From Samuelson’s multiplier-accelerator to bifurcations and chaos in economic dynamics

Abstract

This piece in the series of Milestones starts from the short, clear and highly cited paper by Samuelson (Rev Econ Stat 21(2):75–78, 1939) entitled “Interactions between the Multiplier Analysis and the Principle of Acceleration”, in which the author proposes a linear dynamic model obtained by combining the Keynesian multiplier and the principle of acceleration, together with a proper structure of time lags. Although very simple, it generated a rich and interesting literature as it provides some answers to the question of endogenous business cycles. Moreover, it raises the question of sensitivity with respect to small variations of the model’s parameters, thus sparking the attention of economists and social scientists toward studies on the qualitative theory of non-linear dynamical systems, and the related work on deterministic chaos and bifurcations which appeared between the 1960 s and 1970 s, such as the paper by Lorenz (J Atmos Sci 20:130–141, 1963) in the Journal of Atmospheric Sciences or the one by May (Nature 26:459–467, 1976) in Nature, thereby stimulating an interdisciplinary mathematical approach to dynamic complexity in physics, biology and social sciences.

1 Introduction

Paul Samuelson (1915–2009) published the article “Interactions between the Multiplier Analysis and the Principle of Acceleration” in The Review of Economics and Statistics at the age of 24, two years before finishing his Ph.D. at Harvard University. This paper became very influential in the economic literature: just to give a rough measure, according to Google Scholar it counts about 1740 mentions so far (April 2023) with 280 citations in the 2019–2022 period. It is just four pages long, including two pictures and two tables, written following a heuristic approach. Indeed, after a short description of the equations of the dynamic macroeconomic model representing the time evolution of the national income, some step by step explicit numerical computations of the dynamic variables (consumption and investment) are proposed by using different values of the two economic parameters (the marginal propensity to consume and the relation) which are seen as knobs to tune the dynamic behavior and deliver different long-term trends, such as convergence (monotonic or oscillatory) to a stationary value, or divergence through oscillatory patterns of increasing amplitude. These numerical explorations of the model are used by Samuelson to increase the readers’ curiosity about the rationales behind such different kinds of time patterns, so that some suspense is created, and he even increases the pathos by stressing that “A variety of qualitatively different results emerge in a seemingly capricious manner from minor changes in the hypotheses” (that is, in the values of the two economic parameters). After this, the author reveals that “simple algebraic analysis can be applied which will yield all possible qualitative types of behavior and enable us to verify the results”. Indeed, the mathematical structure of the model is a simple linear second order difference equation; hence, a unique equilibrium point exists and the study of its stability reduces to the localization in the complex plane of the solutions of a second degree algebraic equation (called characteristic equation in this framework), an exercise of undergraduate mathematics. Like in a detective story, the solution of an intriguing question, raised at the very beginning, is only given at the end. Moreover, according to the best storytelling tradition, the paper concludes with some moral issues and open questions. For example, Samuelson stresses that “The limitations inherent in a so simplified a picture as that presented here should not be overlooked”, and consequently claims for a larger view, not limited to a small neighborhood around the equilibrium: in other words, the necessity to consider non-linear terms. Moreover, he remarks that a dynamic representation of an economic system, as well as the mathematical methods employed to analyze it, give rise to interesting interpretations and points of view, more general than the ones dealing with the specific considered model.

So, the value of the paper is to be found not only in the economic phenomena explained, but also in raising more general problems, such as the extreme sensitivity to small variations in parameters, a field of research that will become crucial in the years to come bearing the name of structural stability, with the related concept of bifurcations. Moreover, other sources of sensitivity, typical of non-linear dynamic models, are brought to light, such as the long-run effects of very small changes in the initial conditions (sensitive dependence on initial conditions) also denoted as deterministic chaos, or the problem of the coexistence of several equilibria (or more complex attractors characterizing the long run behavior of economic systems) with the related problem of the delimitation of their basins of attraction. For this reason, starting from the paper by Samuelson, this article also describes other studies on economic dynamics which also quote contributes from physical and biological literature that highlighted the role of bifurcations and deterministic chaos in the study of dynamical systems, such as the paper “Deterministic non-periodic flow” published in 1963 in the Journal of the Atmospheric Sciences by Edward Lorenz, a physicist dedicated to meteorology, as well as “Simple mathematical models with very complicated dynamics” published in 1976 in Nature by Robert May, a physicist dedicated to ecology. Although other authors stressed these typically nonlinear phenomena before, these two papers highly contributed to the diffusion of such topics and were cited not only within the literature steams of scientific disciplines, but even outside the narrow circle of specialists.

In the following, we will also touch on the importance of interdisciplinarity and transdisciplinarity, i.e. the need to look into different research fields and browse through various types of literature to find cross-fertilizations between different approaches and techniques, even at the risk of dissipating one’s energies in sectors that, at least apparently, are not so strictly related to economic and social expertise.

Before we get into these topics, it is worth remembering that Paul Samuelson has been a famous economist, and he was endowed with excellent mathematical skills. He entered the University of Chicago at age 16, during the depths of the Great Depression, and received his Ph.D. in economics from Harvard. Then he became professor of economics at Massachusetts Institute of Technology (MIT) and he spent his entire career there. He also contributed to the growth of Economics Department of MIT by attracting many noted economists, including Nobel Prize awardees like Robert Solow, Franco Modigliani, Robert C. Merton, Joseph Stiglitz, and Paul Krugman. Moreover, Samuelson was the first American to win the Nobel Prize in Economic Sciences, in 1970, and the motivation given by the Swedish Royal Academies states «for the scientific work through which he has developed static and dynamic economic theory and actively contributed to raising the level of analysis in economic science». In other words, not due to a specific result or the solution of a particular problem, but for the research methodology and the approach which became widespread among economists.

In the same spirit, this article is organized as follows: in Sect. 2 a short description of the multiplier-accelerator model of Samuelson is given; in Sect. 3 we offer a minimal vocabulary and an overview of the main features of the qualitative theory of non-linear dynamical systems to introduce the concept of deterministic chaos, and in Sect. 4 we briefly describe its impact on economic modeling; in Sect. 5 a short description of the concepts of structural stability and bifurcations is given and Sect. 6 concludes.

