Financial models with interacting heterogeneous agents ...staff.matapp.unimib.it/~pireddu/Beamer-Insubria-1D.pdf · We will then introduce some basic mathematical tools from discrete

Financial models withinteracting heterogeneous agents:

modeling assumptions and mathematical toolsfrom discrete dynamical system theory.

Minicourse for the PhD Program in Methods and Modelsfor Economic Decisions, Insubria University

Marina Pireddu

University of Milano-BicoccaDept. of Mathematics and its Applications

[email protected]

Marina Pireddu (Univ. of Milano-Bicocca) Discrete-time heterogeneous agent models Insubria Univ., 20/03/2018 1 / 139

Outline

1 Introduction

2 1D discrete dynamical systems

3 First applications: Heterogeneous Agents Models


Introduction

Introduction

Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:

(i) asset prices follow a near unit root process;

(ii) asset returns are unpredictable with almost no autocorrelations;

(iii) the returns distribution has fat tails;

(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.

Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.


Introduction

Introduction








Introduction

Introduction








Introduction

Introduction








Introduction

Introduction








Introduction

Introduction








Introduction

Introduction








Introduction

Introduction








Introduction

Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:

high trading volume is mainly caused by differences in beliefs;

volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.

Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.


Introduction






Introduction






Introduction






Introduction






Introduction

Markets are indeed viewed as complex adaptive systems, where theevolutionary selection of expectations rules or trading strategies isendogenously coupled with the market dynamics.

Being usually highly nonlinear, for instance due to evolutionaryswitching between strategies, the heterogeneous agent models exhibita wide range of dynamical behaviors.

We will then introduce some basic mathematical tools from discretedynamical system theory, which will be applied to analyze simple (1Dand 2D) heterogeneous agent models, like those proposed inWesterhoff (2012) and in Naimzada and Pireddu (2014, 2015a,2015b).


Introduction





Introduction





Introduction

References on heterogeneous agent models:

– De Grauwe P (2012) Lectures on Behavioral Macroeconomics.Princeton University Press, New Jersey.– Hommes CH (2006) Heterogeneous Agent Models in Economicsand Finance. In: L. Tesfatsion and K.L. Judd (Eds.), Agent-BasedComputational Economics, pp. 1109–1186. Handbook ofComputational Economics, vol.2. Elsevier Science, Amsterdam.Sections 1 and 6– Hommes CH (2013) Behavioral Rationality and HeterogeneousExpectations in Complex Economic Systems. Cambridge UniversityPress, Cambridge.– Naimzada A, Pireddu M (2014) Dynamic behavior of product andstock markets with a varying degree of interaction. EconomicModelling 41, 191–197


Introduction

– Naimzada A, Pireddu M (2015a) Introducing a price variation limitermechanism into a behavioral financial market model. Chaos 25,083112. doi: 10.1063/1.4927831– Naimzada A, Pireddu M (2015b) Real and financial interactingmarkets: A behavioral macro-model. Chaos Solitons Fractals 77,111–131– Westerhoff F (2012) Interactions between the real economy and thestock market: A simple agent-based approach, Discrete Dynamics inNature and Society 2012, Article ID 504840


1D discrete dynamical systems

Classification of 1D discrete dynamical systems

A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .

A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.

An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.

We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.













































If f is linear, i.e., f (xt ) = axt + b, for some a,b ∈ R, the system is saidto be linear; if f is nonlinear, i.e., if f is not linear, then the system issaid to be nonlinear.

Examples:

(i) xt+1 =√

2− πxt is linear

(ii) xt+1 = 1.27 xt (1− xt ) is nonlinear




Examples:

(i) xt+1 =√

2− πxt is linear





Examples:

(i) xt+1 =√

2− πxt is linear





Examples:

(i) xt+1 =√

2− πxt is linear




The initial value problem and orbits

To start the system xt+1 = f (xt ), we need to specify the initial conditionx0 ∈ R.

Then O(x0) = {x0, f (x0), f (f (x0)), f (f (f (x0))), . . . } ={x0, f (x0), f 2(x0), f 3(x0), . . . } is the (positive) orbit of x0.

Solving the system xt+1 = f (xt ) with initial condition x0 ∈ R meansfinding a sequence {yt , t ∈ N} such that yt+1 = f (yt ), for all t ∈ N, andy0 = x0.





















Equilibria and stability of 1D dynamical systems

If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.

The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .

Example: the logistic equation

xt+1 = µxt (1− xt ), µ > 0.







xt+1 = µxt (1− xt ), µ > 0.







xt+1 = µxt (1− xt ), µ > 0.







xt+1 = µxt (1− xt ), µ > 0.







xt+1 = µxt (1− xt ), µ > 0.



Consider f (xt ) = µxt (1− xt ), with f : [0,1]→ R

µ = 3.5

The fixed points are x∗ = 0 and x∗ = 1− 1µ ∈ (0,1) for µ > 1.




µ = 3.5





µ = 3.5




In the case of linear discrete dynamical systems xt+1 = axt + b, forsome a,b ∈ R, there exists a unique (acceptable?) equilibriumx∗ = b

1−a , for a 6= 1.

a = 0.3, b = −1



Definition of stability/instability

Given xt+1 = f (xt ), with f : I ⊂ R→ R defined on the interval I, theequilibrium point x∗ ∈ I is stable if ∀ε > 0 ∃δ > 0 such that ∀x0 ∈ I with|x0 − x∗| < δ it holds that |f t (x0)− x∗| < ε, ∀t ∈ N \ {0}.

x∗ is stable



If x∗ is not stable then it is called unstable.

x∗ is unstable



x∗ is unstable



If x∗ is stable and attracting, i.e., there exists η > 0 such that for allx0 ∈ I with |x0 − x∗| < η it holds that limt→+∞ f t (x0) = x∗, for t ∈ N,then x∗ is called locally asymptotically stable.

x∗ is locally asymptotically stable

If η = +∞, then x∗ is called globally asymptotically stable.



