Chaos, Solitons and Fractals

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1 Chaos, Solitons Fractals 4 (009) Contents lists available at ScienceDirect Chaos, Solitons Fractals journal homeage: Kneading theory analysis of the Duffing equation Acilina Caneco a, *, Clara Grácio b, J. Leonel Rocha c a Mathematics Unit, DEETC, Instituto Suerior de Engenharia de Lisboa Rua Conselheiro Emídio Navarro,, Lisboa, Portugal b Deartment of Mathematics, Universidade de Évora CIMA-UE, Rua Romão Ramalho, 59, Évora, Portugal c Mathematics Unit, DEQ, Instituto Suerior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro,, Lisboa, Portugal article info abstract Article history: Acceted March 009 Dedicated to the memory of J. Sousa Ramos The urose of this aer is to study the symmetry effect on the kneading theory for symmetric unimodal mas for symmetric bimodal mas. We obtain some roerties about the kneading determinant for these mas, that imlies some simlifications in the usual formula to comute, exlicitly, the toological entroy. As an alication, we study the chaotic behaviour of the two-well Duffing equation with forcing. Ó 009 Elsevier Ltd. All rights reserved.. Motivation introduction The Duffing equation has been used to model the nonlinear dynamics of secial tyes of mechanical electrical systems. This differential equation has been named after the studies of Duffing in 98 [], has a cubic nonlinearity describes an oscillator. It is the simlest oscillator dislaying catastrohic jums of amlitude hase when the frequency of the forcing term is taken as a gradually changing arameter. It has drawn extensive attention due to the richness of its chaotic behaviour with a variety of interesting bifurcations, torus Arnold s tongues. The main alications have been in electronics, but it can also have alications in mechanics in biology. For examle, the brain is full of oscillators at micro macro level [5]. There are alications in neurology, ecology, secure communications, crytograhy, chaotic synchronization, so on. Due to the rich behaviour of these equations, recently there has been also several studies on the synchronization of two couled Duffing equations [3,4]. The most general forced form of the Duffing equation is x 00 ax 0 ðbx 3 x 0xÞ¼bcosðxt /Þ. Deending on the arameters chosen, the equation can take a number of secial forms. In [9,4], Mira et al. studied this equation with b ¼, x 0 ¼ 0, x ¼ / ¼ 0. They studied the bifurcations sets in the lane for different values of the arameter a, based on the notions of crossroad area, saddle area, sring area, isl, li quasi-li. If the coefficient of x is ositive, this equation reresents the single-well Duffing equation if it is negative, we have the two-well Duffing equation. But there are also studies of a three-well Duffing system with two eriodic forcings [3]. We will study the two-well equation with x 0 ¼, b ¼ / ¼ 0, i.e., x 00 ax 0 x 3 x ¼ b cos ðxtþ: ðþ In [], Xie et al. used symbolic dynamics to study the behaviour of chaotic attractors to analyze different eriodic windows inside a closed bifurcation region in the arameter lane. Symbolic dynamics is a rigorous tool to underst chaotic motions in dynamical systems. In this work, we will use techniques of kneading theory due essentially to Milnor Thurston [8] Sousa Ramos [5 7], alying these techniques to the study of the chaotic behaviour. We will use kneading theory to evaluate the toological entroy, which measures the chaoticity of the system. Generally, the grahic of the return ma is a very comlicated set of oints. Therefore, in order to be able to aly these techniques, we show, in Section, regions in the arameter lane where * Corresonding author. addresses: acilina@deetc.isel.il.t (A. Caneco), mgracio@uevora.t (C. Grácio), jrocha@deq.isel.il.t (J. Leonel Rocha) /$ - see front matter Ó 009 Elsevier Ltd. All rights reserved. doi:0.06/j.chaos

