Chaos, Solitons and Fractals 113 (2018) 238–243 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Review Effects of a parametric perturbation in the Hassell mapping Juliano A. de Oliveira a , b , ∗, Hans M.J. de Mendonça a , Diogo R. da Costa a , Edson D. Leonel a , c a Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas, Departamento de Física, Câmpus de Rio Claro, Av.24A, 1515, SP 13506-900, Brazil b Universidade Estadual Paulista (UNESP), Câmpus de São João da Boa Vista, Av. Profa. Isette Corrêa Fontão, 505, SP 13876-750, Brazil c Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste 34151, Italy a r t i c l e i n f o Article history: Received 9 March 2018 Revised 14 May 2018 Accepted 12 June 2018 Available online 23 June 2018 MSC: 00-01 99-00 Keywords: Perturbed Hassell mapping Convergence to the stationary state Extreming curves Parameter space a b s t r a c t The convergence to the fixed point near at a transcritical bifurcation and the organization of the ex- treming curves for a parametric perturbed Hassell mapping are investigated. The evolution of the orbits towards the fixed point at the transcritical bifurcation is described using a phenomenological approach with the support of scaling hypotheses and homogeneous function hence leading to a scaling law related with three critical exponents. Near the bifurcation the decay to the fixed point is exponential with a relaxation time given by a power law. The extreming curves in the parameter space dictates the organi- zation for the windows of periodicity, consequently demonstrating how the set of shrimp-like structures are organized. © 2018 Elsevier Ltd. All rights reserved. t c b m m a p w e z c t m S t b a c 1. Introduction Discrete time can be used to characterize the evolution of dy- namical systems described by the so called mappings. Investiga- tions might consider dissipative as well as non-dissipative dynam- ics in either 1-D or higher dimensions. The literature on iterated mappings is vast [1–6] and is always increasing. The studies on this sort of dynamical systems increased after the seminal papers of May and co-authors [7,8] . The dynamics reveals a complicated and intricate organization either as function of the control param- eter or time [9,10] , namely periodic orbits, bifurcations of different types including both local – transcritical, tangent, period doubling etc. – and global such as boundary crisis, merging attractor crisis, interior crisis, chaos and its different routes. The convergence to the fixed point at the bifurcation was proved to obey an homoge- neous function characterized by a set of three critical exponents [11,12] , while near the bifurcation the dynamics evolves to the sta- tionary state via an exponential decay [13] whose relaxation time is characterized by a power law on the distance from the bifurca- tion. These four exponents define the class of universality of the bifurcation. After that the Hassell mapping [14,15] was considered ∗ Corresponding author at: Universidade Estadual Paulista (UNESP), Câmpus de São João da Boa Vista, Av. Profa. Isette Corrêa Fontão, 505, SP 13876-750, Brazil. E-mail address: juliano.antonio@unesp.br (J.A. de Oliveira). 2 w N https://doi.org/10.1016/j.chaos.2018.06.017 0960-0779/© 2018 Elsevier Ltd. All rights reserved. o explore the evolution towards the equilibrium near the trans- ritical bifurcation [16] . In this paper we investigate the effects of a parametric pertur- ation in the dynamical properties of a Hassell mapping, with the ain goal of to understand and describe the effects of the para- etric perturbation in the convergence to the fixed point as well s to characterize the organization of the extreming curves in the arameter space [17,18] . The investigation to the asymptotic state ill be made by using an homogeneous function [19] . The param- ter space investigation shall allow us to understand the organi- ation of the returning curves hence given the organization of the haotic and periodic domains as a function of the control parame- ers [17] . This work is organized as follows: Section 2 describes the ap, bifurcation diagram and the convergence to the fixed point. ection 3 is devoted to the analytical studies on the convergence o the fixed point at the transcritical bifurcation and in its neigh- oring. Section 4 is placed to discuss about the parameter space nd the extreming curves. Finally, in Section 5 we present the dis- ussions and conclusions. . The model The model we consider is a time perturbed Hassell mapping ritten as n +1 = N n λ(1 + b n ε) [ 1 + αN n ] −γ , (1) https://doi.org/10.1016/j.chaos.2018.06.017 http://www.ScienceDirect.com http://www.elsevier.com/locate/chaos http://crossmark.crossref.org/dialog/?doi=10.1016/j.chaos.2018.06.017&domain=pdf mailto:juliano.antonio@unesp.br https://doi.org/10.1016/j.chaos.2018.06.017 J.A. de Oliveira et al. / Chaos, Solitons and Fractals 113 (2018) 238–243 239 Fig. 1. Bifurcation diagram of the parametric perturbed Hassell mapping (1) , us- ing α = 1 , γ = 6 , ε = 0 . 