2 The Samuelson’s multiplier-accelerator model

Let $Y_{t}$ be the national income at time period t, $C_{t}$ the consumption expenditure in the same period, $I_{t}$ the private investment and $g_{t}$ the government spending. The model that Samuelson proposes in 1939 to represent a national economy is expressed by the following equations:

$$\begin{aligned} Y_{t}&=g_{t}+C_{t}+I_{t} \end{aligned}$$

(1)

$$\begin{aligned} C_{t}&=\alpha Y_{t-1}\end{aligned}$$

(2)

$$\begin{aligned} I_{t}&=\beta \left( C_{t}-C_{t-1}\right) =\alpha \beta \left( Y_{t-1}-Y_{t-2}\right) \end{aligned}$$

(3)

where $t=0,1,...$, the real parameter $\alpha \in \left[ 0,1\right]$ represents the marginal propensity to consume and $\beta >0$ is the accelerator [or relation, according to the terminology used by Harrod (1936)]. The first two equations express the idea of Keynesian multiplier, formally introduced by the British economist Keynes in his book The General Theory of Employment, Interest, and Money (1936). Indeed, starting from the general balance equation for a closed economy $Y=g+C+I$ (i.e. national income given by the sum the sum of government expenditure g, consumption C and private investment I) and assuming $C=\alpha Y$, Keynes obtained $Y=\frac{1}{1-\alpha }\left( g+I\right)$, where the factor $1/(1-\alpha )>1$ is now called the Keynesian multiplier, which expresses the fact that government spending can raise national income by a larger amount through an increase in consumer demand, an idea proposed after the Great Depression of 1929 and which was at the core of the New Deal and the growth of the welfare state. Instead, the third equation expresses the principle of acceleration, stating that the level of investment is induced by the consumption trend and, consequently, by the lagged growth of income.

Realistic empirical values of the coefficient of proportionality between the investment and the increase of income, given by $k=\alpha \beta$, are between 2 and 4, if an annual time scale is assumed, as usual in Keynesian theory and national accounting of 1930’s, whereas realistic values of the propensity to consume $\alpha$ are in the range $\left( 0.6\text {,}1\right)$ (see e.g. Gallegati et al. 2003). The accelerator principle was not invented by Samuelson (1939): in his paper he attributes the idea to his supervisor Alvin H. Hansen, but he should have also cited at least Harrod, who qualitatively illustrated the idea of combining the effects of the multiplier and the relation (as Harrod called the accelerator) in his book The Trade Cycle, see Harrod (1936) (see also Heertjie and Heemeijer 2002, for an historical overview). Indeed, as Samuelson himself wrote referring to this model, “Scientific theories are like children in that they have a life of their own. But, unlike children, they may have more than one father” (see Samuelson 1959, p. 183). However, the new ideas of Samuelson are expresses by the “dynamization” of the model, i.e. the time structure proposed, with proper time lags. Plugging (2) and (3) into (1), and assuming a constant government expenditure $g_{t}=g_{0}$ (fixed at the reference value $g_{0}=1$ in the Samuelson’s paper) a linear second order difference equation is readily obtained

$$\begin{aligned} Y_{t}=1+\alpha \left( 1+\beta \right) Y_{t-1}-\alpha \beta Y_{t-2}. \end{aligned}$$

(4)

Given two initial values of income, e.g. $Y_{0}$ and $Y_{1}$, the successive income value $Y_{2}$ can be computed according to (4) and so on, inductively. In this way, for any fixed couple of parameters $\alpha$ and $\beta$ a whole trajectory can be obtained, whose fate (i.e. its asymptotic, or long-run, behavior for $t\rightarrow +\infty$) depends on the particular values of the parameters. In particular, some of these behaviors exhibit oscillatory time patterns, that may be damped, i.e. with decreasing amplitude before converging to the unique equilibrium $Y^{*}=\frac{1}{1-\alpha }$, or explosive, i.e. characterized by oscillatory divergence. Periodic self-sustained cycles of constant amplitude can only be obtained at very particular (hence non generic) values of the parameters. However, the presence of oscillations constitutes a first step towards an endogenous business cycle theory, as the combination of the multiplier effect and accelerator is capable of inducing a movement from boom to depression and back in a fully endogenous way. Moreover, the case of unrealistic explosive upswings and downswings can be constrained by imposing a ceiling (e.g. an upper limit due to resources limitation) and a floor (e.g. a lower bound related with non negativity conditions) as proposed by Hicks (1950).

Of course, being the difference equation (4) linear, an analytic solution can be easily obtained such that $Y_{t}-Y^{*}$ is expressed in the form of a combination of exponentials $z_{i}^{t}$, where $z_{i}$ are the eigenvalues, i.e. the (complex) roots of the characteristic equation $P(z)=z^{2}-\alpha \left( 1+\beta \right) z+\alpha \beta =0$ [see any standard textbook on discrete dynamical systems, such as Medio and Lines (2001), Elaydi (2005, 2007), Gandolfo (2007)]. Mathematically able as he was, Samuelson gave this solution in a footnote. On the basis of the three Schur stability conditions (see e.g. Medio and Lines 2001; Gandolfo 2007) $P(1)=1-\alpha >0$ (always true); $P(-1)=1+\alpha \left( 1+2\beta \right) >0$ (always true); $P(0)=\alpha \beta <1$, that guarantee that the characteristic equation has roots inside the unitary circle of the complex plane, together with the condition $\Delta =\alpha ^{2}\left( 1+\beta \right) ^{2}-4\alpha \beta =0$ that separated the case of real and distinct roots ($\Delta >0$) from the case of complex conjugate roots ($\Delta <0$), he could draw a picture where the parameters’ plane $\left( \alpha ,\beta \right)$ is divided by the two curves $\alpha =\frac{1}{\beta }$ and $\alpha =\frac{4\beta }{\left( 1+\beta \right) ^{2}}$ into four regions according to the different asymptotic behaviors obtained: A=monotonic convergence to $Y^{*}$; B=oscillatory convergence to $Y^{*}$; C=explosive oscillations; D=monotonic divergence. Samuelson concludes that if the parameters’ values are close to a boundary that separates regions characterized by qualitatively different asymptotic behavior, then small parameters’ variations may cause a border crossing leading to a different long-run evolution of the economic system, a situation that he denotes as “highly unstable”. Moreover, he stresses that the linear model proposed can be applied to describe small oscillations around the equilibrium, and implicitly suggests that non-linear terms should be considered in order to describe the economic system far from the equilibrium. The short paper is concluded by the following apology of mathematical methods: “Contrary to the impression commonly held, mathematical methods properly employed, far from making economic theory more abstract, actually serve as a powerful liberating device enabling the entertainment and analysis of even more realistic and complicated hypotheses.”