2 530 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) the first return Poincaré ma is close to a one-dimensional object. Indeed, by numerical simulations, we found a region U where the return ma is like a unimodal ma a region B where the return ma is like a bimodal ma, see Fig.. Furthermore, we notice that some arameter values in region U corresond to airs of unimodal mas that are symmetric (see Table ). In the same way, some arameter values in region B corresond to symmetric bimodal mas. In fact, in Table 3,we may see two kinds of symmetric bimodal ma. In Section 3, we briefly describe the kneading theory for a general m-modal ma then we study the effects of symmetry on the kneading determinant for symmetric unimodal mas for symmetric bimodal mas. It is well known that symmetry is an imortant issue has been intensively studied [,]. We rove that the kneading matrix resent some simlifications due to this symmetries. Examles are given, in Section 4, for the Duffing equation, but this method can be alied to others equations, whose return mas have this kind of symmetry.. Poincaré mas, bifurcation diagrams, unimodal bimodal regions We choose the Poincaré section defined by the lane y ¼ 0, since it is transversal to the flow, it contains all fixed oints catures most of the interesting dynamics. For our choice of the Poincaré section, the arameterization is simly realized by the x coordinates of the oints. In terms of these fx i g, a first return Poincaré ma x n x n is constructed. This will be done for eachoice of the arameters a, b w. In the examles resented in Section 4, we will fix the arameter w ¼ :8 we will study the first return Poincaré ma for different values of the arameters ða; bþ. We will denote this Poincaré ma by f a;b. In order to see how the first return Poincaré ma change with the arameters, we make bifurcation diagrams. For examle, in Fig., we lot the variation of the first coordinate of the first return Poincaré ma, x n, versus the arameter b ½0:5; 0:5Š, for a fixed value a ¼ 0:5. We see clearly the growing of comlexity as the arameter b increases. To have a more recise notion of this comlexity, we may comute the toological entroy of the first return Poincaré ma in each region U B. Let us first look, in the arameter lane ða; bþ, for those regions where the first return Poincaré ma behaves like a unimodal or a bimodal ma, because we have an exlicit way to evaluate the toological entroy in those cases. For examle, for a ¼ 0:5, we found that when b ½0:54; 0:60Š the first return Poincaré ma f a;b behaves like a unimodal ma when b ½0:479; 0:483Š the first return Poincaré ma f a;b behaves like a bimodal ma. These range of values corresond to two narrow vertical bs in the bifurcation diagram in Fig.. 3. Kneading theory toological entroy Consider a comact interval I R a m-modal ma f : I I, i.e., the ma f is iecewise monotone, with m critical oints m subintervals of monotonicity. Suose I ¼½c 0 ; c m Š can be divided by a artition of oints P ¼fc 0 ; c ;...; c m g in a finite number of subintervals I ¼½c 0 ; c Š; I ¼½c ; c Š;...; I m ¼½c m ; c m Š, in such a way that the restriction of f to each interval I j is strictly monotone, either increasing or decreasing. Assuming that each interval I j is the maximal interval where the function is strictly monotone, these intervals I j are called las of f the number of distinct las is called the la number,, off. In the interior of the interval I, the oints c ; c ;...; c m are local minimum or local maximum of f are called turning or critical oints of the function. The limit of the n-root of the la ffiffiffiffiffiffiffiffiffiffi number of f n (where f n n denotes the comosition of f with itself n times) is called the growth number of f, i.e., s ¼ lim n ðf n Þ.In[0], Misiurewicz Szlenk define the toological entroy as the logarithm of the growth number h to ðf Þ¼log s. In[8], Milnor Thurston develoed the concet of kneading determinant, denoted by DðtÞ, as a formal ower series from which we can comute the toological entroy as the logarithm of the inverse of its minimum real ositive root. On the other h, Sousa Ramos et al., using homological roerties roved a recise relation between the kneading determinant the characteristic olynomial of the Markov transition matrix associated with the itinerary of the critical oints. In fact, they roved that the toological entroy is the logarithm of the sectral radius of this matrix. Fig.. Unimodal region ðuþ bimodal region ðbþ.