01 and the initial condition N 0 = 0 . 1 . The parameter λ was considered in the range λ∈ [0, 60]. w t b i [ 2 t t a N F i p( y i t p c t t t t γ t γ e i d c t t Fig. 2. Convergence to the fixed point N ∗ = 0 considering as fixed α = 1 , γ = 6 , ε = 0 . 01 and the different initial conditions shown in Fig. Here λc = 1 / √ (1 − ε2 ) . 10-4 10-2 100 N0 100 102 n x Numerical Data Best Fit z=-1.002(1) Fig. 3. Crossover iteration number n x against the initial condition N 0 , yielding through a power law fit z = −1 . β n o i t N N w a z o t N a b i p e N here α > 0, γ > 0 and λ> 0 are control parameters and ε con- rols the amplitude of the parametric perturbation, b n is chosen as n = (−1) n or b n = (−1) n +1 , so that for a given initial condition N 0 t can be set as b 0 = 1 or b 0 = −1 . If ε = 0 the Hassell mapping 14–16] is recovered. The time perturbation is periodic - period -, it implies that, in order to find the fixed points and to study he convergence of the orbits to the stationary state, we have o calculate the map (1) in its 2 th iteration, i.e., N n +2 . Rewriting (n + 2) and n in a convenient way, such that (n + 2) → (m + 1) nd n → m , we obtain m +1 = λ2 (1 − ε2 ) N m × [ (1 + αN m ) + αλ(1 − ε) N m (1 + αN m ) (−γ +1) ]−γ . (2) ig. 1 shows the bifurcation diagram of map (1) and we aim to nvestigate what happens to the behavior of the system at the fixed oint N ∗ = 0 and at the transcritical bifurcation obtained from dN m +1 dN m ) N m +1 = N ∗ = 1 , (3) ielding λc = 1 √ (1 −ε2 ) , where the bifurcation arises (see zoom box n the Fig. 1 ). To describe the convergence to the steady state we analyze he map from second iteration and looking the behavior of N ap- roaching to the fixed point at the transcritical bifurcation. The onvergence depends on the number of iterations n , on the ini- ial condition N 0 and the parameter μ = λc − λ ∼= 0 , which defines he distance from the bifurcation point. The parameter μ = 0 es- ablishes the transcritical bifurcation in the fixed point N ∗ = 0 . In his way, the convergence is shown in Fig. 2 and we choose α = 1 , = 6 , ε = 0 . 01 and different initial conditions N 0 (as labeled in he figure). The procedure used can be made for another values of too. A careful investigation of Fig. 2 , allows us to see that differ- nt initial conditions lead to different curves with similar behav- ors. They have a plateau at short n and a regime of power law ecay for large enough n converging to the same fixed point. The hangeover from the plateau to the decay is marked by a charac- eristic crossover iteration number n x . From the behavior shown in hat figure, we can suppose 1. For a short n typically n � n x , the behavior of N n vs . n is given by N n ∝ N α 0 leading to α = 1 ; 2. For n n x , we notice N n ∝ n β where β is called a decay expo- nent; 3. Finally the crossover iteration number n x is given by n x ∝ N z 0 , where z is a changeover exponent. A power law fitting in the regime of decay in Fig. 2 gives = −0 . 999987(1) . The exponent z is obtained from the plot of x vs . N 0 , as shown in Fig. 3 . Using α = 1 , γ = 6 and ε = 0 . 01 we btained through a power law fitting z = −1 . The behavior shown n Fig. 2 together with the scaling hypotheses allow us to describe he behavior of N n as homogeneous function of the variables n and 0 , when μ = 0 , of the type (N 0 , n ) = l N(l ˜ a N 0 , l ˜ b n ) , (4) here l is a scale factor, ˜ a and ˜ b are characteristic exponents. Doing similar procedure as made in Ref. [11] we obtain that = α β . (5) The knowledge of any two exponents allows us to find the third ne by using Eq. (5) . Moreover, the exponents can also be used o rescale the variables N n and n in a convenient way, such that → N N α 0 together with n → n N z 0 to overlap all curves of N vs. n onto single and hence universal curve, as shown in Fig. 4 . Let us now discuss the behavior of the convergence of the or- its to the fixed point not at the bifurcation, but rather close to t. This is the case when μ = 0 and the decay is marked by an ex- onential law given by Hohenberg and Halperin [19] , and Leonel t al. [20] (n, μ) ∝ e −n/τ , (6) 240 J.A. de Oliveira et al. / Chaos, Solitons and Fractals 113 (2018) 238–243 Fig. 4. Overlap of all curves shown in Fig. 2 onto a single and universal plot, after a convenient rescale of the axis. N s s N c t n t t g t a t p N w 0 e∫ w N t l 4 d t L T d b c t o b a t p F n P f w t t u where τ is the relaxation time given by τ ∝ μδ, (7) here μ defines the relaxation parameter and the exponent δ gives the speed of the decay. From the simulations and fitting δ in a power law, we obtained δ = −1 . 