In fact, the common opinion about business cycle was that exogenous shocks are necessary in order to produce oscillations in an economic system. Instead, the oscillations obtained in the model proposed by Samuelson are endogenous, i.e. induced by an intrinsic self-sustained mechanism, without any need to involve external causes. This striking feature (oscillations without any exogenous cause) had already be mathematically proved in the biological literature by the Italian mathematician (Volterra 1926) to describe the oscillations observed in a prey-predator ecological system. However, the seminal model of Samuelson (1939) triggered several extensions giving rise to the stream of endogenous business cycle modeling. For example, Metzler (1941) published, in the same journal a type of linear multiplier-accelerator model that includes producers’ desire to keep inventory stock proportional to expected sales, and by a proper structure of lags between production and sales proved that inventory policy chosen by producers might have profound effects on the different economic dynamics, in particular the ones characterized by endogenous oscillations. Hicks (1950) noticed that in the ranges of empirically measured values of the parameters included in the Samuelson’s model cycles with increasing amplitude are obtained, implying that national income and related economic variables are driven too far away from their equilibrium values and eventually become negative. To avoid this he suggested adding upper and lower boundaries (ceiling and floor) on the investment part of the multiplier-accelerator model, so that the dynamics are always bounded. From a mathematical point of view, the Hicks model corresponds to a piecewise-linear model, a first step towards non-linearity. Indeed, a suggestion to introduce nonlinear terms is already given by Samuelson (1939) in the last lines of his paper, where he writes “The limitations inherent in so simplified a picture as that presented here should not be overlooked. In particular, it assumes that the marginal propensity to consume and the relation are constants; actually these will change with the level of income”, i.e. a suggestion to consider a marginal propensity to consume that depends on income. This suggestion has been picked up by some authors, such as Westerhoff (2008) that starting from Samuelson’s model, endogenizes the marginal propensity to consume, and proves that business cycles can be obtained even without the accelerator term. Another example can be found in Sheehan (2009). However nonlinear dynamic macroeconomic models are now quite common on the literature, see e.g. the book edited by Puu and Sushko (2006) for several references. Just to quote a few, we mention Westerhoff (2006a, 2006b) who introduces expectations formation, Cavalli et al. (2017) where the real part of an economy, described within a multiplier-accelerator framework, interacts with a financial market, Gardini et al. (2023) stress the role played by animal spirits in a multiplier-accelerator model where investors can be optimistic (pessimistic) when national income increases (decreases). Other recent nonlinear business cycle models, derived from the basic structure of Samuelson’s model, include Gallegati et al. (2003), Puu et al. (2005), Dalla and Varelas (2016), Dassios and Zimbidis (2014), Dassios and Devine (2016), Barros and Ortega (2019), Ortega and Barros (2020), and Tramontana and Gardini (2021). See also Mourao and Popescu (2022, 2023) for a more general overview of the influence of the multiplier-accelerator model in the economic literature.

So, from one side the Samuelson’s seminal paper triggered several extensions in the stream of endogenous business cycle modeling, while on the other side it claimed for methodological advances in the direction of dynamic modeling in economic and social sciences, in particular in the emerging field of qualitative analysis of non-linear dynamical systems, with the related issues of structural stability and deterministic chaos. This parallel stream of literature, mainly developing between 1960 s and 1970 s in the fields of mathematical physics, singularity theory and applied mathematics (such as control theory and mathematical biology), after some pioneering work on non-linear business cycle models, such as those by Goodwin (1951, 1967), became very popular among economists, attracted by the possibility that endogenous cycles and irregular (chaotic) dynamics, resembling stochastic fluctuations, could be generated by simple deterministic equilibrium models of the economy. As stressed by Benhabib (2018), this new literature on non-linear phenomena and chaotic dynamics showed that endogenous irregular cycles were indeed possible under intertemporal arbitrage with complete markets and without any market frictions, both in standard models of overlapping generations as well as in calibrated models of infinitely lived representative agents, as shown in Benhabib and Day (1982), Grandmont (1985), Boldrin and Montrucchio (1986), Benhabib and Rustichini (1990), Hommes (1991, 2013).

Before discussing these implications, it is necessary to introduce a minimal vocabulary of dynamical systems theory in order to define what is meant by a non-linear dynamic model and to understand how deterministic chaos can be generated.

3 Non-linear dynamic models and chaos

Even if the multiplier-accelerator model proposed by Samuelson produces endogenous oscillations of an economic system, its outcome is not satisfactory if compared with real macroeconomic oscillations. This is a consequence of the linearity of the second order difference equation proposed by Samuelson, which can be easily written in the equivalent (and more standard) form of a pair of first order difference equations with two dynamic variables by setting $Y_{t-1}=x_{1}(t)$ and $Y_{t}=x_{2}(t)$ so that

$$\begin{aligned} x_{1}(t)&=x_{2}(t-1)\nonumber \\ x_{2}(t)&=1-\alpha \beta x_{1}(t-1)+\alpha \left( 1+\beta \right) x_{2}(t-1) \end{aligned}$$

(5)

with initial conditions $x_{1}(0)$, $x_{2}(0)$ given. More generally, if the state of an economic system is described by the state vector $\mathbf {x=} \left( x_{1},...,x_{n}\right) \in \mathbb {R}^{n}$, whose components are the dynamic variables, i.e. functions of time $\textbf{x}(t)\mathbf {=}\left( x_{1}(t),...,x_{n}(t)\right)$ then their time evolution is described by local evolution equations, denoted as dynamic equations or laws of motion. In the case of discrete time, these are expressed by a set of difference equations that inductively define a sequence of discrete points starting from a given initial condition

$$\begin{aligned} x_{i}(t)&=f_{i}(\textbf{x}(t-1);\mathbf {\alpha })\text {,} \ \ i=1,...,n\nonumber \\ \textbf{x}(0)&=\overline{\textbf{x}}\text { } \end{aligned}$$