3 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) The intervals I j ¼½c j ; c j Š are searated by the critical oints, numbered by its natural order c < c < < c m. We comute the images by f ; f ;...; f n ;... of a critical oint c j ðj ¼ ;...; m Þ we obtain its orbit n o Oðc j Þ¼ c n j : c n j ¼ f n ðc j Þ; n N : If f n ðc j Þ belongs to an oen interval I k ¼Šc k ; c k ½, then we associate to it a symbol L k with k ¼ ;...; m. If there is an r such that f n ðc j Þ¼c r, with r ¼ ;...; m, then we associate to it the symbol A r. So, to each oint c j, we associate a symbolic sequence, called the address of f n ðc j Þ, denoted by S ¼ S S S n, where the symbols S k belong to the m-modal alhabet, with m symbols, i.e., A m ¼fL ; A ; L ; A ;...; A m ; L m g. The symbolic sequence S ¼ S 0 S S S n can be eriodic, eventually eriodic or aeriodic [7]. The address of a critical oint c j is said eventually eriodic if there is a number N, such that the address of f n ðc j Þ is equal to the address of f n ðc j Þ, for large n N. The smallest of such is called the eventual eriod. To each symbol L k A m, with k ¼ ;...; m, define its sign by eðl k Þ¼ if f is decreasing in I k ðþ if f is increasing in I k eða k Þ¼0, with k ¼ ;...; m. We can comute the numbers s k ¼ Q k i¼0 eðl k Þ for k > 0, take s 0 ¼. The invariant coordinate of the symbolic sequence S, associated with a critical oint c j, is defined as the formal ower series j ðtþ ¼ Xk¼ s k t k S k : The kneading increments of eacritical oint c j are defined by m cj ðtþ ¼ j ðtþ ðtþ with j ¼ ;...m; ð4þ j where ðtþ ¼lim j xc h x ðtþ. Searating the terms associated with the symbols L ; L ;...; L m of the alhabet A m, the increments m j ðtþ, are written in the form j m cj ðtþ ¼N j ðtþl N j ðtþl N jm ð Þ ðtþl m : ð5þ The coefficients N jk in the ring Z½½tŠŠ are the entries of the m ðmþ kneading matrix 3 N ðtþ N ðmþ ðtþ NðtÞ ¼ : ð6þ N m ðtþ N mðmþ ðtþ Fig.. Bifurcation diagram for x n as a function of b with a ¼ 0:5 b ½0:5; 0:5Š. From this matrix, we comute the determinants D j ðtþ ¼det b NðtÞ, where b NðtÞ is obtained from NðtÞ removing the j column ðj ¼ ;...; m Þ, ð3þ

4 53 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) DðtÞ ¼ ðþj D j ðtþ eðl j Þt ð7þ is called the kneading determinant. Here, eðl j Þ is defined like in (). Let f be a m-modal ma DðtÞ defined as above. Let s be the growth number of f, then the toological entroy of the ma f is, see [8], h to ðf Þ¼log s with s ¼ t t ¼ minft ½0; Š : DðtÞ ¼0g: ð8þ 3.. Symmetry effect on the kneading theory for unimodal mas To each value of the arameters ða; bþ, consider a function f a;b : I I, from the comact interval I R to itself, such that I ¼½c 0 ; c Š is divided in two subintervals L ¼½c 0 ; cš R ¼½c; c Š, where c denote the single critical oint of f a;b suose that f a;b j L, f a;b j R are monotone