3. Analytical approach To have a better insight near the fixed point and as an attempt to describe the convergence of the orbits to the stationary state in a more robust way, it turns out to be convenient to expand the map (2) in its second iteration in Taylor series near the fixed point N ∗ = 0 . Considering only the terms of lowest nonlinear contribu- tion, we end up with N m +1 = λ2 (1 − ε2 ) N m [1 − aN m ] , (8) where a = γα[1 + λ(1 − ε)] . We suppose that close to the fixed point the dynamical variable N can be considered as a continuous variable. Then, for the case of μ = 0 , Eq. (8) is written as N m +1 = λ2 c (1 − ε2 ) N m − λ2 c (1 − ε2 ) aN 2 m , N m +1 − N m = N m +1 − N m (m + 1) − m , ≈ dN dm = −aN 2 . (9) Grouping the terms properly we obtain the following first order differential equation dN N 2 = −a dm. (10) The initial condition N 0 is defined by m = 0 , while for variable m we have N m . Applying these variables as integration limits, we have ∫ N(m ) N 0 dN ′ N ′ 2 = − ∫ m 0 a dm ′ . (11) Proceeding with the integration and regrouping we have N(m ) = N 0 [ amN 0 + 1] . (12) Let us then discuss the implications of Eq. (12) for specific val- ues of m . We start with the case amN 0 � 1, which is equivalent to the first scaling hypotheses n � n x . For such a case we obtain that ( m ) ∝ N 0 and allows us to conclude that α = 1 . Second we con- ider the situation amN 0 1, corresponding to n n x in the second caling hypotheses. For such case we obtain that (m ) ∝ m −1 . (13) Comparing this result with second scaling hypothesis, we con- lude that β = −1 . Finally when amN 0 = 1 , which is the case of he third scaling hypotheses n = n x , we obtain x ∝ N −1 0 , (14) herefore leading to z = −1 . Using this procedure we obtained all he three exponents discussed in the previous section, being in a ood agreement with the numerical results. The Fig. 2 show us he decay of the orbits to the fixed point N ∗, with the analytical pproaches indicated by the dashed lines. Those curves were ob- ained from Eq. (12) . We now study the behavior of the convergence to the fixed oint when μ = 0. The mapping is rewritten as m +1 − N m = ( λ λc )2 N m − N m , = N m +1 − N m (m + 1) − m ≈ dN dm , = −μ˜ k N m , (15) here μ = λc − λ and ˜ k = ( λ+ λc λ2 c ) . Considering again that for m = the initial condition is N 0 , we have to integrate the following quation N(m ) N 0 dN ′ N ′ = −μ˜ k ∫ m 0 dm ′ , (16) hich leads to (m ) = N 0 e −μ˜ k m . (17) Comparing this result with Eqs. (6) and (7) , we conclude that he exponent δ = −1 has the same decay speed to the fixed point ike in [16] . . Parameter space investigation This section is devoted to investigate the parameter space. To o that we shall use Lypaunov exponents to discriminate be- ween chaos and regular dynamics. For the considered system, the yapunov exponent is obtained as = lim n →∞ 1 n ln ( ∂N n +1 ∂N n ) . (18) he dynamics is chaotic when is positive while negative values efine regularity, that might include periodic or quasi periodic or- its. Fig. 5 (a)–(d) show the parameter space λ as a function of ε, onsidering mapping (1) for α = 1 and γ = 6 . The colors represent he Lyapunov exponent in (a) and (c), while the period of the rbits is used in (b) and (d). To obtain Fig. 5 (a) we chose 10 6 com- inations of λ and ε. For each pair of control parameters we iter- te the map for up to 10 6 iterations that is completely disregarded herefore being assumed as transient. After that the Lyapunov ex- onent is calculated for the next 10 6 iterations. As shown in ig. 5 (a) periodic structures, called as shrimp structures [18] , have egative values of , marked as the orange to yellow pallet color. ositive Lyapunov represents the chaotic structures and change rom green to blue. The periodic structures can be highlighted hen plotting the period as shown in Fig. 5 (b). One can notice he existence of a pattern of distribution of these structures, where he extreming curves are the responsible for this distribution. The nderstanding of them allows one to predict where the periodic J.A. de Oliveira et al. / Chaos, Solitons and Fractals 113 (2018) 238–243 241 Fig. 5. The parameter space for the mapping (1) is shown. In (a) and (c) the color represents the Lyapunov exponent , while in (b) and (d) the period is highlighted. o d c i i T { l N A N w o h N ε I − r N a u y u S v S u a i c z a q T u a v T A rbits exist in a parameter space. da Costa et al. [17] shows some etails of these ones for the perturbed logistic map as well as the ircle map. To find the superstable orbits and also the extreming curves, it s important to calculate the conditions that lead to → −∞ , and t happens when ∂N n +1 ∂N n = 0 . (19) his equation contains, at least, one of the