by fmily {f } of functions prmetrized by, where A R. (t + ) = f ()... Fied points nd their stbility. Let I be point of the dynmicl system () stisfying f( ) =. Consider trjectory strting t =. It is evident tht the entire trjectory is formed by the unique point, i.e. = t. A point stisfying f( ) = is clled fied point or equilibrium point of the system (). Definition. The trjectory of the system () strting t is the set {, f( ), f(f( ),... }, i.e., the succession of points {} t= determined by the recurrence () with the initil condition () =. We re now interested in the trjectories strting t points which re ner. In order to better understnd the trjectories of the one-dimensionl dynmicl system we introduce the grphicl solution. Consider the grph of the function f(). The bsciss represents nd the ordinte (t + ).
2 FRANCESCA AICARDI (t+) (4) (3) (2) () () () (2) (3) Figure. Grphicl method to obtin trjectory Consider now fied point nd suppose tht f() be smooth t. Then there is neighbourhood of the point where ll trjectories strting t point of remin in nd pproch or ll trjectories strting t point of move wy from nd eit from. In the first cse the fied point is sid to be n ttrcting point or equilibrium point nd in the second cse repelling point or n un equilibrium point. (t+) (t+) () '() () '() Figure 2. Attrcting (left) nd repelling (right) fied point Question. Observe Figures 2 nd 3. Which property of f t the fied point determines whether is ttrcting or repelling?
BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS 3 (t+) (t+) () () Figure 3. Attrcting (left) nd repelling (right) fied point The following theorem nswers the question bove. Theorem. If is fied point of eqution (t + ) = f() nd f t is smooth then is n ttrcting point if f ( ) < nd is repelling point if f ( ) >. Problem. Give grphicl emple where fied point is neither ttrcting nor repelling, nd f is smooth with f ( ) =. Problem 2. Give grphicl emple where fied point is neither ttrcting nor repelling, nd f is not smooth t. Problem 3. For the dynmicl system represented in Figure, find the fied points nd sy if they re ttrcting or repelling. Remrk. If t the fied point the derivtive is or, in order to know the stbility property we hve to investigte higher derivtives of f (if f is smooth t ). In this cse the fied point is sid non hyperbolic. 2. Loss of stbility: bifurctions We hve seen tht the fied points of the dynmicl system () re the points stisfying f() =. Let us suppose tht our system is given by function f like tht of Figure, nd tht such function belongs to fmily f of functions depending continuously on prmeter. Let f = f. Hence, for = there is only one ttrcting fied point nd two repelling fied points t the etremes of the intervl where f is defined. Let us suppose tht ll the functions of the fmily stisfy f() =, f() = nd f () = f () >. By continuous chnge of the function f into f, the ttrcting fied point (stisfying f ( ) = moves continuously. However, for some isolte vlue of the prmeter, something my hppen which chnges the dynmic. 2.. Sddle-node bifurction. As shown in Figure 4, it my hppen tht the grph of f, for some isolted vlue of becomes tngent to the digonl (the grph of the function h() = ). At the point of tngency, sy, the derivtive of the function f is
4 FRANCESCA AICARDI equl to, nd therefore the equilibrium point is non hyperbolic. For > two new fied points eist. Observe tht necessrily one is nd the other one is un. <* =* >* (t+) * tngency * un Figure 4. Sddle-node bifurction in the fmily f. The bifurction digrm is the grph of multivlued function, showing for every vlue of the prmeter in neighbourhood of the bifurction vlue the fied points of f in neighbourhood of. Stble fied points re mrked by continuous line, un points by dotted line. un * * Figure 5. Bifurction digrm of sddle-node bifurction. 2.2. Pitch-fork bifurction. In this cse the derivtive of the fied point of f chnges pssing through the vlue (or ) (see Figure 6). At tht point, sy, the grph of f is tngent to the digonl, with n order-2 tngency. When increses, the point of tngency disppers, the fied point tht ws (derivtive higher thn zero nd less thn one) becomes un (derivtive higher thn one) nd two other fied points eist t right nd t left of the un fied point. These two points re.
BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS 5 <* =* >* (t+) * tngency * un Figure 6. Pitch-fork bifurction in the fmily f. Figure 7 shows the bifurction digrm. At the fied point becomes un nd bout it new fied points pper. * un * Figure 7. Bifurction digrm of pitch-fork bifurction. Eercise. Drw the grph of function with n un fied point tht becomes by pitch-fork bifurction. Drw the corresponding bifurction digrm. 3. Periodic points In order to introduce nother typicl phenomenon of the discrete one-dimensionl systems we study the dynmics determined by the fmily of smooth functions: f = ( ) defined on the unit intervl I = [, ] for (, 4]. Evidently, = is fied point, nd since f () =, it is for ll vlues of less thn. For = the origin is therefore non hyperbolic fied point nd for > it is un. We will denote by the vlue =. The eqution f () = hs s solution, besides =, the point = /, which is in the intervl [, ] for >. The derivtive t such point is ( 2 ) = 2, therefore is for < < 3.