functions. Such a ma is called a unimodal ma. With the above rocedure, we can comute the toological entroy for the unimodal mas. In Table, we show the kneading data the toological entroy associated to unimodal first return Poincaré ma f a;b, obtained for some values of the arameters ða; bþ U w ¼ :8. Note that some sequences of Table have zero toological entroy. They have eriods ower of two, so this result, obtained by numerical comutation, was exected as a consequence of Sharkovsky theorem. Note also, that some sequences of Table are eventually eriodic, see [7]. Looking at Table, we notice that there are airs of mas f a;b f a ;b with kneading sequences such that we can obtain one of them by changing the symbols L by R R by L. They are somehow symmetric they have the same toological entroy. In Table, we have some airs of symmetric unimodal mas, chosen from Table. We wonder if this equality between entroies holds for all symmetric unimodal mas. In order to rove that, we must define more recisely what we mean by mirror symmetric unimodal mas. Definition. Let f a;b f a ;b be two unimodal mas. Assume that f a;b has a critical oint c has a eriodic kneading sequence ðas S S Þ, being A the symbol corresonding to c. Then, we say that f a ;b, with a critical oint c, is the mirror symmetric ma of f a;b if only if f a ;b has the kneading sequence ðb b S b S b S Þ, such that ( c is a minimum if c is a maximum b Sj ¼ R if S j ¼ L; ð9þ c is a maximum if c is a minimum b Sj ¼ L if S j ¼ R; where B is the symbol corresonding to c. In that case, we call to (9) a mirror transformation to the unimodal mas f a;b f a ;b. Lemma. If f a;b f a ;b are two mirror symmetric unimodal mas, in the sense of (9), then N fa;b ðtþ ¼N fa ðtþ N fa;b ðtþ ¼N fa ðtþ: ;b ;b Table Kneading data toological entroy for some unimodal mas. ða; bþ Kneading data for f a;b h toðf a;b Þ ( , ) ðclrlllrlþ 0.0 ( , ) ðclrlllrlþ 0.0 ( , ) ðclrlllrlþ 0.0 ( ,0.3500) ðclrlllrlrlrlþ ( ,0.3500) ðclrlllrlrlrlþ ( , ) CLRLLðLRÞ ( , ) CLRLLðLRÞ ( , ) CLRLLðLRLRÞ ( , ) ðclrlllrlrlþ ( , ) ðclrlllrlrlþ ( , ) ðclrlllþ ( ,0.3580) ðclrlllþ (0.9540,0.800) ðclrlllrlþ 0.0 (0.9540,0.85) ðclrlllrlþ 0.0 (0.9540,0.850) ðcrlrrrlþ (0.9540,0.868) CRðLRRRÞ (0.9540,0.8750) ðcrlrrrþ ( ,0.860) ðclrlllþ ( ,0.8580) CRðLRRRÞ (0.0000,0.900) ðclrlllrlþ 0.0 (0.0000,0.90) ðcrlrrrlrþ 0.0 (0.5000,0.5390) ðclrlllrlllrlllrlþ 0.0 (0.5000,0.550) ðclrlllrlþ 0.0 (0.5000,0.5870) ðclrlllþ