If x∗ is stable and attracting, i.e., there exists η > 0 such that for allx0 ∈ I with |x0 − x∗| < η it holds that limt→+∞ f t (x0) = x∗, for t ∈ N,then x∗ is called locally asymptotically stable.

x∗ is locally asymptotically stable

If η = +∞, then x∗ is called globally asymptotically stable.



How do we check local stability?

1) Graphical method: the cobweb (or stair-step) diagram

Monotone convergence to the equilibrium x∗ ∈ (0,1)⇒ it isasymptotically stable.



How do we check local stability?

1) Graphical method: the cobweb (or stair-step) diagram

Monotone convergence to the equilibrium x∗ ∈ (0,1)⇒ it isasymptotically stable.



Non-monotone convergence to the equilibrium x∗ ∈ (0,1)⇒ it is stillasymptotically stable.



Convergence to a period-two cycle⇒ x∗ ∈ (0,1) is unstable.



Convergence to a chaotic attractor⇒ x∗ ∈ (0,1) is unstable.



2) Analytical method:

TheoremLet x∗ be an equilibrium point of the dynamical system xt+1 = f (xt ),with f continuously differentiable at x∗.

(i) If |f ′(x∗)| < 1, then x∗ is locally asymptotically stable;(ii) if |f ′(x∗)| > 1, then x∗ is unstable;(iii) if |f ′(x∗)| = 1, you need higher derivatives to establish the nature

of x∗.

We will focus on hyperbolic equilibrium points x∗, i.e., with |f ′(x∗)| 6= 1.






of x∗.







of x∗.







of x∗.




Example:Considering again the logistic equation

xt+1 = µxt (1− xt ), µ > 0

and the associated map f (xt ) = µxt (1− xt ), with f : [0,1]→ R, we have

f ′(xt ) = µ(1− 2xt )⇒

f ′(0) = µ and thus x∗ = 0 is asymptotically stable for µ < 1 andunstable for µ > 1;

f ′(1− 1µ) = 2− µ ∈ (−1,1) for µ ∈ (1,3) and thus x∗ = 1− 1

µ isasymptotically stable for µ ∈ (1,3) and unstable for µ ∈ (3,+∞).




xt+1 = µxt (1− xt ), µ > 0


f ′(xt ) = µ(1− 2xt )⇒







xt+1 = µxt (1− xt ), µ > 0


f ′(xt ) = µ(1− 2xt )⇒






µ = 0.7

(Monotone) convergence to the equilibrium x∗ = 0⇒ it isasymptotically stable.



µ = 2

(Monotone) convergence to the equilibrium x∗ = 1− 1µ = 0.5⇒ it is

asymptotically stable.



µ = 2.8

(Non-monotone) convergence to the equilibrium x∗ = 1− 1µ ≈ 0.64⇒

it is still asymptotically stable.



µ = 3.25

Convergence to a period-two cycle⇒ x∗ = 1− 1µ ≈ 0.69 is unstable.

The two-cycle framework is readily revealed by the fact that the systemcycles around a rectangle.



µ = 3.25

Convergence to a period-two cycle⇒ x∗ = 1− 1µ ≈ 0.69 is unstable.

The two-cycle framework is readily revealed by the fact that the systemcycles around a rectangle.



µ = 3.8

Convergence to a chaotic attractor⇒ x∗ = 1− 1µ ≈ 0.74 is unstable.

What precisely happens when x∗ = 1− 1µ becomes unstable (for

µ > 3)?



µ = 3.8

Convergence to a chaotic attractor⇒ x∗ = 1− 1µ ≈ 0.74 is unstable.

What precisely happens when x∗ = 1− 1µ becomes unstable (for

µ > 3)?



Solving linear equations

General form: xt+1 = axt + b, with a, b ∈ R, a 6= 0.

– First case: b = 0 (homogeneous equation)

⇒ xt+1 = axt , with initial condition x0 ∈ R.

We have x1 = ax0, x2 = ax1 = a(ax0) = a2x0, . . . ,xt = axt−1 = a(at−1x0) = atx0.

The unique equilibrium is given by x∗ = 0 and it is globallyasymptotically stable for |a| < 1, and unstable for |a| > 1.

For a = 1 we have xt+1 = xt , with initial condition x0 ∈ R. The solutionis given by xt = x0, ∀t ∈ N.

For a = −1 we have xt+1 = −xt , with initial condition x0 ∈ R. Thesolution is given by xt = (−1)tx0, ∀t ∈ N.









































































(A) (B)Time series in (A) and stair-step diagram in (B) with a = 1.2, x0 = 10



(A) (B)Time series in (A) and stair-step diagram in (B) with a = 0.75, x0 = 10



(A) (B)Time series in (A) and stair-step diagram in (B) with

a = −0.75, x0 = 10



(A) (B)Time series in (A) and stair-step diagram in (B) with a = −1.2, x0 = 10



Example: the Malthusian population growth model (discrete version)

We denote by pt > 0 the size of the population p at time t ∈ N.

Assuming a constant population growth rate pt+1−ptpt

= K > −1⇒pt+1 = (K + 1)pt .

The solution is given by pt = (K + 1)tp0, with p0 > 0.






= K > −1⇒pt+1 = (K + 1)pt .







= K > −1⇒pt+1 = (K + 1)pt .







= K > −1⇒pt+1 = (K + 1)pt .




For K ∈ (−1,0), x∗ = 0 is globally asymptotically stable (exponentialdecay).

K = −0.5, p0 = 7.5



For K > 0, x∗ = 0 is unstable (exponential growth).

K = 0.5, p0 = 0.35



– Second case: b 6= 0 (nonhomogeneous equation)

⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.

• Subcase I: a = 1

⇒ xt+1 = xt + b, with b ∈ R \ {0}, and initial condition x0 ∈ R.