6 FRANCESCA AICARDI < = =.5 (t+) tngency un Figure 8. The logistic mp for < 3 The point becomes un t = 3. A trjectory strting ner the equilibrium point is like tht in Figure 3, left, for < 3 nd like tht in figure 3, right, for > 3. We will denote the vlue = 3 by. <3 =3 >3 (t+) un Figure 9. The logistic mp bout = 3 But the question now is: where the trjectory is going, i.e., does the succession, strting ner, pproch some set of points? In other words, does it eist n ttrcting set, which is not fied point? The nswer is yes. There is vlue 2 > 3 such tht for < < 2, ll trjectories strting t points different from nd non contining re ttrcted towrds cycle of two points (see Figure ).
BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS 7 (t+) =3.4 2-periodic points Figure. The ttrcting 2-cycle for = 3.4 In fct, for every vlue of between nd there re two points nd 2 such tht f( ) = 2 nd f( 2 ) =. The trjectory string t or 2 is therefore formed by {, 2,, 2,, 2,... }. Moreover, lmost ll other trjectories tend to such cycle. How to prove this? We will consider, insted of the mp f, the mp f (2) iterte. It is evident tht nd 2 stisfy := f (f ()), the second i = f (2) ( i ) i =, 2 i.e., they re fied points for this mp. We my pply Theorem to vlute their stbility: if they re ttrcting (repelling) for f (2), the cycle (, 2 ) will be ttrcting (repelling). (t+2) =3.4 2 Figure. The stbility of the ttrcting 2-cycle for = 3.4 In Figure we see tht the bsolute vlue of the slope of f (2) t nd 2 is less thn.
8 FRANCESCA AICARDI =2.5 =3 =3.4 (t+2) 2 Figure 2. The pitchfork bifurction of f (2) t = Eercise. Prove tht, 2 re the roots of the eqution 2 z 2 2 z z + + =. Find these roots. 4. Period doubling bifurction As increses, the bsolute vlue of the slope of f (2) t nd 2 increses (see Figure 2), till the vlue 2 = + 6 3.4495 when it becomes equl to. For such vlue of, for > 2 hs lwys four fied points (,,, 2 ) but they re ll un. Agin, we sk: where the trjectories re going? the 2-cycle (, 2 ) loses stbility. Observe tht f (2) We observe tht, loclly, i.e. in neighbourhood of or of 2, the function f 2 () looks like the function f () bout (its grph intersects the digonl, the slope vrying bout ). Therefore, if we now consider the iterte of f (2) (), i.e. the fourth iterte f (4) () = f (f (f (f ()))), we epect similr phenomenon ner the points nd 2. I.e., for = 2 the function f (4) hs contemporrily 2 pitchforks bifurctions in correspondence of the points nd 2, see Figure 3. =3.4 =3.5 (t+4) (t+4) 2 2 4- periodic points Figure 3. The loss of stbility of nd 2 nd birth of 4 4-periodic points.
BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS 9 A cycle similr to tht of Figure continues to eist, but it is un nd double cycle of 4 points is the ttrcting set (see Figure 4). (t+) =3.5 NSTABLE 2-CYCLE 4-periodic points STABLE 4-CYCLE Figure 4. The 4-cycle nd the un 2-cycle. This phenomenon repets when increses: for vlue 3 > 2 the 4-cycle loses stbility: the mp f (4) (f (4) () = f (8) () hs 4 pitch-fork bifurctions nd 8 new fied points pper (i.e. 8-periodic points for f ). Wht we observe in the behviour of the mp f when vries is not the pitch-fork bifurction (which is visible in the 2 n -iterte of f ), but phenomenon which is clled period doubling bifurction, see figure 5. =3.5 =3.56 (t+) (t+) 4-periodic points 8-periodic points Figure 5. By period doubling bifurction 4-cycle loses stbility nd ppers 8-cycle. This phenomenon occurs for succession of vlues i, (where the 2 i -cycle loses stbility nd the 2 i -cycle ppers), which is converging to vlue = 3.569946..., nd whose first vlues re = 3, 2 3.49949, 3 3.5449, 4 3.5644, 5 3.5687.
FRANCESCA AICARDI un 2 3 4 5 Figure 6. Scheme of the cscde of period doubling bifurctions. 5. niverslity nd Feigenbum constnts This phenomenon of succession of period dding bifurctions is not peculir of the logistic mp. Indeed, Feigenbum proved in 975 tht every fmily F = F () of functions defined on the unit intervl, such tht F is t lest 3 times differentible nd hs unique mimum in [, ], ehibits the sme behviour. Such functions re sid unimodl Moreover, he found two universl constnts, tht re chrcteristics only of the cscde of doubling periods bifurction, nd not depend on the prticulr mp we re using. These constnts re denoted by δ nd α: δ = lim n n n n+ n = 4.669269299678532382... The windows of the prmeter vlues between successive bifurction vlues decreses very rpidly. The constnt α is given by d n α = lim = 2.5297875958928222839287328... n d n+ where d n is the distnce between two brnching points (coming from the preceding bifurction) t the vlue = n. 6. Chos nd other periods At the vlue the periodic cycle is n infinite set of points which is clled Feigenbum ttrctor nd hs frctl dimension equl to.538. This dimension is the sme for unimodl mps. For vlues of > the mp f hs chotic behviour, but there re intervls where there re ttrcting cycles, s shown in this bifurction digrm, where the ttrcting set is plotted versus. The period 3 loses stbility by doubling period cscde, so tht there re ll 3 2 n periodic points, chrcterised by the Feigenbum constnts.
BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS Remrk. The rtio between the dimeters of successive circles on the rel is of the Mndelbrot set converges to the Feigenbum constnt δ.