5 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) Table Some airs of arameter values corresonding to symmetric unimodal mas. ða; bþ ða ; b Þ h to ( , ) (0.0000,0.90) 0 (0.0000,0.90) (0.0000,0.900) 0 (0.0000,0.550) (0.0000,0.90) 0 ( , ) (0.9540,0.850) ( , ) (0.9540,0.8750) (0.9540,0.8750) ( ,0.3580) ( ,0.860) (0.9540,0.8750) (0.5000,0.5870) (0.9540,0.8750) Proof. Let ðas S S Þ ðb b S b S b S Þ be the kneading sequences of f a;b f a ;b, resectively, where c ðrs S S Þ ; c ðrb S b S b S Þ ; c ðls S S Þ ; c ðlb S b S b S Þ : The invariant coordinates of the sequence associated to the critical oint of f a;b f a ;b are, resectively, ðtþ ¼ Xk¼ ðtþ ¼Xk¼ ðtþ ¼ Xk¼ ðtþ ¼Xk¼ s k¼ k c t k S k ¼ R X s k c t k S k k¼ s k¼ k c t k S k ¼ L X s k c t k S k k¼ s k c t kb k¼ Sk ¼ R X s k c t kb Sk k¼ s k c t kb k¼ Sk ¼ L X k¼ s c ; t s c ; t s c ; t s k ðc Þtkb S k s ðc : Þt Note that, s ðc Þ¼s ðc Þ¼s ðc Þ¼s ðc Þ, because eðs kþ¼eð b S k Þ due to (9). Denoting by L k ðc j Þ¼ Xk s i c j t i R k c j ¼ Xk s i c j t i i¼ i¼ S i or b Si ¼L S i or b Si ¼R with j ¼ ;, we notice that L ðc Þ¼R ðc Þ R ðc Þ¼L ðc Þ, because the number of symbols L in c is equal to the number of R in c the number of symbols R in c is equal to the number of L in c. Searating the terms associated with the symbols L R of the alhabet A ¼fA; L; Rg, we may write the invariant coordinates in the following way: ðtþ ¼ R L c L R c R s c ; t ðtþ ¼ L L c L R c R s c t ðtþ ¼ R R c L L c R s c ; t ðtþ ¼ L R c L L c R s c : t Consequently, the entries of the kneading increments for the oints c c are N fa;b ðtþ ¼ L c s c t L c s c t s c t N fa;b ðtþ ¼ R c s c t R c s c t : s c t It follows, as desired, that N fa;b ðtþ ¼N fa ;b ðtþ N fa;b ðtþ ¼N fa ;b ðtþ. h

6 534 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) The above lemma suggests the next result. Proosition 3. The mirror symmetric unimodal mas f a;b f a ;b, under the conditions of the revious lemma, have the same toological entroy. Proof. This statement is a consequence of Lemma, i.e., D fa;b ðtþ ¼N f a;b ðtþ ¼N fa ðtþ ¼D fa ðtþ ;b ;b D fa;b ðtþ ¼N f a;b ðtþ ¼N fa ;b ðtþ ¼D fa ;b ðtþ: By (7) noticing that eðl fa;b Þ¼eðL fa Þ, we have ;b D fa;b ðtþ D D fa;b ðtþ ¼ eðl fa;b Þt ¼ f ðtþ a;b eðl fa;b Þt D fa ðtþ D ;b ¼ eðl fa Þt ¼ f ðtþ a ;b eðl ;b fa Þt ¼ D f a ðtþ: ;b ;b So, the mas f a;b f a ;b entroy. h have the same kneading determinant DðtÞ, consequently, they have the same toological 3.. Symmetry effect on the kneading theory for bimodal mas To each value of the arameters ða; bþ, consider a function f a;b : I I, from the closed interval I to itself, such that I is divided in three subintervals L ¼½c 0 ; c Š, M ¼½c ; c Š R ¼½c ; c 3 Š, where c c denote the critical oints of f a;b suose that f a;b j L, f a;b j M f a;b j R are monotone functions. Such a ma is called a bimodal ma. For a bimodal ma, the symbolic sequences corresonding to eriodic orbits of the critical oints c, with eriod, c with eriod k, may be written as AS S S ; ðbq Q k Þ : In this case, we have two eriodic orbits, but in other cases of bimodal mas we have a single eriodic orbit, of eriod k, that asses through botritical oints (bistable case), for which we write only ðap P BQ Q k Þ : See some examles in Table 3. Now, we