Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate

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Original article

Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate Elisabetta Michetti ∗ Dipartimento di Economia e Diritto, via Crescimbeni 20, Università di Macerata, 62100 Macerata, Italy Received 15 February 2013; received in revised form 30 August 2013; accepted 3 September 2013

Abstract In this paper we study a discrete-time growth model of the Solow type with nonconcave production function where shareholders save more than workers and the population growth dynamics is described by the logistic equation. We prove that the resulting system has a compact global attractor and we describe its structure. We also perform a mainly numerical analysis to show that complex features are exhibited, related both to the structure of the coexisting attractors and to their basins. The study presented aims at showing the existence of complex dynamics when the elasticity of substitution between production factors is not too high (so that capital income declines) or the parameter in the logistic equation increases (so that the amplitude of movements in the population growth rate increases). © 2013 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Triangular system; Critical curves and absorbing areas; Local and global dynamics; Economic growth and population dynamics

1. Introduction The standard one-sector Solow–Swan model (see [34,35]) represents one of the most used framework to describe endogenous economic growth; it shows that the system monotonically converges to the steady state, so neither cycles nor complex dynamics can be observed. In order to investigate the possibility of complex dynamics to be exhibited in optimal growth models, many authors have studied the question whether the different saving propensities of two groups (labor and capital) might influence the final dynamics of the system. The question of differential savings between groups of agents was originally posed within the Harrod–Domar model of fixed portion [23]. Obviously different but constant saving propensities make the aggregate saving propensity nonconstant and dependent on income distribution so that multiple and unstable equilibria may occur. However, qualitative dynamics is still simple. Bohm and Kaas [7] investigated the discrete-time neoclassical growth model with constant but different saving propensities between capital and labor using a generic production function satisfying the weak Inada conditions, i.e. lim f (kt )/kt = 0, lim f (kt )/kt = ∞. The authors showed that instability and topological chaos can be generated in

kt →∞

kt →0

this kind of model. ∗

Tel.: +39 0733 2583237; fax: +39 0733 2583205. E-mail address: [email protected]

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Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Following the contribution by Bohm and Kaas a number of studies have considered the following two questions. First of all, the elasticity of substitution between production factors plays a crucial role in the theory of economic growth since it represents one of the determinants of the economic growth level. Hence the question of how the use of different production functions may affect the qualitative and quantitative dynamics in the long term becomes crucial. Secondly, the assumption that labor force grows at a constant rate is quite unrealistic since it implies that population grows exponentially. Hence different iteration schemes that are able to describe the evolution of the population growth rate must be considered. Several papers have been proposed to give an answer to the first question. Brianzoni et al. [8–10] investigated the neoclassical growth model in discrete time with differential savings and endogenous labor force growth rate while assuming Constant Elasticity of Substitution (CES) production function. The authors proved that multiple equilibria are likely to emerge and that complex dynamics can be exhibited if the elasticity of substitution between production factors is sufficiently low. Later, Tramontana et al. [36] studied the bifurcations related to the presence of a discontinuity point in the map with Leontief technology, representing the limit case as the elasticity of substitution tends to 0. As a further step in this field, Brianzoni et al. [12] firstly introduced the Variable Elasticity of Substitution (VES) production function in the form given by Revankar [31]. The authors proved that the model can exhibit unbounded endogenous growth (differently from CES) and that the production function elasticity of substitution is responsible for the creation and propagation of complicated dynamics, as in models with explicitly dynamic optimizing behavior by the private agents. More recently, Brianzoni et al. [11] considered the nonconcave production function, as firstly formulated by [19,33], proving that similarly to what happens with the CES and VES production function, if shareholders save more than workers and the elasticity of substitution between production factors is low, then the model can exhibit complexity. In order to give an answer to the second question, different iteration schemes have been introduced in economic growth models to describe the evolution of the population growth rate. Refs. [8,16] investigated the Solow–Swan growth model with labor force dynamics described by the Beverton–Holt (BH) equation (see [3]) while assuming CES production function; the results reached have been then generalized in [9] where a generic map having a unique positive globally stable fixed point was proposed to describe the evolution of the population growth rate. More recently the BH equation was also proposed in a growth model with VES production function (see [17]). While in the abovementioned works the authors considered simple population dynamic laws such that the population growth rate converges to a unique globally stable fixed point, in a number of papers the population growth evolution has been formalized with the well-known logistic map, able to exhibit more complex dynamics as cycles of every order or chaos. For instance in [10,15] the economic setup with CES production function was considered, while in [18] the VES production function was taken into account. In the present paper we study the discrete time one sector Solow–Swan growth model with differential savings as in [7], while assuming that: (1) the technology is described by a nonconcave production function in the form proposed by Capasso et al. [14] and (2) the population growth dynamics is described by the logistic equation. The reasons for introducing these two assumptions are the following: (1) For many economic growth models the production function is assumed to be non-negative, increasing and concave, and also to fulfill the so called Inada Conditions, i.e. f(0) = 0, lim f  (kt ) = +∞ and lim f  (kt ) = 0. Let us focus kt →0

kt →+∞

on the meaning of condition lim f  (kt ) = +∞ from an economic point of view. Consider a region with almost no kt →0