solutions ̃ N 1 , ˜ N 2 , . . . , ˜ N i , . . . , ˜ N j , . . . } and in the first iterate it has as so- ution ˜ = 1 α(γ − 1) . (20) n extreming curve is an orbit given by ˜ i = F (k ) ( ̃ N j ) , (21) here ˜ N i and ˜ N j are solutions of Eq. (19) , that also contain ˜ N , as ne of the iterations of the mapping. For the system considered ere, one can obtain the first extreming curve considering N n +1 = n = ˜ N = 1 α(γ −1) , which leads to = 1 b n [ ( 1 + α ˜ N )γ λ − 1 ] . (22) t is important to remember that b n assumes two different values 1 and +1 , and the extreming curve with period 1 and b n = −1 is epresented in Fig. 5 (b) as the curve 1(−) . When we consider n +2 = N n = ˜ N , (23) superstable curve with period 2 is obtained. Considering = λ(1 − ε) and v = λ(1 + ε) , (24) ields immediately that + v = 2 λ and u − v = −2 λε. (25) olution of Eq. (23) gives = 1 u { (1 + αN n ) + αuN n ( 1 + αN n ) (1 −γ ) }−γ . (26) o, to obtain the superstable orbit with period 2 one needs to vary in order to obtain v and after that one can find the values of ε nd λ through Eq. (25) . A extreming curve with period 3 can also be obtained. For this t is necessary to solve N n +3 = N n = ˜ N . To find this solution it is onvenient to define = (1 + αN n ) + uαN n ( 1 + αN n ) (1 −γ ) , (27) nd = u v N n z −γ . (28) herefore, we find = N n q [ 1 + αq ] −γ , (29) nd = q uN n [ (1 + αN n ) + uαN n (1 + αN n ) (1 −γ ) ]−γ . (30) he main idea is to vary the value of q to find u through Eq. (29) . fter that one finds the value of v in Eq. (30) . With these values we 242 J.A. de Oliveira et al. / Chaos, Solitons and Fractals 113 (2018) 238–243 Fig. 6. The parameter space for the mapping (1) is shown. In (a) and (c) the color represents the Lyapunov exponent , while in (b) and (d) the period is highlighted. z g n i o f v l a r p o t l A ( ( ( D 9 b ( obtain the values of λ and ε using Eq. (25) . The extreming curves with period 3 are shown in Fig. 5 (b) and (d) as 3(+) and 3(−) . These ones dictates the organization of many periodic structures of the parameter space. Therefore, they play an important role for a dissipative system. The other high order extreming curves were obtained numerically, for example, making N n +5 = N n = ˜ N high- lights the extreming curves with period 5 (represented as 5(+) and 5(−) ). When two of these curves with different signs inter- cept each other, we have that the period of a shrimp is the sum of the period of each extreming curve. As an illustration, observe the curves 5(−) and 3(+) in Fig. 5 (b). When they intercept each other a shrimp of period 8 (represented as the red color) appears. If two extreming curves with same sign intercept each other, the period of the shrimp is given by subtracting the period of these curves. The intercept of the curve 5(+) and 3(+) in Fig. 5 (b) hap- pens at the same location of a period 2 shrimp (| 5 − 3 | = 2) . Fig. 6 (a)–(d) show the parameter space γ as function of ε for α = 1 and λ = 100 . In Fig. 6 (a) we have a plot of the periodic structures, where in Fig. 6 (b) the colors show the period of these ones, while the colored curves represent the extreming curves ob- tained analytically and numerically. Fig. 6 (c) shows an enlargement in Fig. 6 (a), where another cascade of shrimps can be observed in details. 5. Discussions and conclusions We have studied decay to the fixed point at the transcritical bifurcation in the time perturbed Hassell mapping and the organi- ation of the extreming curves in the parameter space. The conver- ence leads to a set of three critical exponents at the bifurcation, amely α = 1 , β = −1 and z = −1 , therefore the same universal- ty class of the bifurcation observed in the logistic map and in the riginal Hassell map. In fact, the periodic perturbation did not af- ect the critical exponents obtained in the Hassell map. The rele- ant relation of the critical exponent is summarized in the scaling aw z = α/β . Near the bifurcation the convergence is exponential nd with a relaxation time given by τ∝ μδ with δ = −1 . The pa- ameter space was investigated by the use of both Lyapunov ex- onents and period of the orbits. The extreming curves dictate the rganization for the windows of periodicity. We have shown how o obtain these orbits for this mapping both numerically and ana- ytically. cknowledgments JAO thanks CNPq ( 421254/2016-5 ), ( 311105/2015-7 ) and FAPESP 2014/18672-8 )(Brazilian agencies). HMJM acknowledges FAPESP 2015/22062-3 )(Brazilian agency). DRC acknowledges PNPd/ CAPES Brazilian agency). EDL thanks to CNPq ( 303707/2015-1 ), FUN- UNESP and FAPESP ( 2017/14414-2 , 2012/23688-5 , 2008/57528- , 2005/56253-8 )(Brazilian agencies). This research was supported y resources supplied by the Center for Scientific Computing NCC/GridUNESP) of the São Paulo State