(6)

where $\mathbf {\alpha }=\left( \alpha _{1},...\alpha _{m}\right)$ represents a set of real parameters, and the functions in the right hand side define a vector field $\textbf{f}=\left( f_{1},...,f_{n}\right) :M\rightarrow M$ from the phase space $M\subseteq \mathbb {R}^{n}$ into itself such that a single application of the map $\textbf{f}$ represents a “unit time step” for the state of the dynamical system. Its repeated application (or iteration) defines a trajectory, obtained by the composition of the map with itself

$$\begin{aligned} \textbf{x}(1)=\textbf{f}(\textbf{x}(0))\text {;} \ \textbf{x}(2)=\textbf{f} (\textbf{x}(1))=\textbf{f}(\textbf{f}(\textbf{x}(0))=\textbf{f}^{2} (\textbf{x}(0)\text {;...; }\textbf{x}(t)=\textbf{f}^{t}(\textbf{x} (0))\text {...} \end{aligned}$$

A trajectory $\tau (\textbf{x}(0))=\left\{ \textbf{x}(t)\in M\text {: }\textbf{x}(t)=\textbf{f}^{t}(\textbf{x}(0))\text {,... }t\in \mathbb {N} \right\}$ represents a simulated time evolution starting from a given initial state. Each functional relation $f_{i}(x_{1}(t),...x_{n} (t);\mathbf {\alpha })$ gives information about the influence of the same state variable $x_{i}$ (self-control) and of the other state variables $x_{j}$, $j\ne i$ (cross-control) on the value of $x_{i}$ at the next time period, and the parameters $\alpha _{i}\in \mathbb {R}$ are fixed along a trajectory, but can assume different numerical values in order to represent exogenous influences on the dynamical systems, e.g. different policies or effects of the outside environment. The modifications induced in the model after a variation of some parameters $\alpha _{i}$ are called structural modifications, as such changes modify the shape of the functions $f_{i}$, and consequently the properties of the trajectories.

An equilibrium point (or steady state) $\textbf{x}^{*}\in \mathbb {R}^{n}$ is a particular trajectory at which the state remains constant, i.e. $x_{i}(t)=x_{i}^{*}$ for each $t\ge 0$, $i=1,...,n$. It is a fixed point of the map $\textbf{f}$, i.e. $\textbf{f}(\textbf{x}^{*})=\textbf{x}^{*}$. This means that if the initial condition is taken in an equilibrium point, the future state of the system remains there forever. More generally, a subset $A\subset M$ is called invariant if $\textbf{x}(0)\in A$ implies $\textbf{x}(t)\in A$ for each $t\ge 0$, where A may be an equilibrium, a periodic cycle, a closed invariant curve or a more complex compact set. One may wonder what happens if a trajectory starts from an initial condition close to an invariant set, i.e. in a neighborhood of it. It may approach the invariant set as $t\rightarrow +\infty$ (in this case the invariant set is said locally asymptotically stable) or it may remain around it (stable) or it may go far from it (unstable). Sometimes it may occur that a trajectory generated from an initial condition close to an invariant set moves quite far from it and then bounces back to it. In this case the invariant set is unstable and that kind of trajectory is denoted as homoclinic (or snap back repeller, after Marotto 1978). The presence of a homoclinic trajectory, only possible in non-linear dynamical systems, may be an indicator of the presence of deterministic chaos, an extreme form of sensitivity of trajectories with respect to small displacements of the initial conditions (see e.g. Guckenheimer and Holmes 1983; Elaydi 2007; Robinson 2012).

The definition of stability is local, i.e. it concerns the future behavior of a dynamical system when its initial state is in an arbitrarily small neighborhood of an invariant set, hence they can be used to characterize the behavior of a system under the influence of small perturbations from an equilibrium or another invariant set, perturbations always present in real systems. However, in the study of real systems we are also interested in their global behavior, i.e. far from invariant sets, in order to consider finite (and not always small) perturbations and to answer questions like: how far can an exogenous perturbation shift the state of a system from an equilibrium with the certainty that it will spontaneously go back to the originary state? These questions lead to the concept of basin of attraction B(A) of an attractor (i.e. an asymptotically stable invariant set) A, defined as the set of all points $x\in M$ such that $\lim _{t\rightarrow +\infty }\tau (x)\in A$. Generally, the extension of the basin of a given attractor measures its robustness with respect to the action of exogenous perturbations, and if $B(A)=M$, the whole feasible phase space, then A is called global attractor.

Similar definitions can be given in the case of a dynamical system with continuous time $t\in$ $\mathbb {R}$, whose evolution equations are expressed by ordinary differential equations involving the time derivatives of the dynamic variables, i.e. their rates of change:

$$\begin{aligned} \frac{dx_{i}(t)}{dt}&=f_{i}(\textbf{x}(t);\mathbf {\alpha })\text {,} \ \ i=1,...,n\nonumber \\ \textbf{x}(0)&=\overline{\textbf{x}} \end{aligned}$$

(7)

As argued while presenting the Samuelson’s multiplier-accelerator model, discrete time naturally arises in economic and social modeling, where changes in the state of a system occur as a consequence of decisions that cannot be continuously revised (event-driven time), with a characteristic finite time lag $\Delta t$ between decisions and realizations taken as a unit of time, i.e. $\Delta t=1$.

In the case of the Samuelson’s multiplier-accelerator model, the equations of motion (5) are linear, i.e. the functions $f_{i}$ are polynomials of degree 1. A generic linear model has one and only one equilibrium point, whose local stability property is also global, i.e. initial conditions have no role in determining the asymptotic fate of the trajectories. No other invariant sets can exist, and oscillations, if any, can only be damped (contractive map) or explosive (expanding map) except for (non generic) cases of characteristic roots (eigenvalues) located along the unit circle of the complex plane. Of course, the use of linear dynamic models offers the undeniable advantage that explicit solutions can be obtained in an analytical form. It is clear that, within a linear framework, economists usually impose conditions of stability of the unique equilibrium to achieve paths that make sense in an economic context. In conditions of instability (as occurs in the Samuelson’s model when $\alpha \beta >1$), in order to avoid divergent trajectories, they impose barriers to limit the amplitude of fluctuations, i.e. ceiling and floor constraints, like in the Hicks business cycle models.