physical capital, that is there are no machines to produce goods, no infrastructure, etc. Then the previous condition states that it is possible to gain infinitely high returns by investing only a less amount of money. This obviously cannot be realistic since before getting returns it is necessary to create prerequisites by investing a certain amount of money. After establishing a basic structure for production, one might still get only less returns until reaching a threshold where the returns increase greatly to the point where the law of diminishing returns takes effect. In the economic literature this fact is known as poverty trap. In other words, one should expect that there is a critical level of physical capital having the property that, if the initial value of physical capital is lesser than such a level, then the dynamic of physical capital will descend to the zero level, thus eliminating any possibility of economic growth. Following this argument concavity assumptions provide a good approximation of a high level of economic development but it is not always applicable to less-developed countries. Thus it makes sense to assume that only an amount of money greater than some threshold will lead to returns. The first model with nonconcave production function was introduced by [19,33]. Following such works several contributions have then focused on the existence Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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and implications of critical levels (see, among others, [24,25,27]). More recently, Capasso et al. [14] focused on a parametric class of nonconcave production functions which can be considered as an extension of the standard Cobb–Douglas production function; the authors study the Solow growth model in continuous time and show the existence of rich dynamics by mainly using numerical techniques. (2) The hypothesis of constant population growth rate implies that population grows exponentially. This assumption is quite unrealistic since, as described by [28], in many countries a more realistic population growth model would have the following properties: (i) when population is less in proportion to the environmental carrying capacity, then it grows at a positive constant rate, (ii) when population is greater in proportion to the environmental carrying capacity, the resources become relatively more scarce and, as a result, this must affect the population growth rate negatively. Since the logistic map satisfies both properties, it is of interest to describe the population growth using the logistic function rather than the exponential one. Ref. [32] proposed to model population growth with the logistic equation. Other authors made the same choice: [13] analyzed the neo-classical Solow model with growth of population described by a generalized logistic growth law, Accinelli and Brida [2] studied the Ramsey model of optimal economic growth with the logistic equation (Richard’s law). On the basis of the above-mentioned arguments, in this work we study a discrete time growth model with differential saving, nonconcave production function and logistic population growth rate: the resulting system T is a bidimensional triangular dynamic system able to generate endogenous fluctuations for certain values of the parameters. We study the long run behavior of the system by performing both a theoretical and a numerical analysis. More precisely, we first prove that T admits the compact global attractor and we describe its structure; secondly we show how the global dynamics of the model can be analyzed by studying some global bifurcations that change the shape of the chaotic attractor as some parameters vary. These bifurcations are analyzed by using the critical curves method. Finally, we describe the structures and the global bifurcations of the basins of attraction generated by system T. We want to underline the added value of the study herewith conduced w.r.t. those presented in [10,11]. We prove that, as in [11], the presence of a nonconcave production function implies the existence of a poverty trap which can coexist with another attractor. However, since in the present work we add a law describing the evolution of the population growth rate, the final dynamic model is no longer one-dimensional and consequently the analysis of the basins of the two coexisting attractors on the plane becomes prominent. More precisely, the study proposed in [11] is able to describe only the dynamics of system T along its one-dimensional restriction to the set of points with constant population growth rate, i.e. n = n* , while if n varies over time, as it will be proved, there may exist points belonging to the basin of the poverty trap lying outside the set n = n* , and consequently the study of the structure of the basins of attraction for the model proposed in this paper cannot be derived from the study presented in [11]. On the other hand we show that, as in [10], the introduction of a logistic law to describe the evolution of the population growth rate produces complex feature depending on the value of its related parameter. Nevertheless, since we are now considering a nonconcave production function, the new system obtained always admits the poverty trap and consequently the structure of the basins of the coexisting attractors must be investigated. In fact, the study proposed in [11] can give information only about the evolution of system T if we consider initial conditions which do not converge to the poverty trap, while, with coexisting attractors, the global dynamics of T must be described. For instance, as it will be proved, a global final bifurcation occurs after which the poverty trap attracts almost all trajectories so that the results concerning the role of the logistic parameter presented in [10] cannot hold for the model proposed in this paper. From the previous considerations, a study devoted to the analysis of an economic growth model under the assumption of nonconcave production function together with the assumption of logistic population growth evolution must be focused mainly on the global dynamics of the system. Following this approach, it will be shown that the capital per capita and the population growth rate may converge to different attractors, depending on the initial conditions, and that, after a final bifurcation, depending on the parameter values, almost all trajectories will converge to the poverty trap. This study is of great importance in economics since featuring the economic system could be ambiguous with respect to initial conditions close to the basins boundary and perturbations on it. The paper is organized as follows. In Section 2, we present the triangular growth model with differential savings and logistic population growth. In Section 3, we perform a local analysis of the stability of the fixed points and other invariant sets owned by the system. In Section 4, we prove that system T has the compact global attractor and we describe its structure. We also use the critical lines method to describe the attractors of the system and to obtain the invariant absorbing area where asymptotic dynamics is confined. Finally we study the structure of the basins of Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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coexisting attractors and present some numerical simulations to describe the global bifurcations that are responsible for a change in the structure of such basins. We conclude the paper in Section 5. 2. The model We consider the Solow–Swan growth model where the two types of agents, workers and shareholders, have different but constant saving rates as in [7]. The one-dimensional map describing the evolution of the capital per-capita kt is given by 1 (1) [(1 − δ)kt + sw (F (kt ) − kt F  (kt )) + sr kt F  (kt )] 1+n where δ ∈ (0, 1) is the depreciation rate of capital, sw ∈ (0, 1) and sr ∈ (0, 1) are the constant saving rates for workers and shareholders respectively while n is the constant population growth rate. Function F is the production function in intensive form, mapping capital per worker k into output per worker y; we assume a sigmoidal production function (that is it shows an S-shaped behavior) F : R+ → R+ as proposed by [14], given by kt+1 =

y = F (k) =

αkp , 1 + βkp

(2)

where α > 0, β > 0 and p ≥ 2. The resulting one-dimensional dynamical system was studied in Brianzoni et al. [11]: the authors proved that complex features related both to the structure of the coexisting attractors and to their basins are exhibited and that complexity emerges if the elasticity of substitution between production factors is not too high and shareholders save more than workers, confirming the results obtained with concave production functions. Observe that F  (k) =

αpkp−1 (1 + βkp )2

and

F  (k) =

(pαkp−2 )[p(1 − βkp ) − (1 + βkp )] , (1 + βkp )3

(3)

and recall that the Inada Conditions are F(0) = 0, lim F  (k) = +∞ and lim F  (k) = 0. Then function (2) is positive k→0

k→+∞

∀k > 0, strictly increasing and it is a convex-concave production function. In fact F (k) > 0, ∀k > 0 while a k > 0 exists such that F  (k) >(resp. <)0, if 0 < k < k (resp. k > k), being k = ((p − 1)/β(p + 1))1/p the inflection point of F. Furthermore, the production function (2) does not satisfy one of the Inada Conditions since lim F  (k) = 0. k→0

Observe also that the elasticity of substitution between production factors of function (2) depends on the level of the capital per-capita k, as it is given by: σ(k) = 1 +

βpkp p(1 − βkp ) − (1 + βkp )