will study these two cases, with the additional condition that ¼ k the symbols Q j are the symmetric of the symbols P j, in the sense of the following definition. Definition 4. Let f a;b be a symmetric bimodal ma for which the eriodic kneading sequence, with eriod q ¼, is S ¼ ðas S S Þ ; ðb b S b S b S Þ ð0þ or S ¼ðAS S S B b S b S b S Þ ðþ with S ; S ;...; S fl; M; Rg, such that 8 b >< Sj ¼ R if S j ¼ L b Sj ¼ L if S j ¼ R A $ B: ðþ >: b Sj ¼ M if S j ¼ M Table 3 Kneading data for some bimodal mas. ða; bþ Kneading data for f a;b ( ,0.630) ððalrmrlþ ; ðbrlmlrþ Þ ( ,0.640) ðalrmrllbrlmlrrþ ( ,0.6500) ðalrmrlllrmrlmlrbrlmlrrrlmlrmrlþ ( ,0.6600) ðalrmrlllrbrlmlrrrlþ ( ,0.6900) ALRðMRLLLRBRLMLRRÞ ( ,0.6350) ðalrðmrllþ; BRLðMLRRÞÞ ( ,0.6300) ðalrðmrllþ; BRLðMLRRÞÞ

7 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) The bimodal ma f a;b is called a symmetric bimodal ma () is called a mirror transformation for this ma. We call b S j the mirror image of S j. The second half of S is obtained from the first one by the mirror transformation, see []. Note that, this definition of symmetry is just for one bimodal ma, while in Section 3. we defined symmetry between two different unimodal mas. In the next results, we will find some interesting roerties of the kneading theory due to this symmetry. Lemma 5. Let S ¼ððAS S S Þ ; ðb b S b S b S Þ Þ be a symmetric bimodal kneading sequence satisfying the mirror transformation (). Then, we have Proof. Set D ðtþ ¼D 3 ðtþ ¼N ðtþðn 3 ðtþn ðtþþ D ðtþ ¼N 3 ðtþn ðtþ: c ðms S S Þ ; c ðrb S b S b S Þ ; c ðls S S Þ ; c ðmb S b S b S Þ : The invariant coordinates of the sequences associated to the critical oints of f a;b are ðtþ ¼ Xk¼ ðtþ ¼Xk¼ ðtþ ¼ Xk¼ ðtþ ¼Xk¼ s k¼ k c t k S k ¼ M X s k c t k S k k¼ s k¼ k c t k S k ¼ L X s k c t k S k k¼ s k c t kb k¼ Sk ¼ R X s k c t kb Sk k¼ s k c t kb k¼ Sk ¼ M X s k c t kb Sk k¼ s i c j i¼ S i or b Si ¼L s c ; t s c ; t s c ; t s c : t Note that, s ðc Þ¼s ðc Þ¼s ðc Þ¼s ðc Þ, because eðs kþ¼eð b S k Þ due to (). Denoting by L k c j ¼ Xk t i ; M k c j ¼ Xk s i c j t i i¼ R k c j ¼ Xk s i c j i¼ S i or b Si ¼R S i or b Si ¼M t i with j ¼ ; : We notice that L c ¼ R c ; M c ¼ M c ; R c ¼ L c ; L c ¼L c ; M c ¼M c ; R c ¼R c : Searating the terms associated with the symbols L, M R of the alhabet A ¼fA; L; M; B; Rg, the kneading increments for the oints c c are L m c c ðtþ ¼ s c L c L M c t s c t s c M c R c M t s c t s c R c R t s c t m c ðtþ ¼ R c s c R c L t s c t M c s c M c M L c t s c t s c L c R: t s c t So, we have N 3 ðtþ ¼N ðtþ, N ðtþ ¼N ðtþ N ðtþ ¼N 3 ðtþ, consequently, D ðtþ ¼N ðtþn 3 ðtþn 3 ðtþn ðtþ ¼N ðtþðn 3 ðtþn ðtþþ; D ðtþ ¼N ðtþn 3 ðtþn 3 ðtþn ðtþ ¼ðN 3 ðtþn ðtþþðn 3 ðtþn ðtþþ; D 3 ðtþ ¼N ðtþn ðtþn ðtþn 3 ðtþ ¼N ðtþðn 3 ðtþn ðtþþ: For the articular case of bistable symmetric bimodal mas, we get a similar result.