Things are quite different when the functions $f_{i}$ are non-linear. In this case it is difficult to find a general solution, i.e. an analytical form expressing the parametric equations of the trajectories $x_{i}(t)$ for each $t\ge 0$. However, some methods have been proposed to guess the asymptotic (or long-run) behavior of the solutions as $t\rightarrow \infty$, by detecting the presence of invariant sets and their stability properties, according to the qualitative theory of dynamical systems. In the case of non-linear models it is possible to have paths that do not converge to a stable equilibrium. In fact, differently from linear dynamical systems, more complex bounded attractors may exist around an unstable equilibrium, such as periodic or chaotic attractors. This happens, for example, in economic systems characterized by local destabilizing effects, such as positive feedback like the one obtained in the multiplier-accelerator model with $\alpha \beta >1$, associated with self-correcting effects (i.e. negative feedbacks) acting as a recall force when the state of the system is far from equilibrium. This scenario of local expansion and global contraction is also known as the stretching and folding mechanism (equivalent to the existence of snap back repellers) at the basis of self-sustained aperiodic bounded oscillations, like those denoted as deterministic chaos. This apparent oxymoron joining the two words “deterministic” and “chaos” juxtaposes two counterpoised meanings: deterministic means without uncertainty, predictable, regular, where any cause implies clear effects or consequences; chaos is generally referred to confused, unpredictable, irregular systems, where the consequences of a given cause are not clear. Indeed, the laws of motion (6) and (7) are completely and perfectly deterministic, because given an initial condition and the knowledge of the dynamic equations, a unique time evolution (i.e. a trajectory) of the dynamical system is obtained. This allows one to compute the future state of the system for any time (even in the past by replacing t with $-t$) without any uncertainty, as was expressed by the French mathematician (Laplace 1776):

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes”.

The idea expressed by this statement is known as the Laplacian determinism, and the intellect which is assumed to know the equations of motion of the Universe and its exact state at a given time is sometimes called Laplace’s demon.

A remarkable addendum to this statement was expressed by Poincaré (1908) after his attempt to find the trajectories of a three-body system in the presence of the gravitational forces, in order to participate to a contest sponsored in 1887 by the king of Sweden Oscar II in honour of his 60th birthday, where some mathematical questions were proposed. One of the questions in this contest was to prove that the solar system is stable. Poincaré reduced this problem to a non-linear dynamical system describing three planets interacting by gravitational forces, however he did not succeed in giving an answer. Nevertheless, he won an award with the following motivation: “the work of Poincaré cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics.” In practice, in his work Poincaré started the qualitative theory of dynamical systems. One striking feature of the non-linear dynamical system studied by Poincaré was an extraordinary sensitivity of trajectories with respect to arbitrarily small, even negligible, variations of the initial conditions. His description of such phenomenon is one of the most famous pages of mathematical literature (see Poincaré 1908):

“If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have an apparently fortuitous phenomenon. ”

In other words, deterministic non-linear systems may exhibit behaviors that appear as governed by stochastic influences. On the one side, this phenomenon imposes severe limits to making predictions using dynamic models; on the other side, it suggests that apparently irregular time patterns may be described by deterministic dynamical systems. This statement of Poincaré was not noticed outside the narrow circle of scholars in the field; however, the phenomenon was re-discovered by the meteorologist Edward Lorenz while studying dynamical systems applied to weather forecasting. In 1963 he published a famous seminal paper (another milestone) in the Journal of the Atmospheric Sciences, entitled “Deterministic non-periodic flow”, where he quoted Poincaré and described very clearly the sensitive dependence of trajectories on initial conditions, an effect which quite soon became popularly known as the “butterfly effect”, after the title of the lecture given by Lorenz (1972) entitled “Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”, where he highlighted how small differences in initial conditions (such as those due to empirical measures of even rounding errors in numerical computation) yield different outcomes for non-linear dynamical systems, rendering long-term prediction impossible in general. This was summarized by Lorenz in the sentence: “the present determines the future, but the approximate present does not approximately determine the future”.

So, the common statement claiming that if one knows the equations of motion then reliably forecast of the future states of a system can be obtained from the knowledge of its state at a given time, is not generally true in practice. This has a strong impact in economics, where the paradigm of the rational agent (a cornerstone of the mainstream economic theory) even it is mathematically correct in a deterministic framework, is based on the assumption that economic agents have correct expectations about future states of the economy because they know the equations of motion of the economic systems.

Of course, a non-linear dynamic model can behave regularly (converging to an equilibrium or to a periodic orbit) for some sets of parameters and may exhibit chaotic dynamics for different parameter values, so a goal of the qualitative study of non-linear dynamical systems is the detection of the parameter values leading to chaotic behavior.

The discovery (or, better, the re-discovery after the clear statement of Poincaré in 1908) of these kind of trajectories in deterministic models, called chaotic after the paper “Period three implies chaos” by Li and Yorke (1975), opened in the Sixties and Seventies of $20^{th}$ century a huge stream of literature in the field of the theory of dynamical systems, and caused a sort of revolution in several disciplines, including physics, chemistry, engineering, biology, sociology, economics. The so called “chaos theory” even entered fiction, cinema and philosophical debates, see e.g. the popularization book by Gleick (1987). Moreover, concerning discrete time dynamic models, the paper titled “Simple mathematical models with very complicated dynamics” published in 1976 by Robert M. May, a physicist working on dynamic models in ecology, greatly contributed to consolidating such popularity. Indeed, this article proved in a very detailed and didactical style that the iteration of a one-dimensional quadratic map (a parabola) known as logistic map

$$\begin{aligned} x(t)=\alpha x(t-1)(1-x(t-1)){, \ \ \ \ }\alpha >0 \end{aligned}$$

(8)

can generate periodic trajectories of any period as well as chaotic dynamics. The paper of May ends with an “evangelical plea for the introduction of these difference equations into elementary mathematics courses, so that students’ intuition may be enriched by seeing the wild things that simple non-linear equations can do. [...] The elegant body of mathematical theory pertaining to linear systems, and its successful application to many fundamentally linear problems in the physical sciences, tends to dominate even moderately advanced University courses in mathematics and theoretical physics. The mathematical intuition so developed ill equips the student to confront the bizarre behavior exhibited by the simplest of discrete non-linear systems [...] I would therefore urge that people be introduced to [logistic map] early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about non-linear systems. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple non-linear systems do not necessarily possess simple dynamical properties.”