We have x1 = x0 + b, x2 = x1 + b = (x0 + b) + b = x0 + 2b, . . . ,xt = xt−1 + b = (x0 + (t − 1)b) + b = x0 + tb.

In this case there are no equilibria.




⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.








⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.








⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.








⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.








⇒ xt+1 = axt + b, with a, b ∈ R \ {0}.







a = 1, b = 3, x0 = −7



a = 1, b = −5, x0 = 9




• Subcase II: a 6= 1

xt+1 = axt + b, with a, b ∈ R \ {0}, a 6= 1, and initial condition x0 ∈ R.

The system has x∗ = b1−a as unique equilibrium.

x∗ = b1−a is globally asymptotically stable for |a| < 1, and unstable for

|a| > 1.

Since xt+1 − x∗ = axt + b − x∗ = axt − ab1−a = a(xt − x∗),

then setting yt = xt − x∗ for t ∈ N we obtainyt+1 = ayt ⇒ yt = aty0, i.e., xt − x∗ = at (x0 − x∗) ⇒xt = x∗+at (x0−x∗) = atx0+x∗(1−at ) = atx0+b 1−at

1−a = atx0 + b∑t−1

i=0 ai

(recalling that 1− at = (1− a)(1 + a + a2 + · · ·+ at−1), for t ∈ N)

If b = 0, we find again xt = atx0, with the unique equilibrium x∗ = 0globally asymptotically stable for |a| < 1, and unstable for |a| > 1.








|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai










|a| > 1.



1−a = atx0 + b∑t−1

i=0 ai





a = 2, b = −0.5, x0 = 0.4



a = −2, b = −0.5, x0 = −0.2



a = 0.5, b = 0.4, x0 = −1



a = −0.7, b = 0.4, x0 = −1



Example: the cobweb model

Demand qdt at time t depends on the current price pt , while the supply

qst at time t depends on planting, which in turn was governed by the

price pt−1 the farmer received in the last period.

The market is cleared in any period⇒ qdt = qs

t , ∀t .

Assuming linear demand and supply curves, the model is:

qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t

from which a− bpt = c + dpt−1, or pt = −db pt−1 + a−c

b .

The solution is given by pt =(−d

b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.








t , ∀t .


qdt = a− bpt , a, b > 0

qst = c + dpt−1, c ∈ R, d > 0,

qdt = qs

t


b .


b

)tp0 + a−c

b∑t−1

i=0(−d

b

)i.



The unique equilibrium p∗ = a−cb+d is globally asymptotically stable for

db < 1, and unstable for d

b > 1.



a = 1.7, b = 1.3, c = −0.3, d = 1, p0 = 0.5



a = 1.7, b = 1, c = −0.3, d = 1.3, p0 = 0.5



Summarizing:

Solution for xt+1 = axt + b, with initial condition x0 ∈ R :

• if b 6= 0 and a = 1⇒ xt = x0 + tb.


• In all other cases⇒ xt = atx0 + b∑t−1

i=0 ai .

The unique equilibrium x∗ = b1−a is globally asymptotically stable for

|a| < 1, and unstable for |a| > 1.

Rmk: the more general case xt+1 = atxt + bt can be handled similarly,but the solution has a more complex formulation.



Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Summarizing:


• if b 6= 0 and a = 1⇒ xt = x0 + tb.



i=0 ai .






Two important points:

A) Why do we use derivatives to check the local stability of equilibria?

If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:

f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.

Ignoring the remainder term, we obtain

f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).

The rhs is a linear equation in x with slope f ′(x∗).

If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.

Such stability condition can be used at any equilibrium.






f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).









f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),

with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.


f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).








Even if we confine ourselves only to stable equilibria, with nonlineardynamical systems there may be many.

This leads to some new and interesting policy implications.

The welfare attached to each equilibrium will be probably different.

If this is so, then it is possible for governments to choose between thevarious equilibrium points.

With linear systems in which only one equilibrium exists, suchquestions are meaningless.































B) With nonlinear dynamical systems, the local stability of anequilibrium does not give any information on the global dynamics.



Periodic points and their stability

If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.

If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .

Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .

Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.

In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.



































Let x be a periodic point of f with minimal period k . Then we say that:

x is asymptotically stable if it is an asymptotically stable fixed pointof f k .

x is unstable if it is an unstable fixed point of f k .

Hence, studying the stability of k -periodic solutions of xt+1 = f (xt )reduces to studying the stability of the equilibrium points of theassociated difference equation yt+1 = f k (yt ).





















Since by the chain rule we have that

ddx

f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),

we find the following practical criterion to check the stability of periodicpoints:

TheoremGiven the discrete dynamical system xt+1 = f (xt ), let x be a periodicpoint of f with minimal period k . If f is continuously differentiable atevery point of O(x), then it holds that:

(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then x is locallyasymptotically stable;

(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then x is unstable.




ddx

f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),








ddx

f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),








ddx

f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),







Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.

Hence, the conclusions in the previous result can be equivalentlyrewritten as:

(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;

(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.



























Example:Let us consider again the logistic equation

xt+1 = µxt (1− xt ), µ > 0

and the associated map f (xt ) = µxt (1− xt ), with f : [0,1]→ R.

We saw that:

x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1;

x∗ = 1− 1µ ∈ (0,1) for µ > 1; it is asymptotically stable for

µ ∈ (1,3) and unstable for µ ∈ (3,+∞).

What precisely happens for µ > 3?




xt+1 = µxt (1− xt ), µ > 0


We saw that:








xt+1 = µxt (1− xt ), µ > 0


We saw that:








xt+1 = µxt (1− xt ), µ > 0


We saw that:








xt+1 = µxt (1− xt ), µ > 0


We saw that:







Computing f 2(x) = x , we find µ2x(1− x) (1− µx(1− x)) = x .