8 536 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) Lemma 6. Let S ¼ðAS S S B b S b S b S Þ be a symmetric bimodal kneading sequence with eriod q ¼, satisfying the mirror transformation (). Then, we have D ðtþ ¼D 3 ðtþ ¼N ðtþðn 3 ðtþn ðtþþ D ðtþ ¼N 3 ðtþn ðtþ: Proof. The kneading matrix NðtÞ has entries obtained from m cj ðtþ (5) m cj ðtþ is defined by (4). Set c ðms S S M b S b S b S Þ c ðls S S R b S b S b S Þ ; c ðrb S b S b S LS S S Þ c ðmb S b S b S MS S S Þ : Note that, due to the symmetry of the ma, the right side of AðMÞ, is the mirror of the left side of BðMÞ the left side of AðLÞ, is the mirror of the right side of BðRÞ. The mirror of S k is b S k, for all k ¼ ;...;. Thus, we have ðtþ ¼ k¼ M X k¼ k¼ ðtþ ¼ L X k¼ s k c t k S k s k¼ c t M X s k c t kb Sk k¼ s k c t k S k s k¼ c t R X s k c t kb Sk k¼ t t : Attending to the eriodicity of S, these formal ower series are geometric series have a common ositive ratio t, because the number of symbols L, M R are even in each sequence of consecutive terms of the series. Denote by L k ¼ Xk c j s i i¼ S i ¼L c j t i ; M k c j ¼ Xk i¼ S i ¼M s i c j t i R k ¼ Xk c j s i i¼ S i ¼R c j t i with j ¼ ; : Looking to the first symbols of the sequences of c j c j, we see that the only difference is in S 0, which have oosite signs, so we have, s k ðc Þ¼s j kðc j Þ, for 0 < k <, j ¼ ;. Analogously, we see that the difference between the comlete sequences of c j c j is in S 0 S, which have the same sign, so we have, s k ðc Þ¼s j kðc j Þ. This imlies that L ¼ X t i ¼ X t i ¼L ; L q M R c j c j i¼ S i ¼L s i ¼ X c j c j c j s i i¼ S i ¼L c j i¼ S i ¼L s i t i ¼ X c j s i c j i¼ S i ¼L ¼M c j ; M q c j ¼ M q ¼R R q ¼ R q c j c j c j t i ¼ L q c j c j c j ; ; with j ¼ ; : So, the kneading increment is m c ðtþ ¼ L c L s c t M c M s c t R c R : ð3þ t Notice that, due to eðs i Þ¼eð b S i Þ, for all i, we have s c ¼ s c s c ¼ s c ; consequently, L c ¼R c ; M c ¼M c R c ¼L c : ð4þ In the same way, we comute the invariant coordinates the kneading increment of c by equalities (4), we may write m c ðtþ ¼ s c t R c L s c t M c M L c R : ð5þ t

9 From (3) (5), it follows immediately the kneading matrix the desired result. As a consequence of the above results, we have Proosition 7. Let f a;b be a symmetric bimodal ma of tye (0) or () in the sense of Definition 4 DðtÞ the kneading determinant (7). Let s ¼ =t be the growth number of f a;b, where t ¼ min ft ½0; Š : N 3 ðtþn ðtþ ¼0g: Then, the toological entroy of the ma f a;b is log s. Proof. Considering (8) by formula (7), the roots of DðtÞ are also the roots of D ðtþ, D ðtþ D 3 ðtþ. From Lemmas 5 6, we have D ðtþ ¼D 3 ðtþ ¼N ðtþ½n 3 ðtþn ðtþš D ðtþ ¼½N 3 ðtþn ðtþš½n 3 ðtþn ðtþš: So, the smaller real ositive root of DðtÞ occurs when the common factor of D ðtþ, D ðtþ D 3 ðtþ vanish. Notice that, although the sequence S has terms, it suffices the first terms to determine its dynamics the symbols M does not matter in the evaluation of the toological entroy. This suggests that the behaviour of a symmetric bimodal ma is determined by some unimodal ma, as ointed out in []. 4. Duffing alication A. Caneco et al. / Chaos, Solitons Fractals 4 (009) h h Examle 8. Let us take the Duffing equation () with the arameter values a ¼ 0:954 b ¼ 0:875. In this case, the attractor the unimodal Poincaré return ma are shown in Fig. 3. The symbolic sequence is ðcrlrrrþ, so we have c ðrrlrrrþ c ðlrlrrrþ : The invariant coordinates of the sequence S associated with the critical oint c are ðtþ ¼ t t L t t3 t 4 t 5 R; 6 t 6 t ðtþ ¼ t L t t3 t 4 t 5 R: 6 t 6 The kneading increment of the critical oint, m c ðtþ ¼ ðtþ ðtþ, is m c ðtþ ¼ t t 6 t L t t3 t 4 t 5 t 6 t R: So, the kneading matrix is NðtÞ ¼½N ðtþ N ðtþš ¼ ½D ðtþ D ðtþš, i.e., NðtÞ ¼ t t 6 t t 3 t 4 t 5 t 6 t t xk xk Fig. 3. Duffing attractor Poincaré return ma for a ¼ 0:954 b ¼ 0:875.