The discovery that even very simple dynamic models, like the iterated application of a second-degree function, are able to generate deterministic chaos, together with the observation that models of this kind can be easily obtained with completely standard assumptions of general economic equilibrium (with perfect competition, complete information and rational expectations), has shaken the foundations of many of the ideas to which economists had become accustomed to. The combination of simple deterministic models that generate chaotic dynamics and standard assumptions of the economic theory has broken the link between determinism and predictability, between equilibrium in the economic sense and equilibrium in the mathematical sense. Indeed, as noted by Grandmont (1985) and emphasized by Chiarella (1990), if a chaotic economic model is obtained assuming that economic agents have perfect information, including knowledge of the model (as a physicist knows the equations governing a certain phenomenon), in reality they can in no way achieve the infinite precision required to avoid the effects of the extreme sensitivity of chaotic dynamics in their forecasts. So, economic science might accept that sometimes economists are not able to make correct predictions. Moreover, chaotic fluctuations in the economy may be a necessary condition to achieve Paretian efficiency. Therefore, the classical paradigm according to which economic policies must always try to eliminate or dampen oscillations may be misleading, because an economy seeking maximum profits in the presence of chaotic fluctuations may go so far as to say - to paraphrase the Beatles - ’let it be’ (see Matsumoto 2003). Of course, other considerations may be made, related to the social consequences of economic fluctuations, when policies have to be decided. For this reason, several methods to avoid chaotic time patterns have been proposed in the literature, based on the introduction of stabilizing feedback policies in order to bring a dynamical system towards a stable equilibrium or periodic cycle. These methods are included in a growing stream of literature dealing with chaos control (see e.g. Schöll and Schuster 2008). In the same spirit, a pioneering paper proposing some methods to stabilize an oscillatory pattern was published by Baumol (1961). His results, based on properly modified Samuelson’s models, state that none of the contracyclical policies proposed can definitively rule out economic fluctuations.

However, even in the presence of chaotic behavior some regularities can be detected. For example, if the trajectories are represented in the phase space, one can see that the shape of the attracting set where the chaotic trajectories are confined may be characterized by interesting topological properties. Indeed, when a chaotic trajectory of the three-dimensional Lorenz model is represented in the phase space $\left( x_{1},x_{2},x_{3}\right)$, a structure like the one shown in Fig. 1 is obtained. If a trajectory starts from an initial condition inside that set then it remains there and covers any point of it as time goes on, and if a trajectory starts outside it (not too far, i.e. inside its basin of attraction) then it moves towards that set, where it exhibits irregular non-periodic time patterns and sensitive dependence on initial conditions.

Fig. 1

The Lorenz chaotic attractor

For this reason such an invariant set is denoted as “chaotic attractor”. The shape and extension of such attracting compact sets may give useful information about the long-run behavior of a dynamical system, being that it is bounded by intrinsic upper and lower bounds (endogenous ceiling and floor) for each dynamic variable, even if any time series is quite irregular. The same holds for the chaotic trends generated by the May’s parabola. In fact, one can note, for example, that even in the presence of chaotic dynamics sometimes the trajectories obtained do not cover the entire codomain of the function but accumulate in particular subintervals. Sometimes it is a single interval, other times several disjoint intervals, which are cyclically visited by the sequences generated iterating the map. The presence of such intervals is linked to the existence of the vertex of the parabola, since they are delimited by the maximum value and its images through the function (see Fig. 2).

Fig. 2

Trapping intervals for the logistic map, bounded by the maximum value c and its images $c_{1}$, $c_{2}$ and $c_{3}$

4 Chaotic dynamics in economic models

In this section we briefly describe some economic models (obtained from standard economic assumptions) represented by simple discrete dynamical systems capable of generating chaotic dynamics. Let us begin with the model proposed by Benhabib and Day (1982) which describes an economic system consisting of a population of consumers growing at a rate g, considering time (with an infinite horizon) divided into adjacent time periods with the assumption that each individual lives for two such periods: in the first (when young) she consumes a quantity $c_{1}(t)$ and earns $w_{1}$, in the second (when old) she consumes a quantity $c_{2}(t+1)$ and gains $w_{2}$. Therefore in each period t there coexist young consumers (born in that period) and elderly consumers, leading to the name “overlapping generations model”. The authors assume that each representative consumer is able to solve, as a young person, the problem of determining the ’consumption plan’ of one’s life in order to maximize the utility function $U(c_{1}(t),c_{2}(t+1))$ considering the budget constraint $w_{2}+r[w_{1}-c_{1}(t)]=c_{2}(t+1)$, where r is a factor of interest. With appropriate assumptions on the utility function, Benhabib and Day show that periodic or chaotic trajectories are obtained. Moreover, the assumptions on which the model is based allow them to state such oscillating economic equilibrium values are Pareto-efficient. This last statement has important consequences in terms of economic policies. Indeed, economists used to believe that, to make an economic system efficient, one must bring it to equilibrium (which in the linear mentality is synonymous with stationarity). In the presence of fluctuations, it seemed obvious that every effort should be made to dampen them as soon as possible, whereas Benhabib and Day (1982) show that this is not necessarily so.

Grandmont (1985), using a similar overlapping generations model, shows that chaotic fluctuations are obtained under the assumption of perfect consumer prediction. In other words, from a mathematical point of view, the hypothesis of perfect prediction is compatible with economic equilibrium characterized by chaotic trends. So, the author stresses that one cannot state that consumers are able to predict trends practically indistinguishable from random sequences.

Another kind of dynamic model that in the same years have greatly influenced the economic literature was proposed by Boldrin and Montrucchio (1986). They examine a typical problem in which a representative consumer maximizes her utility function over an infinite horizon with an appropriate discount rate. Considering two goods (i.e. two production sectors) with one consumption good and one holding as capital, labour as input for the production of the goods, the authors prove the path of optimal capital accumulation can present chaotic trends. They show how periodic or chaotic dynamics can be obtained provided that the discount rate is sufficiently small, i.e. consumers tend to be impatient because they consider future consumption much less important than the present one, and they give a constructive proof to state that, for any trend in the equilibrium dynamics of such an economic system, economically plausible utility and production functions can be proposed that allow to achieve the desired trends (however these may be complicated, even chaotic).