Discarding the already known solution x∗ = 0, we have to solveµ2(1− x) (1− µx(1− x)) = 1, i.e., x3 − 2x2 + x

(1 + 1

µ

)− 1

µ + 1µ3 = 0.

Knowing that x∗ = 1− 1µ is a solution of such equation, we can use the

polynomial long division to obtain the following factorization:

x3 − 2x2 + x(

1 +1µ

)− 1µ

+1µ3 =

=

(x − 1 +

1µ

)(x2 − x

(1 +

1µ

)+

1µ

(1 +

1µ

)).

The solutions to x2 − x(

1 + 1µ

)+ 1

µ

(1 + 1

µ

)= 0 are given by

x1,2 = 12

(1 + 1

µ ±√

1− 2µ −

3µ2

), which are real and distinct for µ > 3.

Hence, for µ = 3 the period-two cycle {x1, x2} arises.





(1 + 1

µ

)− 1

µ + 1µ3 = 0.



x3 − 2x2 + x(

1 +1µ

)− 1µ

+1µ3 =

=

(x − 1 +

1µ

)(x2 − x

(1 +

1µ

)+

1µ

(1 +

1µ

)).


1 + 1µ

)+ 1

µ

(1 + 1

µ

)= 0 are given by

x1,2 = 12

(1 + 1

µ ±√

1− 2µ −

3µ2







(1 + 1

µ

)− 1

µ + 1µ3 = 0.



x3 − 2x2 + x(

1 +1µ

)− 1µ

+1µ3 =

=

(x − 1 +

1µ

)(x2 − x

(1 +

1µ

)+

1µ

(1 +

1µ

)).


1 + 1µ

)+ 1

µ

(1 + 1

µ

)= 0 are given by

x1,2 = 12

(1 + 1

µ ±√

1− 2µ −

3µ2







(1 + 1

µ

)− 1

µ + 1µ3 = 0.



x3 − 2x2 + x(

1 +1µ

)− 1µ

+1µ3 =

=

(x − 1 +

1µ

)(x2 − x

(1 +

1µ

)+

1µ

(1 +

1µ

)).


1 + 1µ

)+ 1

µ

(1 + 1

µ

)= 0 are given by

x1,2 = 12

(1 + 1

µ ±√

1− 2µ −

3µ2







(1 + 1

µ

)− 1

µ + 1µ3 = 0.



x3 − 2x2 + x(

1 +1µ

)− 1µ

+1µ3 =

=

(x − 1 +

1µ

)(x2 − x

(1 +

1µ

)+

1µ

(1 +

1µ

)).


1 + 1µ

)+ 1

µ

(1 + 1

µ

)= 0 are given by

x1,2 = 12

(1 + 1

µ ±√

1− 2µ −

3µ2





To investigate its stability, we have to solve |f ′(x1) · f ′(x2)| < 1.

Since f ′(x) = µ(1− 2x), we simply find|f ′(x1) · f ′(x2)| = |µ2(1− 2x1)(1− 2x2)| = | − µ2 + 2µ+ 4| < 1, which isfulfilled for µ ∈

(3,1 +

√6)≈ (3,3.449).

Thus, the period-two cycle {x1, x2} is asymptotically stable forµ ∈ (3,3.449) and unstable for µ > 3.449.





(3,1 +

√6)≈ (3,3.449).






(3,1 +

√6)≈ (3,3.449).




First iterate of f for µ = 3.4

The period-two cycle {x1, x2} is asymptotically stable.



First iterate of f for µ = 3.4

The period-two cycle {x1, x2} is asymptotically stable.



(A) (B)Second iterate of f for µ = 3.4



Third iterate of f for µ = 3.4

The period-three cycle does not exist yet for µ = 3.4.



Third iterate of f for µ = 3.4

The period-three cycle does not exist yet for µ = 3.4.



Before the emergence of the period-three cycle for µ = 3.8284, alleven-period cycles emerge. Why?

As we shall see, the answer is given by the Sharkovsky theorem.

But let us first look at the bifurcation diagram of the system.













Bifurcation diagrams

The behavior of some systems suddenly changes in a dramaticmanner at certain parameter values.

This led to the study of bifurcation theory.

The parameter values at which the system’s behavior changes arecalled bifurcation values.

Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),

where f is a nonlinear function depending also on the parameter µ ∈ R.

For instance, we may consider the logistic map fµ(xt ) = µxt (1− xt ),with fµ : [0,1]→ R.







Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),









Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),









Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),









Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),









Let us consider

xt+1 = f (xt ;µ) = fµ(xt ),





For certain values of the parameter µ the system settles down to aperiodic cycle.

However, for the remaining values of µ the system is irregular orchaotic.

Bifurcation diagrams are very useful in showing the occurrence ofchaotic behavior of dynamical systems.

In the bifurcation diagrams we plot the relationship between theparameter values and the stable equilibrium points (or the attractors) ofthe system.

The horizontal axis represents the parameter µ and the vertical axisrepresents suitable forward iterates f n

µ (x0) of a certain initial point x0.

If the considered ns are large enough, the diagram will show thelimiting behavior of the orbit of x0.
















































The computer-generated bifurcation diagram is obtained by thefollowing procedure:

0) Fix a suitable interval [µ1, µ2] for the parameter values.

1) Choose an initial value x0 from the domain of f and iterate, say,500 times to find

x0, fµ(x0), f 2µ (x0), . . . , f 500

µ (x0).

2) Drop the first, say, 400 iterations x0, fµ(x0), . . . , f 400µ (x0) and plot

the iterations f 401µ (x0), . . . , f 500

µ (x0) in the bifurcation diagram.

3) The procedure in 1)-2) is repeated for several values ofµ ∈ [µ1, µ2], usually taking increments of 1/100.






x0, fµ(x0), f 2µ (x0), . . . , f 500

µ (x0).










x0, fµ(x0), f 2µ (x0), . . . , f 500

µ (x0).










x0, fµ(x0), f 2µ (x0), . . . , f 500

µ (x0).










x0, fµ(x0), f 2µ (x0), . . . , f 500

µ (x0).