,

(4)

so that also the sigmoidal production function belongs to the class of VES production functions. Concerning the new ingredient of the study herewith proposed, we assume that the labor force growth rate is not constant, so that we take into account the possibility of fluctuations in the population growth rate. More precisely, we describe the evolution of the population growth rate with the logistic map that satisfies some economic desired properties, as explained in the Introduction of this work. The final two-dimensional dynamic system T = (n , k ), n ∈ [0, 1] and k ≥ 0, describing the capital per-capita (k) and the population growth rate (n) evolution is then given by: ⎧  n = f (n) = μn(1 − n) ⎪ ⎪ ⎪ ⎨ (5) T :=    ⎪ 1 αkp sr − sw ⎪  ⎪ ⎩ k = g(n, k) = sw + p (1 − δ)k + 1+n 1 + βkp 1 + βkp where μ ∈ (1, 4] for the dynamics generated by the logistic map being economically interesting, i.e. not explosive. We also assume sr > sw , i.e. s = sr − sw > 0, that is shareholders save more than workers, for function g not being negative. Furthermore, this case is more realistic from the economic point of view and it can be compared with other studies considering differential saving rates. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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System T is a discrete-time dynamical system described by the iteration of a triangular map of the plane, with g and f continuous and smooth functions for all k ≥ 0 and n ∈ [0, 1]. 3. Fixed points, invariant sets and their stability We first consider the question of the existence and number of fixed points or steady states of system (5) and then we discuss about their stability depending on all the parameter values. The equilibrium points (or steady states) of map T are all the solutions of the algebraic system T(n, k) = (n, k) where T is given by (5). The first equation says that the fixed points belong to the lines n = 0 and n = n , where n = (μ − 1)/μ. From the second equation we have that the k-values are the fixed points of the one-dimensional maps g0 (k) : = g(0, k) and gn (k) := g(n , k). The establishment of the number of steady states of such one-dimensional maps is not trivial to solve, considering the high variety of parameters. As a general result, the one-dimensional map gn (k) : = g(n, k), for any given n ∈ [0, 1], always admits the fixed point characterized by zero capital per capita, i.e. k = 0 is a fixed point for any choice of the parameters. Anyway, steady states which are economically interesting are those characterized by positive capital per worker. In order to determine the positive fixed points of gn (k) for any given n value, we recall the following result proved in [11]. Proposition 1. Let kp−1 G(k) := 1 + βkp



sr − sw sw + p 1 + βkp

 ,

k > 0.

(6)

Then a k˜ > 0 does exist such that: (i) if (ii) if (iii) if

n+δ α n+δ α n+δ α

˜ gn (k) has a unique fixed point given by k = 0; > G(k), ˜ gn (k) has two fixed points given by k = 0 and k = k* > 0; = G(k), ˜ gn (k) has three fixed points given by k = 0, k = k1 and k = k2 , 0 < k1 < k2 . < G(k),

According to the previous Proposition it follows that map gn (k) always admits the equilibrium k = 0, moreover up to two additional (positive) fixed points can exist according to the parameter values, hence multiple equilibria are exhibited. Since  1/p M (7) k˜ = β √ 2 where M = −b− 2ab −4ac > 0 being a = −sw < 0, b = sw (p − 2) − s(p2 + p) and c = sw (p − 1) + s(p2 − p) > ˜ does not depend on parameters n, δ and α. 0, then G(k) The above-mentioned arguments prove the following Proposition stating the number of fixed points of T. Proposition 2. Let G(k) given by (6) and k˜ > 0 given by (7). ˜ < δ then T admits two fixed points E00 = (0, 0) and En∗ 0 = (n∗ , 0); (i) If G(k) α ˜ < n∗ +δ then T admits four fixed points E00 = (0, 0), E0k = (0, kA ), E0kB = (0, kB ) and En∗ 0 = (ii) if αδ < G(k) A α (n∗ , 0), 0 < kA < kB ; ∗ ˜ > n +δ then T admits six fixed points E00 = (0, 0), E0k = (0, kA ), E0kB = (0, kB ) and En∗ 0 = (n∗ , 0), (iii) if G(k) A α En∗ k1 = (n∗ , k1 ), En∗ k2 = (n∗ , k2 ), 0 < kA < k1 < k2 < kB . ˜ = At G(k)

δ α

˜ = and G(k)

n∗ +δ α

a tangent bifurcation occurs.

As it can be noticed, system T is such that T(0, k) = (0, k ) and T(1, k) = (0, k ), ∀k ≥ 0; furthermore T(n* , k) = (n* ,

k ), ∀k ≥ 0, and T(n, 0) = (n , 0), ∀n ∈ [0, 1]. These properties prove the following Proposition on the invariant sets of T. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Proposition 3. Let D = [0, 1] × [0, + ∞). The following sets are invariant for system T : D → D: N0 = {(n, k) ∈ D : n = 0}, Nn∗ = {(n, k) ∈ D : n = n∗ }, K0 = {(n, k) ∈ D : k = 0}; furthermore set N1 = {(n, k) ∈ D : n = 1} is mapped into N0 at the first iteration. For the local stability analysis of the fixed points, denote with J(n, k) the Jacobian matrix of system T and recall the following property. Property 4. The eigenvalues of J(n, k) are always real, given by λ1 = f (n) and λ2 = ∂g ∂k (n, k). Any fixed point of T is therefore either a node or a saddle. The same is true for any cycle Om = {(ni , ki ), i = 1, 2, . . ., m} of T whose eigenvalues

m ∂g  are given by λ1 = m i=1 f (ni ) and λ2 = i=1 ∂k (ni , ki ). The local stability analysis of a fixed point can be carried out by studying the localization of the eigenvalues of the Jacobian matrix in the complex plane, and it is well known that a sufficient condition for the local stability is that both the eigenvalues are inside the unit circle in the complex plane. The triangular structure of system T simplifies the analysis, since, according to Property 4, the Jacobian matrix of T has real eigenvalues, located on the main diagonal, given by ∂g (n, k) = gn (k). ∂k Consider first the eigenvalues of T evaluated into points belonging to the invariant sets defined in Proposition 3. Then the following Proposition holds. λ1 (n) = μ − 2μn,

λ2 (n, k) =

Proposition 5. Set K0 attracts all trajectories starting from initial conditions (i.c.) (n0 , k0 ) having k0 sufficiently ˜ < δ then set K0 attracts all the trajectories. small. If G(k) α Proof. The eigenvalues of T restricted to the set K0 are λ1 (n) = μ − 2μn,

λ2 (n, 0) = gn (0).

Firstly notice that function gn (k) may be written in terms of function G(k) as gn (k) =

1 [(1 − δ)k + αkG(k)], 1+n

hence gn (k) =

1  1+n [1 − δ + α(G(k) + kG (k))].

