Critical Slowing Down at a Fold and a Period Doubling Bifurcations for a Gauss Map

Brazilian Journal of Physics https://doi.org/10.1007/s13538-019-00706-0

STATISTICAL

Juliano A. de Oliveira1,2 · Hans M. J. de Mendonc¸a2 · Anderson A. A. da Silva2 · Edson D. Leonel2

Received: 2 July 2019 © Sociedade Brasileira de F´ısica 2019

Abstract The convergence to the stationary state is described using scaling arguments at a fold and a period doubling bifurcation in a one-dimensional Gauss map. Two procedures are used: (i) a phenomenological investigation leading to a set of critical exponents defining the universality class of the bifurcation and; (ii) analytical investigation that transforms, near the stationary state, the difference equation into an ordinary differential equation that is easily solved. The novelty of the procedure comes from the fact that it is firstly applied to the Gauss map and critical exponents for the fold bifurcations are defined.

Keywords Gauss map · Bifurcations · Scaling law · Critical exponents

1 Introduction under the dynamics that mostly undertakes the effects of transients [21]. A transient corresponds to a length of time A important description to represent the growth of biologi- to where the dynamics evolves from an initial condition cal populations using nonlinear mapping was introduced by until reaches the stationary state. The time evolution of the May [1]. After his work, many other applications of map- dynamics may exhibit intricate dynamics including chaotic pings can be found in problems of the physics, chemistry, transient [22], when a tangent bifurcation is observed, an biology, engineering, mathematics, and many others [2–17] algebraic decay to the stationary state when bifurcations The one-dimensional mappings can be described by are from periodic to periodic dynamics [23]. Our goal in a dynamical variable and control parameters allowing to this paper is to apply a recent developed procedure [24, built the orbit diagrams. The local bifurcations in the orbit 25] used in a family of logistic-like maps to a Gaussian diagrams can be investigated by the stability of either fixed map. The nonlinearity of the Gauss map is exponential, points or from higher period orbits [18, 19]. As soon as the and investigation using local expansion must be needed to control parameter is varied and the periodic orbit changes determine the dynamical properties of the convergence to its stability, therefore changing locally the dynamics of the the stationary state. flow of solutions, a bifurcation is in course. If the variation In this work, we investigate two specific bifurcations, is from a single parameter, the bifurcation is called co- namely fold and period doubling, in the Gauss mapping dimension one [20]. If two parameters are varied than a co- [21] with the goal of understand and describe the behavior dimension two, bifurcation is observed [20]. Generally the of the convergence to the fixed point at the bifurcations. stationary state is obtained by imposing specific conditions We confirm that, at the bifurcations, the convergence of the distance from the stationary state is algebraic and described by a homogeneous function leading to a set of critical Juliano A. de Oliveira [email protected] exponents that can be used to characterize the type of bifurcation. The paper is organized as follows. The model and some 1 Universidade Estadual Paulista (UNESP), Campusˆ de Sao˜ dynamical properties as the orbit diagram and fixed points Joao˜ da Boa Vista, Av. Profa. Isette Correaˆ Fontao,˜ 505, Sao˜ Joao˜ da Boa Vista, SP, 13876-750, Brazil are discussed in Section 2. Discussion of the convergence to the steady state at the two different bifurcations is made 2 Departamento de F´ısica, Instituto de Geocienciasˆ e Cienciasˆ Exatas, Campusˆ de Rio Claro, Universidade Estadual Paulista in Section 3. The analytical findings are treated in Section 4 (UNESP), Av.24A, 1515, Rio Claro, SP, 13506-900, Brazil while concluding remarks are drawn in Section 5. Braz J Phys