Focusing on the logistic map fµ(xt ) = µxt (1− xt ), with fµ : [0,1]→ R, inorder to have fµ([0,1]) ⊆ [0,1], we need 0 ≤ µ ≤ 4.

Indeed, fµ(1/2) = µ/4 ≤ 1⇔ µ ≤ 4.

Let us then consider [µ1, µ2] = [0,4] and choose, e.g., x0 = 0.5.




Indeed, fµ(1/2) = µ/4 ≤ 1⇔ µ ≤ 4.





Indeed, fµ(1/2) = µ/4 ≤ 1⇔ µ ≤ 4.




Bifurcation diagram of fµ for µ ∈ [0,4] and x0 = 0.5



We saw that:

x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1

µ is asymptotically stable for µ ∈ (1,3).

For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two

stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1

µ .

The period-two cycle {x1, x2}, with x1,2 = 12 + 1

2µ ±12

√1− 2

µ −3µ2 ,

is asymptotically stable for µ ∈ (3,3.449).

For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.



We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





We saw that:





µ .


2µ ±12

√1− 2

µ −3µ2 ,





What happens next?

Period-doubling bifurcations (leading from a stable period-four cycle toa stable period-eight cycle; from a stable period-eight cycle to a stableperiod-sixteen cycle, and so on) occur until µ ≈ 3.57.

After this value, the system starts exhibiting aperiodic or chaoticbehavior, i.e., behavior that, although generated by a deterministicsystem, has all the characteristics of randomness.



What happens next?





What happens next?





Chaotic attractor of fµ for µ = 3.65 and x0 = 0.5

For instance, we have sensitive dependence on initial conditions andthus predictions become virtually impossible.Indeed, given arbitrarily small differences in initial conditions, then thesystem will after time behave very differently.











Time series of fµ for µ = 3.65, T = 40, and x0 = 0.1 in blue, x0 = 0.15in green



Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5

The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.



















Bifurcation diagrams can be easily generated using E&F Chaossoftware, available at:

http://cendef.uva.nl/software/ef-chaos/ef-chaos.html



Sharkovsky theorem

Why do periodicity windows emerge in that order?

Let us define the Sharkovsky ordering asS0 � S1 � S2 � · · · � Sk � · · · � 24 � 23 � 22 � 2 � 1

where a � b means “a precedes b in the Sharkovsky ordering” and

S0 3 � 5 � 7 � . . . odd numbers (except 1)S1 2 · 3 � 2 · 5 � 2 · 7 � . . . 2·odd numbersS2 22 · 3 � 22 · 5 � 22 · 7 � . . . 22·odd numbers

...Sk 2k · 3 � 2k · 5 � 2k · 7 � . . . 2k ·odd numbers−− · · · � 24 � 23 � 22 � 2 � 1 Powers of 2 in descending order



Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Sharkovsky theorem








Theorem (Sharkovsky)

Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.

As a corollary, we have the following result:

Theorem (Li and Yorke)

Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.































If a continuous function f over the closed interval [a,b] has a period-5cycle, then it has cycles of all periods with the possible exception ofperiod-3.

Notice that the possibility of a period-3 is not ruled out.

If f has no period-2 orbits, then there do not exist higher-order periodicorbits, including chaos.

The Sharkovsky theorem, and to some extent the Li-Yorke theorem,show that even systems that exhibit chaotic behavior still have astructure.





















More about bifurcations

The logistic map presents a cascade of period-doubling bifurcationsleading to chaos.

The first period-doubling bifurcation occurs for the logistic map atx∗ = 1− 1

µ for µ = 3, at which x∗ = 1− 1µ becomes unstable and a

stable period-two cycle emerges.



More about bifurcations

The logistic map presents a cascade of period-doubling bifurcationsleading to chaos.

The first period-doubling bifurcation occurs for the logistic map atx∗ = 1− 1

µ for µ = 3, at which x∗ = 1− 1µ becomes unstable and a

stable period-two cycle emerges.



Period-doubling bifurcation for fµ at x∗ = 1− 1µ = 2

3 for µ∗ = 3



The graph of the logistic map fµ for µ in a neighborhood of 3



The graph of f 2µ for µ in a neighborhood of 3



From a mathematical viewpoint, a period-doubling bifurcation for amap g(x ;µ) at the fixed point (x∗, µ∗) is characterized by∂g∂x (x∗, µ∗) = −1 and other conditions on higher-order derivatives.

The logistic map displays also a transcritical bifurcation at(x∗, µ∗) = (0,1), where x∗ = 0 loses stability in favor of x∗ = 1− 1

µ ,which enters the interval (0,1).



From a mathematical viewpoint, a period-doubling bifurcation for amap g(x ;µ) at the fixed point (x∗, µ∗) is characterized by∂g∂x (x∗, µ∗) = −1 and other conditions on higher-order derivatives.

The logistic map displays also a transcritical bifurcation at(x∗, µ∗) = (0,1), where x∗ = 0 loses stability in favor of x∗ = 1− 1

µ ,which enters the interval (0,1).



Transcritical bifurcation for fµ at x∗ = 0 for µ∗ = 1



The graph of the logistic map fµ for µ in a neighborhood of 1



From a mathematical viewpoint, a transcritical bifurcation for a mapg(x ;µ) at the fixed point (x∗, µ∗) is characterized by ∂g

∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) = 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.

Indeed, for the logistic map fµ we have ∂2f∂x2 (0,1) = −2.

Finally, the logistic map displays also a triple saddle-node (tangent, orfold) bifurcation of the third iterate when the period-three cycleemerges.




∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) = 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.






∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) = 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.