Being lim G(k) = 0 and lim kG (k) = 0, then gn (0) = k→0+

k→0+

1−δ 1+n

∈

˜ < δ , T has two fixed points both located on K0 hence set (0, 1), ∀n ∈ [0, 1]. Hence set K0 is locally attracting. If G(k) α K0 is globally attracting.  According to Proposition 5 trajectories starting close to the invariant set K0 asymptotically tend to K0 ; such a result holds for all i.c. in D if T admits only two fixed points (both located on K0 ). This evidence is due to the fact that, as in [14], the use of the S-shaped production function implies the existence of a poverty trap. Recall that in models previously proposed in which the production function is concave, set K0 is always a locally unstable set, hence the economy will converge in the long run to positive growth rates (eventually with periodic or even a-periodic dynamic features). Differently, in this new setup, set K0 is locally stable, hence economies starting from a sufficiently low level of capital per-capita may be captured by the poverty trap. More precisely, there exists a critical level of physical capital having the property that, if the initial value of physical capital is less than such a level, then the dynamics of physical capital will converge to zero, thus eliminating any possibility of economic growth. About the restriction of system T to the set k = 0, notice that the properties of the trajectories embedded in the invariant axis k = 0 can be easily deduced from the well-known properties of the standard logistic map. More specifically n = 0 is a fixed point and for n0 = 1 the trajectory converges to zero at the second step, while every initial condition n0 ∈ (0, 1) generates bounded trajectories which converge to a unique attractor included in a trapping interval (i.e. a closed region positively invariant). ˜ does not depend on α and being α > 0 we can conclude that ∃α such that K0 attracts all trajectories Notice that, as G(k) ∀α ∈ (0, α) (this means that the production function upper bound is less enough). In this case the poverty trap cannot be avoided and the system will converge to a zero growth rate. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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About the other invariant sets of T observe that, trivially, λ1 (0) > 1 (hence set N0 is a repellor) so that fixed points belonging to the line n = 0 can be both saddle points or unstable nodes while |λ1 (n )| < 1 if and only if μ ∈ (1, 3), so that in this last case the line n = n attracts trajectories starting from initial conditions (n0 , k0 ), with n0 ∈ (0, 1) (i.e. set Nn∗ is an attracting set). Consider now λ2 (n, k). In [11] the following result about the local stability of the fixed points of the one-dimensional map gn (k) was proved. Proposition 6. The equilibrium k = 0 is locally stable for gn (k); if gn (k) admits three fixed points, 0 < k1 < k2 , then k1 is locally unstable while gn (k2 ) < 1. Taking into account the dynamics of the logistic map (see [20,29]) and Proposition 6 it follows that the equilibrium E00 is a saddle point while En∗ 0 is a stable node (resp. saddle point) as long as μ ∈ (1, 3) (resp. μ ∈ (3, 4]); similarly E0kA is an unstable node while En∗ k1 is a saddle point (resp. unstable node) if μ ∈ (1, 3) (resp. μ ∈ (3, 4]). Consider case (i) of Proposition 2, i.e. gn (k) has only one fixed point at the origin. Since gn (0) ∈ (0, 1) then if μ ∈ (1, 3), En∗ 0 attracts all trajectories having initial conditions n0 ∈ (0, 1) while if μ ∈ (3, 4] the attractor of the system may be a cycle or a more complex set belonging to the invariant set K0 . The dynamics of T restricted to K0 is governed by the logistic law. If Proposition 2 (ii) holds, then the dynamics of the system for all initial conditions with n0 ∈ (0, 1) are the same of case (i), i.e. they converge to En∗ 0 or to a more complex set located on the invariant set K0 , depending on the value of μ. Anyway, differently from case (i), also the dynamics involved in the invariant set N0 can be complicated and converging to a complex attractor (the dynamics of system T along the invariant line n = 0 is governed by the one-dimensional map g0 (k) that was completely studied in [11]). Given the above-mentioned considerations, in the rest of the paper we focus on the study of the dynamics of T when Proposition 2 (iii) holds, i.e. T admits six fixed points. Observe that, a sufficient condition for T having six fixed points is α great enough. In this case for any given value of the other parameters, multiple equilibria exist and their basins of attraction have to be discussed. Obviously the fixed point E0kB can be a saddle point or an unstable node while  (k ) that can be En∗ k2 can be a node or a saddle depending on the eigenvalues λ2 (E0kB ) = g0 (kB ) and λ2 (En∗k2 ) = gn∗ 2 discussed while considering the one-dimensional map gn (k). In [11] it was proved that gn (k) can be strictly increasing or bimodal in k, for all n values, and sufficient conditions were pursued. We recall this result. Proposition 7. Define p A = p−1 (sw − ps) < 0, β p

B=

p β

and Mm =

−[(2p − 1)A − (2p + 1)B] +

p−1 p

(sw + ps) > 0



[(2p − 1)A − (2p + 1)B]2 + 4(p2 − 1)AB > 1. 2(−1 − p)A

(i) If sw − ps≥0, then map gn (k) is strictly increasing. 2 +BM . (ii) Let sw − ps < 0 and define H(M) = AM 1 (a) If (b) If

M p (1+M)3 δ−1 H(Mm )≥ α then gn (k) is strictly increasing; H(Mm ) < δ−1 α then gn (k) admits a maximum point kM

and a minimum point km such that 1 < kM < km .

From Proposition 7 a sufficient condition for λ2 (E0kB ) ∈ (0, 1) or λ2 (En∗ k2 ) ∈ (0, 1) can be obtained, in fact if gn (k) is strictly increasing then, taking into account Proposition 6, it must be g0 (kA ) ∈ (0, 1) and gn ∗ (k2 ) ∈ (0, 1). Observe that if s is less enough then condition (i) of Proposition 7 holds (a s does exist such that gn (k) is strictly increasing ∀s < s). In this case, if μ ∈ (1, 3) the dynamics of T are quite simple: both the structure of the attractors (fixed points) and that of their basins (connected sets) are simple. More precisely, the economic system converges to a steady state characterized by a zero (poverty trap) or a positive capital per capita growth rate, while the population growth rate converges to n* for all i.c. n0 ∈ (0, 1). In fact if gn (k) is strictly increasing and μ ∈ (1, 3) the fixed points Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Fig. 1. (a) Fixed points of T and sets B1 (gray region) and B2 (white region) if α is sufficiently high (i.e. six fixed points are owned), s is low enough (i.e. gn (k) is strictly increasing) and μ ∈ (1, 3) (that is n* is stable for map f). Four trajectories are depicted. Parameter values: δ = 0.6, α = 10, β = 0.8, sw = 0.2, sr = 0.21, p = 10 and μ = 2.8. (b) For μ = 3.5, two attractors namely 1 and 2 coexist: they consist of a four period cycle, represented respectively by the red and the blue points, while the black points are fixed points of T. (c) For μ = 3.6, both 1 and 2 are two-piece chaotic attractors. (d) For μ = 3.9, 1 and 2 are complex connected coexisting attractors.