10 538 A. Caneco et al. / Chaos, Solitons Fractals 4 (009) x k x k Fig. 4. Duffing attractor Poincaré return ma for a ¼ 0:5 b ¼ 0:79. the kneading determinant is DðtÞ ¼ ð tþð t t 4 Þ t : The smallest ositive real root of D ðtþ is t ¼ 0:786..., so the growth number is s ¼ =t ¼ :7... the toological entroy is h to ¼ 0: Examle 9. Let us take, for examle, the case a ¼ 0:5 b ¼ 0:79 in the Duffing equation (). See in Fig. 4 the attractor the return ma, which behaves, in this case, like a symmetric bimodal ma. The symbolic sequence is bistable symmetric ðalrmrlllrbrlmlrrrlþ. Thus we have c ðmlrmrlllrmrlmlrrrlþ ; c ðllrmrlllrrrlmlrrrlþ ; c ðrrlmlrrrlllrmrlllrþ ; c ðmrlmlrrrlmlrmrlllrþ : Comuting the invariant coordinates ðtþ the kneading increments m i c ðtþ (i ¼ ; ), we obtain the kneading matrix, i from which we have D ðtþ ¼D 3 ðtþ ¼ ð tþ ð 3t t Þð t t Þð t 3 t 6 Þð t 3 t 6 Þ : t 8 From D ðtþ ¼0, we get t ¼ 0: , s ¼ : h to ðf Þ¼0: See in Table 3, the kneading data associated to symmetric bimodal mas f a;b for some values of ða; bþ B w ¼ :8. For all these examles, we have chaotic behaviour the toological entroy has exactly the same value References [] Duffing G. Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre Technische Beduetung. Vieweg Braunschweig 98. [] Xie Fa-Gen, Zheng Wei-Mou, Hao Bai-Lin. Symbolic dynamics of the two-well Duffing equation. Commun Theor Phys 995;4:43 5. [3] Jing Z, Huang J, Deng J. Comlex dynamics in three-well Duffing system with two external forcings. Chaos, Solitons & Fractals 007;33: [4] Khammari H, Mira C, Carcassés J-P. Behaviour of harmonics generated by a Duffing tye equation with a nonlinear daming. Part I. Int J Bifurc Chaos 005;5(0):38. [5] Lamreia JP, Sousa Ramos J. Symbolic dynamics for bimodal mas. Portugaliae Math 997;54(): 8. [6] Leonel Rocha J, Sousa Ramos J. Weighted kneading theory of one-dimensional mas with a hole. Int J Math Math Sci 004;38: [7] Leonel Rocha J, Sousa Ramos J. Comuting conditionally invariant measures escae rates. Neural Parallel Sci Comut 006;4:97 4. [8] Milnor J, Thurston W. On iterated mas of the interval. Lect notes in math, vol. 34. Berlin: Sringer; [9] Mira C, Touzani-Qriquet M, Kawakami H. Bifurcation structures generated by the nonautonomous Duffing equation. Int J Bifurc Chaos 999;9(7): [0] Misiurewicz M, Szlenk W. Entroy of iecewise monotone maings. Studia Math 980;67: [] El Naschie MS. Hierarchy of kissing numbers for excetional Lie symmetry grous in high energy hysics. Chaos, Solitons & Fractals 008;35:40. [] El Naschie MS. A derivation of the fine structure constant from the excetional Lie grou hierarchy of the micro cosmos. Chaos, Solitons & Fractals 008;36:89. [3] Njah AN, Vincent UE. Chaos synchronization between single double wells Duffing Van der Pol oscillators using active control. Chaos, Solitons & Fractals 008;37: [4] Wu X, Cai J, Wang M. Global chaos synchronization of the arametrically excited Duffing oscillators by linear state error feedback control. Chaos, Solitons & Fractals 008;36: 8. [5] Zeeman EC. Duffing s equation in brain modeling. J Inst Math Al 976:07 4.