University (UNESP). https://doi.org/10.13039/501100003593 https://doi.org/10.13039/501100001807 https://doi.org/10.13039/501100002322 J.A. de Oliveira et al. / Chaos, Solitons and Fractals 113 (2018) 238–243 243 R [ eferences [1] Hilborn RC . Chaos and nonlinear dynamics: an introduction for scientists and engineers. New York: Oxford University Press; 1994 . [2] Zang WB . Discrete dynamical systems, bifurcations and chaos in economics. Elsevier Science; 2006 . [3] Martelli M . Introduction to discrete dynamical systems and chaos. New York: Wiley; 1999 . [4] Devaney RL . A first course in chaotic dynamical systems: theory and experi- ment (studies in nonlinearity). Cambridge: Westview Press; 1992 . [5] Galor O . Discrete dynamical systems. Heildelberg: Springer; 2007 . [6] Devaney RL . An introduction to chaotic dynamical systems. Cambridge: West- view Press; 2003 . [7] May RM . Science 1974;86:645 . [8] May RM , Oster GA . Am Nat 1976;110:573 . [9] Luo ACJ , O’Connor DM . System dynamics with interaction discontinuity (non- linear systems and complexity). Springer; 2015 . [10] Ott E . Chaos in dynamical systems. Cambridge: Cambridge University Press; 2002 . [11] Teixeira RMN , Rando DS , Geraldo FC , Costa Filho RN , de Oliveira JA , Leonel ED . Phys Lett A 2015;379:1246 . [12] Leonel ED , Teixeira RMN , Rando DS , Costa Filho RN , de Oliveira JA . Phys Lett A 2015;379:1796 . [13] Hirsch JE , Huberman BA , Scalapino DJ . Phys Rev A 1982;25:519 . [14] Hassell MP . J Anim Ecol 1975;44:283 . [15] Panik MJ . Growth curve modeling: theory and applications. Wiley; 2013 . [16] de Mendonça HMJ , Leonel ED , de Oliveira JA . Physica A 2017;466:537 . [17] da Costa DR , Hansen M , Guarise G , Medrano-T RO , Leonel ED . Phys Lett A 2016;380:1610 . [18] Gallas JAC . Phys Rev Lett 1993;70:2714 . [19] Hohenberg PC , Halperin BI . Rev Mod Phys 1977;49:435 . 20] Leonel ED , da Silva JKL , Kamphorst SO . Int J Bifurc Chaos 2002;12:1667 . http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0001 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0001 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0002 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0002 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0003 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0003 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0004 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0004 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0005 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0005 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0006 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0006 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0007 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0007 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0008 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0008 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0008 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0009 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0009 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0009 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0010 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0010 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0011 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0012 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0013 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0013 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0013 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0013 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0014 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0014 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0015 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0015 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0016 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0016 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0016 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0016 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0017 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0018 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0018 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0019 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0019 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0019 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0020 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0020 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0020 http://refhub.elsevier.com/S0960-0779(18)30404-1/sbref0020 Effects of a parametric perturbation in the Hassell mapping 1 Introduction 2 The model 3 Analytical approach 4 Parameter space investigation 5 Discussions and conclusions Acknowledgments References