En∗ k2 and En∗ 0 are both locally stable. In such a case two different sets exist, namely B1 (n* , 0) ⊂ [0, 1] × [0, + ∞) and B2 (n* , k2 ) ⊂ [0, 1] × [0, + ∞) (such that B1 ∩ B2 = ∅), and a curve C which separates such sets, passing through the saddle points E0kA and En∗ k1 , such that trajectories starting from an i.c. in B1 will approach En∗ 0 while trajectories starting from an i.c. in B2 will approach En∗ k2 (see Fig. 1(a)). Consequently, featuring the economic system could be ambiguous with respect to i.c. close to C and perturbations on it. To summarize no cycles or complex features are observed if μ ∈ (1, 3) and the difference between the two propensities to save is low enough, confirming what proved in [8,10,12] in which concave production functions were taken into account Consider now case (ii) of Proposition 7. Condition p > sw /s is necessary for gn (k) being bimodal. Furthermore, lim H(Mm ) = −∞ so that a p1 > 0 does exist such that gn (k) is bimodal ∀p > p1 . Let p = max{2, sw /s, p1 }, then p→+∞

the following Proposition holds. Proposition 8. A p > 0 does exists such that gn (k) is bimodal ∀p > p. Observe that if Proposition 8 applies and μ ∈ (1, 3) then all trajectories starting from an i.c. having n0 ∈ (0, 1) converge to the set Nn∗ and consequently the asymptotic evolution of the capital per-capita is governed by the onedimensional map gn∗ (k). Again, we recall that this map was studied in [11] where it was proved that gn∗ (k) may present a complex attractor and that its basin can have a complex structure. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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We now want to explain the role played by parameter p and to show that it affects the elasticity of substitution between production factors. As showed in Eq. (4), the elasticity of substitution between production factors is not constant but it varies with the capital per-capita k. As a consequence, in order to understand how the elasticity of substitution moves when parameter p is increasing, we have to take into account that also the asymptotic values of k depend on p. We can distinguish between two cases: (a) If the system converges to a fixed point (n* , k* (p)), we are interested in the value that the elasticity of substitution assumes at the fixed point k* (p), i.e. σ(k* (p)). If we do not know the capital per-capita equilibrium value, we can only numerically compute σ(k* (p)) for different given values of p. Following this procedure, several numerical computations have shown that the computed values σ(k* (p)) decrease as p increases, that is the elasticity of substitution evaluated at the fixed point decreases as p is increased, once fixed the other parameter values. On the other hand, if the economy converges to the poverty trap, then k* (p) = 0, i.e. it does not depend on p; in this case, keeping fixed the level of the capital per-capita, it can be observed from Eq. (4) that the elasticity of substitution between production factors is decreasing w.r.t. p. (b) If the system converges to a periodic cycle or to a more complex attractor, we have to measure the elasticity of substitution in a different way. We propose the following procedure. Let 2 be the attractor of system and assume that 2 is composed by two or more points. Then we can numerically compute the elasticity of substitution associated to any k values of 2 . To do that we use the following procedure: we fix all the parameter values, then, starting from a point (n0 , k0 ) belonging to the basin of attraction of 2 , we iterate system T an high number of times (we choose N = 50,000) and then we calculate the elasticity of substitution associated to the last 1000 k-values. In this way we can suggest the following definition of the elasticity of substitution associated to an attractor 2 , that is, σ( 2 ) = max{σ(ki ) : (ni , ki ) = T i (n0 , k0 ), i = 49,000, 49,001, . . ., 50,000}. Observe that, for any given p value, σ( 2 ) represents the maximum value that the elasticity of substitution computed at the asymptotic k-values can assume. Observe also that the elasticity of substitution associated to a fixed point corresponds to the elasticity of substitution associated to an attractor if the attractor consists in a fixed point. Following this procedure, we computed σ( 2 ) for several p values and we observed that also the elasticity of substitution associated to an attractor decreases as p is increased. This evidence holds for several choices of the other parameters of the model. The numerical procedure herewith proposed enable us to observe that by increasing the value of p the elasticity of substitution decreases. In the rest of the paper we focus on the study of the open cases, listed in the following Remark, combining analytical tools and numerical experiments. Remark 9. The following cases are open: (a) T has six fixed points (i.e. α great enough) and gn (k) is strictly increasing (i.e. s less enough), when μ ∈ (3, 4]; (b) T has six fixed points (i.e. α great enough) and gn (k) is bimodal (i.e. p great enough), when μ ∈ (3, 4]. 4. Global dynamics In this section we study the global dynamics of system (T, D). Our main goal is to prove the existence of a compact global attractor and to describe its structure. We also study the structure and the global bifurcations of the basins of the coexisting attractors. 4.1. Existence of a compact global attractor Recall that a nonempty compact set C ⊂ D is the global attractor of the dynamical system (T, D) if C is invariant with respect to (T, D) and C attracts all the bounded subsets from D (see [18]). The following Proposition concerning the existence of the compact global attractor of system (T, D) can be proved. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Proposition 10. The dynamic system (T, D) given by (5) admits the compact global attractor  ⊂ [0, 1] × [0, N], where N is a positive real number. Proof. Notice that the map f acts from interval [0, 1] into itself, in other words [0, 1] is a trapping region, i.e. a closed region positively invariant (a set E ⊆ R2+ is positively (resp. negatively) invariant if Tt (E) ⊆ E (resp. Tt (E) ⊇ E), ∀t ∈ Z+ ; E is called invariant when it is both positively and negatively invariant). Consequently f admits a compact global attractor J ⊆ [0, 1]. Observe that the map g can be written as follows

1 g(n, k) = (1 − δ)k + j(k) 1+n where j(k) is continuous and given by   αkp sr − sw sw + p . j(k) := 1 + βkp 1 + βkp Being j(0) = 0 and lim j(k) = sw βα , then ∃P > 0 such that g(n, k) < [(1 − δ)k + P], ∀k ≥ 0. Consequently k→+∞

kt = gt (n0 , k0 ) < (1 − δ)t k0 + P

t−1  i=0

(1 − δ)i →

P 1−δ

as

t → +∞,

(8)