2 The Model distance to the fixed point an orbit has at the instant n. Of course the dynamics depends on the initial condition We study the evolution of orbits converging to the stationary x0, which gives the initial distance of the fixed point, and state at and near at two bifurcations in the Gauss map which of a parameter giving the distance from the bifurcation is written as point. For the investigation of the convergence to the fold − 2 bifurcation, the control parameter μ1 = κc − κ = 0for = νxn + 1 xn+1 e κ, (1) ∗ ≈− ≈− = x1 0.10804821, κc1 1.05244900, ν 4.9 and where ν and κ are the control parameters, x the dynamical different initial conditions are shown in Fig. 2. variable, and n denotes the recurrence term hence the From Fig. 2, we see that depending on the initial discrete time. Given an initial condition x0, a fixed ν and condition x0, the orbits stay confined in a region of constant considering a set of different values of κ, the asymptotic regime until they reach a crossover iteration number nx dynamics is shown in Fig. 1 through the orbit diagram. therefore bending towards a regime of decay marked by The bifurcations are indicated by arrows in Fig. 1 and a power law describes characterized by an exponent β. our goal is the investigation of the convergence to the steady From behavior observed in Fig. 2, we suppose the following state of orbits at and near at the bifurcations. The period one scaling hypothesis: ∗ fixed points are obtained from condition xn+1 = xn = x 1. For a short n typically n nx, the behavior of (x − leading to ∗ − ∗ ∝ − ∗ α x1 )vs. n is given by (xn x1 ) (x0 x1 ) leading to ∗ − ∗2 α = 1; x − e νx − κ = 0. (2) 2. For n nx, the dynamical variable is described as dx + ∗ Using the condition n 1 | ∗ = 1, assuming fixed ν = 4.9 (x − x ) ∝ nβ where β is called a decay exponent; dxn x n 1 and solving numerically the transcendental (2), we obtain 3. Finally, the crossover iteration number nx is given by ∗ ≈− ≈ ∝ − ∗ z the fold bifurcation at x1 0.10804821 and κc1 nx (x0 x1 ) ,wherez is a changeover exponent. dxn+1 −1.05244900, while considering |x∗ =−1, the period dxn The exponent β can be obtained by fitting a power law ∗ ≈ doubling bifurcation is obtained as x2 0.10804821 and to the decay regime. As shown in Fig. 2, we obtained κc ≈−0.83635258, where the index c represents a critical 2 β =−0.9999789(2). Figure 3 shows nx as a function of control parameter. − ∗ (x0 x1 ). The critical exponent z givenbyapowerlawfit is z ≈−1. The behavior shown in Fig. 2 together with the 3 Convergence of the Orbits to the Steady scaling hypotheses allows the following homogeneous and State generalized function to describe the convergence To understand the decay of the convergence to the steady ∗ = b − ∗ c ∗ ∗ x x0,x1 ,n lx l (x0 x1 ), l n , (3) state x1 at a fold bifurcation and x2 at a period doubling bifurcation, we investigate the behavior of x, denoting the -2 10 -2 x0=10 -3 x0=10 -4 x0=10 -4 -5 10 x0=10 Analytical Approach Best Fit * 1 -x n x -6 n β 10 x =-0.9999789(2)

-8 10

0 2 4 6 8 10 10 10 10 10 n ∗ ≈− Fig. 2 Plot of the convergence to the fixed point x1 0.10804821 = ≈− = Fig. 1 Orbit diagram for the Gauss map (1) considering ν 4.9 and a considering κc1 1.05244900, ν 4.9, and the different initial fixed initial condition x0 = 0.01 conditions as labeled in the figure Braz J Phys

-2 10 4 Numerical Data 10 Best Fit -2 x0=10 -3 x0=10 -4 x =10 3 -3 0 10 10 Analytical Approach Best Fit x * n 2

2 -x 10 n -4 β z=-1.000(1) x 10 = -0.499937(1)

1 10 -5 10 -5 -4 -3 -2 -1 10 10 10 10 10 x -x * 0 2 4 6 8 0 1 10 10 10 10 10 n Fig. 3 Plot of the crossover iteration number nx against the initial ∗ condition (x0 − x ) yielding a slope z =−1 Fig. 5 Plot of convergence to the stationary state at a period doubling 1 ∗ ≈ ≈− = bifurcation x2 0.10804821 considering κc2 0.83635258, ν 4.9 for different initial conditions. Dashed lines correspond to the where l is a scaling factor, b and c are characteristic analytical procedure discussed in Section 4 exponents. Using a similar procedure made in ref. [24], we end up with the following scaling law period doubling bifurcation. The convergence depends of α the number of iterations n, initial conditions x0, and param- z = , (4) = − = β eter μ2 κc2 κ 0, which defines the distance from the bifurcation point. The second iteration of the map (1)is such that knowledge of any two exponents allow one to given by find the third. The exponents are also used to rescale the ∗ ∗ − 2 (x − x ) n (x − x ) → (x − νx + variables n 1 and such that n 1 n xn+2 = e n 1 + κ.(5) ∗ − ∗ α → − ∗ z x1 )/(x0 x1 ) and n n/(x0 x1 ) and overlap all curves − ∗ − ∗ of (xn x1 )vs. n onto a single and hence universal plot as Figure 5 shows the behavior of (xn x2 ) as a function ∗ ≈ ≈− shown in Fig. 4. of n considering x2 0.10804821, κc2 0.83635258, Let us now discuss the convergence of the orbits to the ν = 4.9, and different initial conditions as shown in the ∗ period doubling bifurcation in x2 . To describe the conver- figure. The scenario is very similar as the previous case and gence to the steady state, we analyze the second iteration leads to the critical exponent β =−0.499937(1). Figure 6 − ∗ ≈− of the map and looking the behavior of x approaching to the shows nx as a function of (x0 x2 ), furnishing z 2.