Finally, let us mention a class of models which have been subject to considerable study in the literature of mathematical economics, consisting of oligopoly models. The first example dates back to Augustine Cournot (1838) who proposed a model with linear demand and costs functions whose dynamic equilibrium is an anticipation of what is now called Nash Equilibrium, after Nash (1950). This equilibrium point is always stable with two competing firms (duopoly) but loses stability (and we know what this means in a linear context) with more than two. Rand (1978) showed that, with non-linear models, even in the presence of unstable equilibria one can obtain bounded sequences of productions characterized by chaotic fluctuations. Economically sound models, which led to oligopoly models similar to those proposed by Rand, have been proposed in the literature (see e.g. Puu 1991; Dana and Montrucchio 1986). We recall that traditional models assume rational players and one-shot games, where each player is assumed to have enough information and computational skills to choose a Nash equilibrium. However, real agents are sometimes not so smart nor informed, and they behave following adaptive methods, i.e. learning-by-doing or trial-and-error practices. If players replace one-shot optimal decisions with repeated myopic or adaptive decisions, the dynamic process may or may not converge to a Nash equilibrium. This is also the case in the evolutionary game approach, based on the principle that the choices of better performing economic agents are imitated by other agents, see e.g. the influential paper by Nowak and May (1992). Moreover, when a game has several Nash equilibria represented by equilibrium points of the dynamical system, then a step-by-step dynamic process may act as a selection device, i.e. the stability of the equilibria suggests which of them will prevail in the long-run (see e.g. Bischi and Kopel 2001; Bischi et al. 2010), and if several equilibrium points are stable, then the study of their basins of attraction will give information about the path dependence, i.e. how the convergence will depend on historical accidents (represented by exogenous shifts of initial conditions).

A remarkable result regards whether a repeated boundedly rational (or trial-and-error) decision leads to an adaptive (or myopic) process that converges, in the long run, to the same equilibrium chosen by rational players. An external observer may conclude that some invisible hand led them to the ’optimal’ outcome, whereas it just emerged spontaneously, and this may be seen as an evolutionary explanation of the outcome of a Nash equilibrium. Sometimes such repeated adaptive processes never converge, and continue to move around an equilibrium point following some periodic or chaotic time patterns, or they may even irreversibly depart from it. Such evolutions can be expressed by saying that players are not able to learn how to play a Nash equilibrium, and some dynamic analysis may be required to understand the kind of time evolutions that characterize the long run behavior of the repeated game. However, while analyzing a simple duopoly model, Matsumoto (2003) showed that producers may achieve higher profits in a market with chaotic trends than those obtained in a market with stationary equilibrium values, thus undermining the classical paradigm according to which stable oligopoly systems are always preferable than those with continuous ups and downs.

5 Structural stability and bifurcations

Chaotic oscillations are characterized through the feature of sensitivity with respect to small variations of the initial conditions, whereas the question raised by Samuelson (1939) mainly concerns the role of small changes in the parameters of the model. In that case, as the model is linear, a variation of the parameters $\alpha$ and $\beta$ can only affect the kind of stability (or instability) of the unique equilibrium. Instead, in the case of non-linear dynamic models, small variations of the parameters, besides stability switches may lead to creation or destruction of equilibrium points or other kinds of invariant sets. Indeed, in a non-linear model, the transitions from stability to instability (or vice versa) as a parameter’s value is varied are generally associated with the appearance or disappearance or merging of invariant sets, thus leading to qualitatively different dynamic scenarios, through what has been called a bifurcation.

Before giving some general definitions and historical developments of concepts related to bifurcations in non-linear dynamical systems, it is worth stressing that in economics the consequences of small changes of the parameters in equilibrium situations have been investigated in the framework of comparative statics, a type of static analysis that aims to compare how the position of an equilibrium is affected by small variations of a parameter. Such tool of analysis in microeconomics and macroeconomics was formalized by Hicks (1939) and Samuelson (1941). Comparative statics results are usually derived by using the implicit function theorem to calculate a linear approximation to the system of equations that defines an equilibrium, under the assumption that the equilibrium is stable. That is, if we consider a sufficiently small change in some exogenous parameter, we can calculate how each endogenous variable changes using only the first derivatives of the terms that appear in the equilibrium equations.

In a dynamic setting, parameters’ variations reveal not only quantitative effects, associated with proportional displacements of trajectories and invariant sets, but also qualitative transitions leading to different dynamic scenarios, denoted as bifurcations, due to a stability switch and/or appearance/disappearance of attractors. In the language of mathematics, the term bifurcation is commonly used to denote a qualitative change of a mathematical object described by an equation or a system of equations (sets of points, curves or surfaces) as a function of one or more coefficients (or parameters) on which the properties of the object depend. A field intensively studied in the 18th century, by Euler among others, gave rise to singularity theory, which shows that the global properties of a curve or a surface can be deduced from the knowledge of certain singular points, consisting of special folds or non smooth points like cusps. This theory has undergone considerable development during the 20th century, thanks to the work in the 1930 s of Marston Morse, in the 1940 s by Hasser Whitney, and in the 1960 s by René Thom and John Mather, leading to the conclusion that the generality of the possible bifurcations can be reduced to a limited number of canonical cases (see e.g. Thom 1972; Arnold 1981), also called elementary catastrophes (a term proposed by Christopher Zeeman commenting on the work of René Thom). However, the most systematic and comprehensive study of bifurcations was carried out within the qualitative analysis of dynamical systems proposed by Poincaré at the beginning of the 20th century, which paved the way for the modern theory of non-linear dynamical systems, within which the fundamental concept of structural stability was introduced thanks to the pioneering work of Russian scholars, especially that of the school of Aleksandr Andronov and Lev Pontryagin in the 1950 s, up to the more recent work by Arnold (see 1937, 1967, 1992).

One of the features of the qualitative theory of dynamical systems consists in having classified different types of invariant sets and acquired the ability of characterizing the phase diagram by just knowing these particular singular sets. When the parameters vary, the vector field changes, and consequently the set of curves that constitute the phase diagram is deformed. In general, a small change in a parameter will result in a small quantitative deformation of the phase diagram (that is, it produces a qualitatively equivalent phase diagram) and then it is said that the system is structurally stable. However, there are situations in which very small variations of a parameter lead to qualitative changes in the phase diagram, such as changes in stability and/or creation/disappearance of equilibrium points or other invariant sets (e.g. closed curves along which periodic motion occurs, or other types of attractors). In these cases it is said that the system is structurally unstable and it is close to a bifurcation that separates two qualitatively different classes of dynamic scenarios.