∀k0 ≥ 0. As a consequence the trajectory starting from a point (n0 , k0 ) ∈ D intersects the set [0, 1] × [0, N], N = P/(1 − δ), at least one time, and never leaves it. Finally, since [0, 1] × [0, N] is a compact, positively invariant and attracting set for T, then  = ∩ t≥0 Tt ([0, 1] × [0, N]) is a compact invariant set which attracts [0, 1] × [0, N], i.e. ∀ > 0 ∃ t such that Tt ([0, 1] × [0, N]) ∈ B(, ), ∀t > t , where B(, ) is a -neighborhood of .  ˜ where By taking into account the Proposition 10 it is easy to conclude that for any initial condition (n0 , k0 ) ∈ D, ˜ := [0, 1] × [0, N], all the images Tt (n0 , k0 ) of any rank t belong to the set D. ˜ Hence, in order to investigate the D ˜ i.e. the subsystem (T, D). ˜ structure of , we consider the restriction of the system on D, Since T is triangular, its dynamics are closely related to those of the one-dimensional map f (see [26,22] for a wider discussion). More precisely, it is easy to see that any bifurcation of the one-dimensional map f gives a bifurcation of system T. For instance, a fold bifurcation of f creates a couple of cyclical trapping lines of T (one repelling and one attracting) while at a flip bifurcation of a cycle of f, trapping cyclical vertical lines from attracting (for T) become repelling and new cyclical attracting lines are created. Taking into account these properties and the fact that f has an attractive m-period cycle for μ ∈ (1, μ∞ ), it is easy to deduce that if μ ∈ (1, μ∞ ), T admits m trapping cyclical attracting lines given by Jμ = ∪m i=1 [ni ] × [0, N] (in Fig. 1(b), T has 4 cyclic attracting lines as f admits a 4-period cycle). 4.2. Critical curves and absorbing areas The main purpose of this section is to analyze the properties of critical curves in order to define compact regions of the phase plane that act as trapping bounded sets, inside which asymptotic trajectories are confined (other works in the topic are [4,5,21]). Taking into account the well-known properties of the logistic map, we recall that f admits a trapping interval I = [c1 , c], where c = μ4 is the maximum value (critical point of rank-1) and c1 = f(c) is the critical point of rank-2 of f. Since I is a trapping interval for the logistic map, it is possible to state the following proposition. Proposition 11. Let μ ∈ (1, 4). The set A : = I × [0, N] is globally attracting and positively invariant for system (T, ˜ D ), being D = (0, 1) × [0, N] ⊂ D. Proof. Taking into account Proposition 10, the proof follows immediately from the fact that the interval I is a trapping region for f in (0, 1) if μ ∈ (1, 4).  Let ⊂ A be the attractor of (T, D ), then we want to describe the structure of to which all trajectories having / 0 and n0 = / 1 converge. Denote with B( ) the set of points (n0 , k0 ) ∈ D which generate trajectories converging n0 = to , then B( ) = D . Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Since the two-dimensional map (n , k ) = T(n, k) is non-invertible, the rank-1 preimages (n, k) = T−1 (n , k ) may be more than one, this means that the plane can be subdivided into regions Zj , j ≥ 0, whose points have j distinct rank-1 preimages and such regions are characterized by the presence of at least two coincident preimages. Let LC be the set of points having two, or more, coincident rank-1 preimages and denote by LC−1 the curve of merging preimages. Then for the two-dimensional map (T, D ), the locus LC−1 is given by the set of points such that |J(n, k)| = 0, where J(n, k) is the Jacobian matrix of the map T (see [30,1] for more details). The Jacobian matrix of system (T, D ) is given by ⎛  ⎞ 0 f (n) ⎠ J(n, k) = ⎝ ∂g (9) ∂g (n, k) (n, k) ∂n ∂k so that   1  [1 − δ + α(G(k) + kG (k))] |J(n, k)| = μ(1 − 2n) (10) 1+n hence, since T is triangular, the following proposition holds. Proposition 12. The locus LC−1 of the phase plane is made up of two curves LCa−1 and LCb−1 such that: (i) LCa−1 is the vertical line of equation n = 21 (ii) LCb−1 is the set of points (n, k) belonging to the horizontal lines k = km and k = kM , where km and kM are the minimum and the maximum points of gn (k), or LCb−1 = {∅}. Observe that n = c = μ4 is the critical value while n = c−1 = 21 is the critical point of the logistic map. Hence, a point n < c has two preimages and no preimages if n > c, thus the region Z2 is the set of points below c, the region Z0 the set of points above c and the critical value c is the boundary separating the two regions. This point has two merging preimages n1 = n2 = c−1 = 21 . Furthermore, consider that the locus ∂g (11) (n, k) = 0 ∂k is given by the set of points (n, k) such that n ∈ (0, 1) and k solves (11). Since function g can be bimodal or strictly monotonic in k then LCb−1 may be composed of two horizontal lines of equations k = km and k = kM , or LCb−1 = {∅}. Consider first the open case (a) of Remark 9. In Fig. 1(b)–(d) the case with LCb−1 = {∅} is presented, i.e. function gn (k) is strictly increasing. We consider the same parameter values as in Fig. 1(a), that is α > α and s < s, with μ > 3 so that the fixed points En∗ 0 and En∗ k2 are saddle points. Set B1 is given by the gray region while set B2 is given by the white region. Two different attractors coexist: an attractor 1 ⊂ B1 located on the invariant set K0 and an attractor 2 ⊂ B2 ; since gn (k) is strictly increasing, the basins of the two coexisting attractors B1 and B2 consist of simple connected sets. In this case the structure of the two coexisting attractors may be complex according to the value of parameter μ while the structure of their basins is simple. In this situation an economic policy trying to increase investment can be able to push the economy out of the poverty trap toward a positive long run economic growth rate. Consider now the open case (b) of Remark 9, i.e. in the following we assume α > α and p > p so that map gn (k) is bimodal in k. The set of points for which the determinant of the Jacobian matrix vanishes is given by the union of b2 three straight lines denoted by LCa−1 , LCb1 −1 and LC −1 (see Fig. 2(a)). Being LC the rank-1 image of LC−1 , i.e. LC = T(LC−1 ), then LCa is the vertical line of equation n = μ4 , while LCb is described by two parabolas. The rank-1 image LC is the union of three branches, LCa = T (LCa−1 ), LCb1 = T (LCb1 −1 ) ). and LCb2 = T (LCb2 −1 In Fig. 2(b) the three branches of LC are shown for the case in which g(n, k) is bimodal in k. Since the one-dimensional map k = gn (k) is of the kind Z1 − Z3 − Z1 while the logistic map if of the kind Z2 − Z0 then system T is non-invertible and of (Z6 − Z4 − Z2 − Z0 )-type. If we consider the critical sets of rank-h, i.e. LCh−1 = Th (LC−1 ) = Th−1 (LC) where LC0 = LC, then segments of critical curves of rank-h, h = 0, 1, . . ., can be used to define trapping regions of the phase Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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(a) 2.5