In general, when a saddle-node bifurcation for a map g(x ;µ) at thefixed point (x∗, µ∗) occurs, a stable (the node) and an unstable (thesaddle) fixed points arise.

Saddle-node bifurcation for a map g(x ;µ) at the fixed point (x∗, µ∗)



With the triple saddle-node (tangent, or fold) bifurcation of the thirditerate of the logistic map, a stable and an unstable period-three cyclesemerge.

The graph of the logistic map fµ for µ in a neighborhood of1 +√

8 = 3.8284



The graph of f 3µ for µ in a neighborhood of 1 +

√8 = 3.8284



From a mathematical viewpoint, a saddle-node bifurcation for a mapg(x ;µ) at the fixed point (x∗, µ∗) is characterized by ∂g

∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) 6= 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.

With the logistic map fµ, we have g(x ;µ) = f 3µ (x).

There exists a last kind of 1-d bifurcation, not displayed by the logisticmap, i.e., the pitchfork bifurcation.




∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) 6= 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.






∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) 6= 0, ∂

2g∂x2 (x∗, µ∗) 6= 0.





At a pitchfork bifurcation, a fixed point loses stability in favor of two newbranches of fixed points.

Pitchfork bifurcation for gµ at x∗ = 0 for µ∗ = 1



For the map g(x ;µ) = µx − x3 a pitchfork bifurcation occurs at(x∗, µ∗) = (0,1), where x∗ = 0 loses stability in favor ofx∗1,2 = ±

√µ− 1.

The graph of gµ(x) = µx − x3 for µ in a neighborhood of 1



From a mathematical viewpoint, a pitchfork bifurcation for a mapg(x ;µ) at the fixed point (x∗, µ∗) is characterized by ∂g

∂x (x∗, µ∗) = 1,∂g∂µ(x∗, µ∗) = 0, ∂

2g∂x2 (x∗, µ∗) = 0.



Summarizing, for a map g(x ;µ) at the fixed point (x∗, µ∗) we have:



References on 1D discrete dynamical systems:

– Elaydi SN (2007) Discrete Chaos, Second Edition: With Applicationsin Science and Engineering. CRC Press, Taylor & Francis Group,Boca Raton, Florida. Chapters 1-2, Paragraphs 1.2, 1.4–1.9, 2.5, 2.6

– Shone R (2002) Economic Dynamics. Phase Diagrams and TheirEconomic Application, second ed. Cambridge University Press,Cambridge. Chapter 3, Paragraphs 3.1–3.5


First applications: Heterogeneous Agents Models

Introduction to Heterogeneous Agents Models (HAMs)

We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.

We will build our models by blocks of increasing complexity:

1) We will consider just the financial sector.

2) We will consider just the real sector.

3) We will jointly consider the two sectors via the interaction degreeapproach.

4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.

For 3) and 4) we will need to analyze 2D and 3D systems.































































1) The financial sector

At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).

Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.

The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).

Pt+1 − Pt = γg(Dt ),

where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.











































The simplest case: a linear price adjustment mechanism, i.e.,g(Dt ) = σDt = σ(F − Pt ), where σ > 0 is the fundamentalists reactivityparameter.

Hence we obtain:Pt+1 = Pt + γσ(F − Pt ).

We call γ = γσ > 0 the joint reactivity of the financial market.

The unique equilibrium is given by P∗ = F and it is stable for1− γ ∈ (−1,1), i.e., for γ < 2.

In such linear setting, when P∗ = F becomes unstable, the systemdiverges.































In order to obtain a parabola recalling the logistic equation, we couldconsider a multiplicative price adjustment mechanism

Pt+1 − Pt

Pt= γg(Dt ).

If g(Dt ) = σDt = σ(F − Pt ), setting γ = γσ, we get

Pt+1 = Pt (1 + γg(Dt )) = Pt (1 + γF − γPt ).

The equilibria are P∗ = 0 (not acceptable) and P∗ = F .Since for φ(P) = P(1 + γF − γP), we have φ′(P) = 1 + γF − 2γP, itholds that

φ′(0) = 1 + γF > 1 and thus P∗ = 0 is always unstable (differentlyfrom the logistic equation);φ′(F ) = 1− γF ∈ (−1,1) for γ < 2

F .




Pt+1 − Pt

Pt= γg(Dt ).





F .




Pt+1 − Pt

Pt= γg(Dt ).





F .




Pt+1 − Pt

Pt= γg(Dt ).





F .




Pt+1 − Pt

Pt= γg(Dt ).





F .




Pt+1 − Pt

Pt= γg(Dt ).





F .



In addition to fundamentalists, we can also deal with chartists in thefinancial market (see Day and Huang, 1990).

In a bull market chartists buy stocks, while in a bear market they sellstocks.

According to Tramontana et al. (2009), we assume that the chartists’demand is linear, i.e., DC

t = η(Pt − F ), where η > 0 is the chartistsreactivity parameter.

Justified by increasing profit opportunities, they assume the nonlinearfundamentalists’ demand DF

t = σ(F − Pt )3.

Hence the price dynamic equation becomes:

Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).

Introducing Xt = Pt − F , it is possible to rewrite it in deviations from thefundamental value as

Xt+1 = Xt + γ(ηXt − σX 3t ) = Xt (1 + γ(η − σX 2

t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).








t = σ(F − Pt )3.


Pt+1 = Pt + γ(η(Pt − F ) + σ(F − Pt )3).



t )).



The steady states are X ∗ = 0 and X ∗ = ±√

ησ .

They are all positive for F >√

ησ .

Since for ϕ(X ) = X (1 + γ(η − σX 2)), we haveϕ′(X ) = 1 + γη − 3γσX 2, it holds that

ϕ′(0) = 1 + γη > 1 and thus X ∗ = 0 is always unstable.

ϕ′(±√

ησ

)= 1− 2γη ∈ (−1,1) for γη < 1.




ησ .


ησ .