0 6 10 10 Numerical data Best Fit 5 10 -2 α 10 4 *)

1 10 -x 0

x 3 n 10

*)/(x -4 1 10 -x n 2 z=-2.006(1)

(x 10

-6 1 10 10

-3 0 3 6 -4 -3 -2 -1 10 10 10 10 10 10 10 10 z x -x * n/(x0-x1*) 0 2

Fig. 4 Overlap of all curves shown in Fig. 2 onto a single and universal Fig. 6 Plot of the crossover iteration number nx against the initial − ∗ =− plot, after a convenient rescale of the axis condition (x0 x2 ) yielding z 2 Braz J Phys

Figure 7 shows the overlap of all curves plotted in Fig. 5 Rewritten (8) properly, we obtained the convergence of the − ∗ → − ∗ after the scaling transformations (xn x2 ) (xn orbits to the steady state x1 given by ∗ − ∗ α → − ∗ z x2 )/(x0 x2 ) and n n/(x0 x2 ) , onto a single curve. ∗ ∗ (x0 − x ) (x − x ) = 1 n 1 + − ∗ .(9) 1 4.09812765n(x0 x1 ) 4 Analytical Approach to the Equilibrium Let us discuss the implications of (9) to different ranges Points of n. We begin with the case to n nx that implies − ∗ 4.09812765n(x0 x1 ) 1, which is equivalent to one of Near the bifurcation, the convergence to the steady state the scaling hypothesis. Such case we obtained is very slow. Such a property allows us to transform the − ∗ ≈ − ∗ discrete map into a differential equation that must be solved (xn x1 ) (x0 x1 ), (10) analytically. To do that however, the mapping must be first Taylor expanded near the equilibrium. Considering only the and while compared with the first scaling hypothesis allow = terms of the lowest nonlinear contribution, we end up with us to conclude that α 1. Now considering n nx,where − ∗ 4.09812765n(x0 x1 ) 1. Such case, we obtained ∗ ∗ ∗ ∗ F (x ) ∗ 2 F(x) ≈ F(x )+F (x )(x−x )+ 1 x − x +O(2). − ∗ ≈ −1 1 1 1 2! 1 (xn x1 ) n , (11) (6) and when comparing with the second scaling hypothesis, we =− − ∗ = Considering terms until the second order and solv- conclude β 1. The case of 4.09812765n(x0 x1 ) 1 ∗ ≈− ≈ leads to the crossover n furnishing z =−1. ing numerically (6)forx1 0.10804821, κc1 x F (x∗) − = 1 = Now we consider the second case about decay of the 1.05244900, and ν 4.9, we obtained 1! 1and ∗ ∗ orbits to the fixed point x at period doubling bifurcation F (x1 ) 2 ! ≈−4.09812765. Let the discrete time series with = = ≈− 2 using μ2 0, when κ κc2 0.83635258. respect x be considered as a continuous series, from that, we To analyze the convergence of the orbits to the period can rewrite (6) as follows: doubling bifurcation in the second iteration of the map (1), − xn+1 xn =− − ∗ 2 furnishes 4.09812765(xn x1 ) , (n + 1) − n 2 −νF(xn) F(F(xn)) = xn+2 = e + κ, (12) dx ∗ ≈ =−4.09812765(x − x )2.(7) dn 1 −νx2 with F(xn) = xn+1 = e n + κ. Expanding the (12)in = ∗ The initial condition x0 is defined to n 0, then for an Taylor series around of x2 , we obtain arbitrary n, we have x(n). Integrating (7), we obtain ∗ ∗ F (F (x )) ∗ x(n) n ≈ + 2 − dx F(F(x)) x2 (x x2 ) =−4.09812765 dn .(8) 1! − ∗ 2 ∗ x0 (x x1 ) 0 F (F (x )) ∗ + 2 (x − x )2 2! 2 ∗ 0 F (F (x )) ∗ 10 + 2 (x − x )3 + O(4). (13) 3! 2 Using the same procedure as described in the previous

α -1 case however considering the second iterated of the 10 *)

2 mapping, we obtain -x 0 (x − x∗) − ∗ = 0 2 (xn x2 ) . (14) *)/(x -2 ∗ 2 2 + − 10 1 86.03112533n(x0 x2 ) -x n

(x Let us discuss some implications with respect to the (14) to some ranges of n. The first one is when n nx,where -3 10 − ∗ 2 86.03112533n(x0 x2 ) 1, which is equivalent to first scaling hypothesis shown previously. Then we obtain -8 -4 0 4 10 10 10 10 z ∗ ∗ (xn − x ) ≈ (x0 − x ), (15) n/(x0-x2*) 2 2