(b) 2.5

LCa −1

a

b2

LC

LCb1

LC

Z

2

Z

4

Z

k

k

−1 6

Z

0

Z

4

b2 LC−1

LCb1

Z

2

0 0

1/2 n

1

0 0

n

µ/4

1

Fig. 2. (a) Critical curves of rank-0, LC−1 for system T and the following parameter values: δ = 0.2, α = 1, β = 0.9, sw = 0.1, sr = 0.7, p = 8 and μ = 2.5. (b) Critical curves of rank-1, LC = T(LC−1 ), for the same parameter values as in panel (a). These curves separate the plane into the regions Z6 , Z4 , Z2 and Z0 , whose points have different number of preimages.

plane. In fact an absorbing area A is a bounded region of the plane whose boundary is given by critical curve segments (i.e. segments of the critical curve LC and its images) such that a neighborhood U ⊃ A exists whose points enter in A after a finite number of iterations and then never leave it, i.e. T (A) ⊆ A. In the following, some numerical simulations are presented and the absorbing area is depicted in order to describe the bifurcations which increase the complexity of the asymptotic dynamic behavior of the system. Consider set D and assume that Proposition 8 applies, thus we are referring to the case in which shareholders save more than workers and p is great enough. In this case g is bimodal in k and system T admits a compact global attractor: a feasible trajectory may converge to the positive steady state En∗ k2 or En∗ 0 or to other more complex attractors inside D . We fix the values of the following parameters: δ = 0.2, α = 1, β = 0.9, sw = 0.1, sr = 0.7. Consequently observe that only the two parameters μ and p must be analyzed: both such parameters play an important role in our model for complicated dynamics to be observed. We consider initial conditions belonging to D and observe the attractors owned by T. In Fig. 3(a), the attractor 1 is depicted in blue while 2 is depicted in black; they consist of 2-cyclic chaotic attractors: in fact, since μ = 3.6, the attractor of the logistic map belongs to two disjoint intervals ( 1 is given by the union of 2 trapping sets belonging to the invariant line k = 0). In this situation the long-run evolution of the system is characterized by cyclical behavior of order 2, but in each period the exact state cannot be predicted. Observe that the strange attractor increases in size as μ increases: the two cyclic chaotic attractor gives rise to a connected chaotic attractor (see Fig. 3(b)) after a contact of the two chaotic areas: a cyclic (although chaotic) behavior is replaced by a totally erratic evolution that covers a wide area of the phase space of the dynamical system (in this picture μ is fixed at the value 3.7 so that the attractor of the logistic map is complicated). Following [30], in order to obtain the boundary of the chaotic area containing the attractor 1 , namely ∂ , let γ = ∩ LC−1 be the portion of critical curve of rank-0 inside , then for a suitable integer r ∂ ⊆ ∪rh=1 T h (γ). An example is shown in Fig. 3(c) where we numerically determine the boundaries of the attractor 2 shown in Fig. 3(b): the boundary of the chaotic area is obtained by the images, up to rank 3, of the portion γ of LC−1 marked in bold black in Fig. 3(c). The previous simulations show that for μ ∈ (3, 4) the attractors 1 and 2 can be quite complicated. Anyway their basins still have a simple structure (connected sets). 4.3. Contact bifurcations and complex basins As previously discussed, system (T, D ) can admit two coexisting attractors, 1 ⊂ K0 and 2 , hence in this section we study the global bifurcations occurring as some parameters are varied that are responsible for changes in the properties of their basins of attraction (that may consist of infinitely many non-connected sets). This kind of bifurcation requires Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Fig. 3. Attractors 1 (depicted in blue) and 2 (depicted in black) of system T for the following parameter values: δ = 0.2, α = 1, β = 0.9, sw = 0.1, sr = 0.7, p = 8.7. (a) μ = 3.6. (b) μ = 3.7. (c) Boundary of the attractor 2 represented in panel (b): set γ is depicted in bold black. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

an analysis of the global dynamical properties of the system, that is, an analysis which is not based on the linear approximation of the map. Recall that system T always admits trajectories converging to the invariant set K0 . Let B1 ⊂ D be the set of points generating trajectories converging to the invariant line k = 0 where the attractor 1 exists. Furthermore, for certain parameter values, T admits an attractor 2 located on the interior of set D , let B2 be the basin of attraction of 2 (i.e. the set of initial conditions generating trajectories converging to 2 ). Then, B2 = Int(D /B1 ) being Int(M) the interior points of set M. In Figs. 1 and 3 the attractors of T are presented and their own basins B1 and B2 are respectively depicted in gray and white. In Fig. 2 the critical curves LC are presented; it can be shown using numerical simulations that the curve LCb1 moves downwards as parameter p increases so that, given the other parameter values, a threshold value p˜ does exist such that a contact bifurcation (i.e. a contact between a critical curve and the basin boundary) occurs (about this kind of bifurcation see, among others, [1]). At this parameter value LCb1 is tangent to the basin boundary ∂B1 (see Fig. 4(a)) and a global bifurcation occurs causing the transformation of B1 from connected to non-connected, i.e. it is given by an infinite sequence of non-connected regions (or holes) inside B2 . This bifurcation is due to the fact that a portion of the basin B1 enters in a region characterized by a higher number of primages (and hence the primages of any rank of such a portion also belong to B1 ). In fact, when a parameter variation causes a crossing between a basin boundary and a critical set a portion of a basin enters in a region where a higher number of inverses is defined, then new components of the basin suddenly appear after the contact (see [6]). Obviously a subset B0 ⊂ B1 , with 1 ⊂ B0 , exists such that trajectories starting from B0 converge to 1 (immediate basin), hence if the economy starts from a low Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Fig. 4. (a) Immediately before the contact bifurcation (p = 10.4524), the tangency between the critical curve and the basin boundary is shown. (b) Basins of attraction for p = 10.47 of 1 (the white region) and 2 (the gray region) after the contact bifurcation: gray holes are depicted. (c) Basins of attraction for p = 11. (d) Basins of attraction for p = 11.5. (e) Basins fractalization for p = 11.72.