For example, in a one-dimensional map a fold bifurcation leads to the creation of a pair of equilibrium points, one stable and one unstable, where the unstable one acts as boundary of the basin of attraction of the stable one. Another kind of bifurcation is the so-called pitchfork, characterized by the transition from a single equilibrium point to three distinct equilibria when the initial equilibrium loses stability. The final outcome is a situation of bistability, with two newborn attractors, one high and one low, separated by a watershed (the central unstable equilibrium) that separates the two basins of attraction. An interesting bifurcation, occurring only in discrete time dynamical systems, is the flip (or period doubling) at which a stable equilibrium or a stable cycle loses its stability and a stable cycle of double period is created around it. A sequence of such bifurcations as a parameter is gradually increased constitutes a typical route leading to chaotic patterns (period doubling route to chaos), see e.g. May (1976), Devaney (1987).

By increasing the dimension of the dynamical system, that is, the number of dynamical variables, we find other interesting bifurcations. For example, with two or more dynamical variables a bifurcation that creates stable closed invariant orbits, along which self-sustained oscillatory motion occurs, has attracted the attention of economists interested in the theory of endogenous business cycle. This is called Andronov–Hopf bifurcation in continuous time models and Neimark–Sacker in discrete time ones. It occurs when an equilibrium point towards which there is a convergence through damped oscillations (a convergent spiral) becomes unstable (a divergent spiral). If the dynamical system considered is non-linear, then this change of stability is generally associated with the creation of an invariant closed orbit, along which the system moves indefinitely with a periodic motion. Such a closed orbit can be stable and surround the unstable equilibrium (supercritical case, see Fig. 3) or be unstable around the stable equilibrium, delimiting the basin of attraction (subcritical case). Thus, the supercritical case provides a mechanism for the creation of stable, or self-sustaining, oscillations. This is a very important case in practical applications, since it describes stable cyclical trends that repeat periodically in time, such as phenomena observed in many systems in the fields of physics (for instance, convection in fluids), ecology (the predator–prey systems) and economics (the endogenous business cycle). In the subcritical case we have instead a gradual narrowing of the basin of attraction of the equilibrium, until the unstable curve delimiting it collapses on the point making it unstable and causing a catastrophic transition towards another, distant attractor.

These are just some examples, for a more complete treatment we refer to Iooss and Joseph (1980), Guckenheimer and Holmes (1983) and Kuznetsov (2004).

Fig. 3

Andronov–Hopf or Neimark–Sacker bifurcation

6 Conclusions

This piece in the Milestones series is dedicated to Samuelson’s seminal paper of 1939. It also introduces other seminal papers coming from parallel literatures, such as Lorenz (1963) from mathematical physics and May (1976) from mathematical biology, because these papers influenced very much the introduction of a dynamic approach in economic modeling along the lines suggested by Samuelson. In fact, the linear dynamic equations proposed by Samuelson to describe a macroeconomic model where a Keynesian multiplier interacts with an accelerator mechanism, both endowed with a suitable structure of time lags, provide a dynamic approach to generate endogenous oscillations by properly tuning the model’s parameters.

Of course, the current literature on business cycle has moved away from the simplistic multiplier-accelerator model to include the role of frictions and financial intermediation in micro-founded models, see e.g. Bernanke et al. (1999); Gertler and Kiyotaki (2010), or Brunnermeier and Sannikov (2014). In particular, the last work, and its related macro-financial literature of the last 10 years, show that a complete explanation of business (and financial) cycles is still an open issue, and the economic profession is still in need of advanced mathematical methods, as it was for Samuelson in the Thirties.

However, even if too simple and unsatisfactory in many respects, Samuelson’s work made the economists aware about the problem of sensitivity of long-run dynamics with respect to small variations of the parameters or initial conditions, and claimed for more general models, especially non-linear ones, dealing with global dynamics, i.e. far from the equilibrium. So, the short and incisive paper of Samuelson, besides opening a research stream on endogenous oscillations generated by deterministic dynamic models, can be seen as an invitation to explore issues of non-linear dynamics, leading to sensitivity with respect to parameters (structural stability and bifurcations) as well as initial conditions (deterministic chaos), topics that have been at the center of a flourishing literature in the fields of mathematical physics and other related disciplines, such as chemistry, engineering and mathematical biology.

A lesson to be learned concerns the importance, for the advancement of a research field, of a comparative interdisciplinary approach based on different kinds of literatures. In this paper, we have stressed how the discovery of the mathematical concept of deterministic chaos led to a rethinking of the basic assumptions of neoclassical economics. In fact, on the one hand, it suggests the possibility that in economic systems there may be endogenous mechanisms capable of creating the disorder observed in the real economy; on the other hand, it may suggest that the assumption of rational agents able to predict the future states of the economy can sometimes be questionable.

It could be argued that the topics and the kinds of literature analyzed in this paper are distant from the problems and methods of economics, and it looks quite strange that an economist reads papers dealing with the growth of insects or the atmospheric sciences applied to weather forecasting. But a closer look reveals that human populations sometimes behave not too differently from insect populations, with cooperation or prey-predator relationships or competition for limiting resources, such as food and space. Concerning weather forecasts and economic predictions, of course these are quite different as in an economic context expectations, beliefs and preferences of people play an important role. However, nonlinearity of the models and, consequently, their complexities, are similar. A common joke states that economics and weather forecasting are two very common topics of conversation and both are generally not taken very seriously. This may be a consequence of the non-linear laws governing both these disciplines. However, it is equally true that despite the irregularity of weather conditions the global study of attractors provides information on the climate and its global changes, analogously in economics, even in the presence of irregular fluctuations, one can obtain from non-linear dynamic models useful global information on the long-run effects of variations in parameters representing economic policies. Moreover, the study of non-linear dynamic models, and the methods and terminology of complexity theory in general, have by now accustomed us to make use of interdisciplinary aspects through continuous comparisons among the behavior of economic, physical, chemical and biological systems.