ϕ′(±√

ησ

)= 1− 2γη ∈ (−1,1) for γη < 1.




ησ .


ησ .



ϕ′(±√

ησ

)= 1− 2γη ∈ (−1,1) for γη < 1.




ησ .


ησ .



ϕ′(±√

ησ

)= 1− 2γη ∈ (−1,1) for γη < 1.




ησ .


ησ .



ϕ′(±√

ησ

)= 1− 2γη ∈ (−1,1) for γη < 1.



What happens when introducing a nonlinear price adjustmentmechanism which determines a bounded price variation in every timeperiod, as done in Naimzada and Pireddu (2015a)?

Pt+1 − Pt = γg(Dt ) = γa2

(a1 + a2

a1 exp(−Dt ) + a2− 1),

with a1,a2 positive parameters.

With this choice, g is increasing in Dt and it vanishes when Dt = 0.

Moreover, g is bounded from below by −a2 and from above by a1.





(a1 + a2

a1 exp(−Dt ) + a2− 1),








(a1 + a2

a1 exp(−Dt ) + a2− 1),








(a1 + a2

a1 exp(−Dt ) + a2− 1),








Hence, the price variations are gradual and the presence of the twohorizontal asymptotes prevents the dynamics of the stock market fromdiverging and helps avoiding negativity issues.

The above adjustment mechanism may be implemented assuming thatthe market maker is forced by a central authority to behave in adifferent manner according to the excess demand value.

In order to avoid overreaction phenomena, he/she has to be morecautious in adjusting prices when excess demand is large, whilehe/she has more freedom when excess demand is small, i.e., whenthe system is close to an equilibrium.

Since we allow a1 and a2 to be possibly different, the market makercan react in a different manner to a positive or to a negative excessdemand.





















Introducing Xt = Pt − F , we rewrite

Pt+1 = Pt + γa2

(a1 + a2

a1 exp(−(η(Pt − F ) + σ(F − Pt )3)) + a2− 1)

in deviations from the fundamental value as

Xt+1 = Xt + γa2

(a1 + a2

a1 exp(−(ηXt − σX 3t )) + a2

− 1

).

The steady states, as in Tramontana et al. (2009), are again thesolutions to Dt = 0, ∀t , i.e., X ∗ = 0 and X ∗ = ±

√ησ .


ησ .




Pt+1 = Pt + γa2

(a1 + a2

a1 exp(−(η(Pt − F ) + σ(F − Pt )3)) + a2− 1)


Xt+1 = Xt + γa2

(a1 + a2

a1 exp(−(ηXt − σX 3t )) + a2

− 1

).


√ησ .


ησ .




Pt+1 = Pt + γa2

(a1 + a2

a1 exp(−(η(Pt − F ) + σ(F − Pt )3)) + a2− 1)


Xt+1 = Xt + γa2

(a1 + a2

a1 exp(−(ηXt − σX 3t )) + a2

− 1

).


√ησ .


ησ .



Since for ψ(X ) = X + γa2

(a1+a2

a1 exp(−(ηX−σX 3))+a2− 1), we have

ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)

(a1 exp(−(ηX − σX 3)) + a2)2 exp(ηX − σX 3),

it holds that

ψ′(0) = 1 + γa1a2ηa1+a2

> 1, implying that X ∗ = 0 is always unstable.

ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.

For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.

When a1 or a2 are sufficiently small, the map ψ is strictly increasing.

This prevents the existence of interesting dynamics.




(a1+a2


ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)


it holds that

ψ′(0) = 1 + γa1a2ηa1+a2


ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.







(a1+a2


ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)


it holds that

ψ′(0) = 1 + γa1a2ηa1+a2


ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.







(a1+a2


ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)


it holds that

ψ′(0) = 1 + γa1a2ηa1+a2


ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.







(a1+a2


ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)


it holds that

ψ′(0) = 1 + γa1a2ηa1+a2


ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.







(a1+a2


ψ′(X ) = 1− γa1a2(a1 + a2)(3σX 2 − η)


it holds that

ψ′(0) = 1 + γa1a2ηa1+a2


ψ′(±√

ησ

)= 1− 2γη(

1a1

+ 1a2

) ∈ (−1,1) for γη < 1a1

+ 1a2.






However, for intermediate values of a1 or a2, it is possible that,although X ∗1 and X ∗3 are locally asymptotically stable, they coexist withperiodic or chaotic attractors.



The bifurcation diagram w.r.t. a1 ∈ [1.3,2.8] for ψ with a2 = 2,γ = 2.14, η = 0.2, σ = 1, with initial conditions X (0) = 0.1 for the

green dots, X (0) = −0.1 for the red dots, X (0) = 1.26 for the blue dots



The graph of ψ for a1 = 2.3



The bifurcation diagram w.r.t. a1 ∈ [1,2.8] for ψ with a2 = 2,γ = 2.14, η = 0.65, σ = 1, with initial conditions X (0) = 0.1 for the

green dots, X (0) = −0.1 for the red dots, X (0) = 1.26 for the blue dots



2) The real sector

We consider a model with a Keynesian good market of a closedeconomy with public intervention.

The Keynesian equilibrium condition is given by

Y = C + I + G

with

I = I, G = G, C = C + cY ,

where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.



2) The real sector



Y = C + I + G

with

I = I, G = G, C = C + cY ,




2) The real sector



Y = C + I + G

with

I = I, G = G, C = C + cY ,




2) The real sector



Y = C + I + G

with

I = I, G = G, C = C + cY ,




2) The real sector



Y = C + I + G

with

I = I, G = G, C = C + cY ,




In a dynamic framework we assume a dependence of consumption attime t on the income in the same time period, i.e., Ct = C + cYt .

The dynamic behavior in the real economy is represented by anadjustment mechanism depending on the excess demand.