level of economic growth it will fall in the poverty trap. On the other hand, after the contact bifurcation, B0 admits new preimages given by B−1 = {(n, k) : T(n, k) = B0 } and consequently initial conditions belonging to B−1 also generate trajectories converging to 1 , as B−1 is mapped into set B0 after one iteration. The previous procedure can be repeated while considering the preimages of rank-2 of the set B0 , namely B−2 . Again, initial conditions belonging to the set B−2 generate trajectories converging to B0 after two iterations. The story repeats and a set of non-connected portion is Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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created, so that the contact between the critical set and the basin boundary marks the transition from simple connected to non-connected basins. Finally the basin of attraction of the poverty trap is given by B1 = B0 ∪i≥1 B−i . In such a case an economic policy trying to push up the investment does not guarantee the exit from the poverty trap. By using numerical computations and fixing δ = 0.2, α = 1, β = 0.9, sw = 0.1, sr = 0.9 and μ = 3.57, the bifurcation value p˜  10.4524 is obtained and in Fig. 4(b) the situation occurring immediately after this global bifurcation is shown. Observe that the attractor 2 does not disappear after this bifurcation. This depends on the fact that the portion of curve LC involved in the contact bifurcation does not belong to the boundary of the absorbing area inside which the attractor 2 exists. The basin structure increases in complexity if p further increases as shown in Figs. 4(c) and (d) and the gray area increases too (i.e. the set of initial conditions generating sequences converging to the poverty trap). The invariant absorbing area is involved when the basin boundary has a contact with the boundary of the invariant absorbing area and this happens as parameter p is further increased. In Fig. 4(e) this situation is represented and the distribution of white and gray points appears quite complicated. Finally it can be noticed that, as p crosses a final bifurcation value p  11.9, the attractor 2 disappears and almost all trajectories converge to the poverty trap. Another final bifurcation occurs when μ is increased, so that the bifurcation value μ = 4 is crossed. This global bifurcation is characterized by a contact between the boundary of D and the critical curve n = μ/4 at which attractor

disappears. The previous arguments show that the bifurcations concerning both the structure of the attractors 1 and 2 and the structure of their basins are strictly related to the values of the two key parameters μ and p as previously discussed: the latter informs about the elasticity of substitution (which decreases as p increases), while the former takes into account the amplitude of the fluctuations in the population growth rate. The joint analysis of the dynamics w.r.t. p and μ explains how the elasticity of substitution in the non-concave production function affects the final long run dynamics of the growth model for different values of the amplitude of the fluctuations in the population growth rate, when shareholders save more than workers. In Fig. 5(a) a two dimensional bifurcation diagram showing the periodicity of the attractor 2 for different choices of the couple (μ, p) is presented. Each color represents a long-run dynamic behavior for a given point in the parameter plane (μ, p) and for an initial conditions with k0 not too small. A great diversity of cycles of different order is exhibited. The red region represents the parameter values for which high period cycles or complex dynamics are exhibited; this situation occurs if p is not too small. Also in the gray region of Fig. 5(c) the system fails to converge to the poverty trap. Observe that once μ is fixed, complex dynamics can be observed in our model for intermediate values of p, till a contact bifurcation occurs, providing that in order for fluctuations to arise the elasticity of substitution between production factors must not be too low. Notice that, differently from previous studies (see [8,10,12]), after the contact bifurcation occurring at a high value of p, the system will converge again to the poverty trap. Global dynamics in economic models are quite relevant, especially when more than one attractor co-exist. In fact, as it has been discussed, the economic system may converge to the poverty trap or to another attractor belonging to the interior of its domain, depending on the initial condition, and consequently the basins of attraction have to be studied. Interesting considerations in terms of economic policy may be conducted. For instance, in order to avoid the poverty trap, the State may try to push the system to an initial state characterized by a sufficiently high level of capital per-capita (i.e. using a policy to push up the investment) in a way such that the initial condition belongs to B2 . This policy can be adopted if the elasticity of substitution associated to the attractor is not low and the basins structure is simple. Anyway, for low values of the elasticity of substitution, after the occurrence of the global bifurcation causing the transformation of the basin from connected to non-connected, even starting with an initial condition quite close to the attractor 2 , it is not possible to exclude that the trajectory will converge to the poverty trap, as the basins structure is quite complex. Consequently, featuring the economic system could be ambiguous with respect to initial conditions close to an attractor and perturbations on it, so that the dynamic evolution of the economic variables can become unpredictable even when the current initial state is known. In this case an opportune economic policy is quite difficult to be adopted as the structure of the basins of attraction may be very complicated, and the economy may converge to a zero growth rate also starting from a situation with high initial level of capital per-capita. Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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Fig. 5. (a) Two dimensional bifurcation diagrams of map T in the plain (μ, p) for the same parameter values of Fig. 4 and initial condition k0 = 1.98 and n0 = 0.3. (b) An enlargement. (c) The attractor belongs to K0 for parameter values belonging to the gray region.

5. Conclusions In this paper we investigated the dynamics of the Solow growth model with differential saving and logistic population growth rate in the case of nonconcave production function. The fixed points and other invariant sets of T were determined and the local stability analysis was conducted. About the global properties of the system, we first proved that the model admits a compact global attractor inside which asymptotic states are confined. We then described its structure as parameters of the system vary showing that complex features emerge as μ is increased (so that the amplitude of fluctuations in population growth rate increases) or p is not too low (so that the elasticity of substitution between production factors is not too high). This evidences confirms the results obtained in previous works with concave production functions. Anyway, differently from other studied cases, with non-concave production function the invariant set characterized by zero capital per capita is an attracting set for all parameter values so that the system may converge to the poverty trap. As the structure of the basins of attraction may also be very complicated, the economy may fail to converge to a zero growth rate also starting from a situation with high initial level of capital (i.e. from i.c. that do not belong to the immediate basin of k = 0). Furthermore, a different feature is due to the fact that map T may admit coexisting attractors Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001

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(consisting of fixed points, cycles or more complex sets) and that a final bifurcation occurs at which complicated dynamics is ruled out for very low values of the elasticity of substitution. These problems lead to different routes to complexity, one related to the complexity of the attracting sets which characterize the long run time evolution of the dynamic process, the other one related to the complexity of the boundaries which separate the basins when several coexisting attractors are present. These two different kinds of complexity are not related in general, in the sense that very complex attractors may have simple basin boundaries, whereas boundaries which separate the basins of simple attractors, such as coexisting stable equilibria, may have very complex structures. A further interesting question is what happens introducing population dynamics with an economic feedback, i.e. where f is a function of both n and k, but we leave such study to future research lines.

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Please cite this article in press as: E. Michetti. Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate, Math. Comput. Simul. (2013), http://dx.doi.org/10.1016/j.matcom.2013.09.001