Fig. 7 Overlap of all curves shown in Fig. 5 onto a single and universal a quickly comparison with the first scaling hypothesis, plot, after a convenient rescale of the axis gave us the critical exponent α = 1. Making another Braz J Phys consideration, when n nx, implying that 86.03112533n References − ∗ 2 (x0 x1 ) 1, we have − ∗ ≈ −1/2 1. R.M. May, Science 86, 645 (1974) (xn x2 ) n . (16) 2. C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. 48, 1507 (1982). Comparing then this result with the second scaling Physica D 7, 181 (1983) =− 3. J.R. Pounder, T.D. Rogers, Nonlinear Analysis, Theory, Methods & hypothesis shown previously, we obtain β 1/2. Finally, Applications 10, 415 (1986) − ∗ 2 = when 86.03112533n(x0 x1 ) 1 and a comparison with 4. M. Joglekar, E. Ott, A. Yorke, Phys. Rev. Lett. 113, 084101 (2014) third scaling hypothesis gives z =−2. Moreover, for the 5. J.A.C. Gallas, Phys. Rev. Lett. 2714, 70 (1983) Gauss map, the values of the exponents α, β, z at the period 6. P. Collet, J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980) doubling bifurcation are the same as the ones observed in 7. R.M. May, G.A. Oster, Am. Natural 110, 573 (1976) the logistic-like map [24, 25]. The dashed lines shown in 8. K. Hamacher, Chaos 22, 033149 (2012) Fig. 5 correspond to the analytical results obtained in (14). 9. M. McCartney, Chaos 21, 043104 (2012) 10. P. Philominathan, M. Santhiah, I.R. Mohamed et al., Int. J. Bifurcations and Chaos 21, 1927 (2011) 11. M. Santhiah, P. Philominathan, Pramana J. Phys. 75, 403 (2010) 5 Conclusions 12. Y.-G. Zhang, J.-F. Zhang, Q. Ma et al., International Journal of Nonlinear Sciences and Numerical Simulation 11, 157 (2010) We have considered the convergence to the steady state 13. H. Wen, Z. Guang-Hao, Z. Gong et al., Acta Phys. Sin. 17, 170505 (2012) in the Gauss map in a bifurcation point near of the 14. M. Urquizu, A.M. Correig, Chaos, Solitons & Fractals 33, 1292 fold bifurcation. At the bifurcation point, we used a (2007) phenomenological description to show that the convergence 15. G. Livadiotis, Advances in Complex Systems 8, 15 (2005) to the fixed point is described using a homogeneous function 16. D. Ilhem, K. Amel, Discrete Dynamics in Nature and Society with three critical exponents β, α, and z. They are related by 2006. Article ID 15840 (2006) 17. T.Y. Li, J.A. Yorke, Am. Math. Monthly 82, 985 (1975) a scaling law z = α/β. Our results provide α = 1, β =−1, 18. R.L.A. Devaney, First Course in Chaotic Dynamical Systems: and z =−1. Theory and Experiment (Studies in Nonlinearity) (Westview Press, Further investigation was also made at a period doubling Cambridge, 1992) bifurcation. The exponents found were α = 1, β =−1/2, 19. K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: an Introduction to Dynamical Systems =− (Springer, New York, 1997) and z 2 in well agreement with results discussed in ref. 20. S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications [24, 25]. These results were also obtained by transforming to Physics, Biology, Chemistry, and Engineering (Westview Press, the difference equation into a differential one which was Cambridge, 2001) solved analytically, giving support to the results obtained 21. R.C. Hilborn, Chaos and Nonlinear Dynamics: an Introduction for Scientists and Engineers (Oxford University Press, New York, from numerical simulations. 1994) 22. C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. 48, 1507 (1982) Funding Information JAO thanks CNPq (303242/2018-3, 421254/ 23. J.E. Hirsch, B.A. Huberman, D.J. Scalapino, Phys. Rev. A 25, 519 2016-5, 311105/2015-7), and FAPESP (2014/18672-8, 2018/14685-9) (1982) (Brazilian agencies). HMJM acknowledges FAPESP (2015/22062- 24. R.M.N. Teixeira, D.S. Rando, F.C. Geraldo, R.N. Costa Filho, J.A. 3) (Brazilian agency). AAAS thanks CAPES (Brazilian agency). de Oliveira, Phys. Lett. A 379, 1246 (2015) EDL thanks CNPq (303707/2015-1), FUNDUNESP, and FAPESP 25. E.D. Leonel, R.M.N. Teixeira, D.S. Rando, R.N. Costa Filho, J.A. (2017/14414-2, 2012/23688-5, 2008/57528-9, 2005/56253-8) (Brazil- de Oliveira, Phys. Lett. A 379, 1796 (2015) ian agencies). This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the Sao˜ Publisher’s Note Springer Nature remains neutral with regard to Paulo State University (UNESP). jurisdictional claims in published maps and institutional affiliations.