If aggregate excess demand is positive (negative), productionincreases (decreases), that is,

Yt+1 = Yt + µf (Dt ),

where

µ > 0 is the real market speed of adjustment between demandand supply;

f is an increasing function with f (0) = 0 and Dt = Zt − Yt is theexcess demand, with Zt the aggregate demand in a closedeconomy, defined as

Zt = Ct + It + Gt .






Yt+1 = Yt + µf (Dt ),

where



Zt = Ct + It + Gt .






Yt+1 = Yt + µf (Dt ),

where



Zt = Ct + It + Gt .






Yt+1 = Yt + µf (Dt ),

where



Zt = Ct + It + Gt .






Yt+1 = Yt + µf (Dt ),

where



Zt = Ct + It + Gt .



For any such map f , the unique steady state, corresponding to Dt = 0,is given by

Y ∗ =A

1− cwhere A = C + I + G is aggregate autonomous expenditure and 1

1−c isthe Keynesian multiplier.

Imposing a linear adjustment mechanism, we obtain

Yt+1 = Yt +µ(Zt−Yt ) = Yt +µ(C+I+G−(1−c)Yt ) = Yt +µ(A−(1−c)Yt ).

In order to have a converging behavior towards Y ∗, we need

−1 < 1− µ(1− c) < 1,

which is satisfied for µ < µ = 21−c .

For larger values of µ we have instead a diverging behavior, while forµ = µ we have period-two cycles.




Y ∗ =A






−1 < 1− µ(1− c) < 1,






Y ∗ =A






−1 < 1− µ(1− c) < 1,






Y ∗ =A






−1 < 1− µ(1− c) < 1,






Y ∗ =A






−1 < 1− µ(1− c) < 1,





With the linear formulation, when Dt limits to ±∞, the same doesYt+1 − Yt .

However, this is an unrealistic assumption because of the materialconstraints in the production side of an economy.



With the linear formulation, when Dt limits to ±∞, the same doesYt+1 − Yt .

However, this is an unrealistic assumption because of the materialconstraints in the production side of an economy.



Assuming instead, like in Naimzada and Pireddu (2014a), that theadjustment mechanism is S-shaped, we specify the function f as

f (Dt ) = a2

(a1 + a2

a1e−Dt + a2− 1),


Hence, f is bounded from below by −a2 and from above by a1.

Thus the income variations are gradual and this prevents the realmarket from diverging and it may create a real oscillator.

Namely, more realistic assumptions lead in this case to more realisticoutcomes.




f (Dt ) = a2

(a1 + a2

a1e−Dt + a2− 1),








f (Dt ) = a2

(a1 + a2

a1e−Dt + a2− 1),








f (Dt ) = a2

(a1 + a2

a1e−Dt + a2− 1),







The dynamic equation of the real market in the nonlinear framework isgiven by

Yt+1 =Yt +µa2

(a1 + a2

a1e−Dt + a2− 1)

=Yt +µa2

(a1 + a2

a1e−(A−Yt (1−c)) + a2− 1).

The unique steady state is still Y ∗ = A1−c .



The dynamic equation of the real market in the nonlinear framework isgiven by

Yt+1 =Yt +µa2

(a1 + a2

a1e−Dt + a2− 1)

=Yt +µa2

(a1 + a2

a1e−(A−Yt (1−c)) + a2− 1).

The unique steady state is still Y ∗ = A1−c .



Setting Φ(Y ) = Y + µa2

(a1+a2

a1e−(A−Y (1−c))+a2− 1), we have

Φ′(Y ∗) = 1− µa1a2(1− c)

a1 + a2∈ (−1,1) for

µ < µ =2

1− c

(1a1

+1a2

).

We notice that µ = 21−c < µ when a1 and a2 are small enough.

Indeed, when reducing a1 and a2, we decrease the current variation ofoutput, enlarging the stability region.

With the introduction of the sigmoidal adjustment mechanism, in theinstability regime we have the emergence of an absorbing interval, i.e.,an invariant interval which eventually captures all forward trajectories.




(a1+a2

a1e−(A−Y (1−c))+a2− 1), we have

Φ′(Y ∗) = 1− µa1a2(1− c)

a1 + a2∈ (−1,1) for

µ < µ =2

1− c

(1a1

+1a2

).







(a1+a2

a1e−(A−Y (1−c))+a2− 1), we have

Φ′(Y ∗) = 1− µa1a2(1− c)

a1 + a2∈ (−1,1) for

µ < µ =2

1− c

(1a1

+1a2

).







(a1+a2

a1e−(A−Y (1−c))+a2− 1), we have

Φ′(Y ∗) = 1− µa1a2(1− c)

a1 + a2∈ (−1,1) for

µ < µ =2

1− c

(1a1

+1a2

).






The bifurcation diagram w.r.t. µ ∈ [0,1] for the map Φ with A = 12,a1 = 50, a2 = 11 and c = 0.6

We observe a cascade of period-doubling bifurcations leading tochaos.The chaotic regime is interrupted, e.g., by a period-three periodicitywindow.











3) Real and financial sectors: the interaction degreeapproach

We start presenting the model in Westerhoff (2012).

The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.

Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.

Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains

Yt+1 = Yt + µ(Zt − Yt ) = Yt + µ(C + I + G − (1− c)Yt + αPt )= Yt + µ(A− (1− c)Yt + αPt ) = A + cYt + αPt .



































The financial sector in Westerhoff (2012) coincides with that inTramontana et al. (2009), when setting γ = 1 and replacing F with

Ft = dYt ,

where d > 0 is a parameter capturing the relationship between thenational income and the perceived fundamental stock market value.

The dynamic equation of the stock market is then:

Pt+1 = Pt +γ(η(Pt−Ft )+σ(Ft−Pt )3) = Pt +η(Pt−dYt )+σ(dYt−Pt )

3.



The financial sector in Westerhoff (2012) coincides with that inTramontana et al. (2009), when setting γ = 1 and replacing F with

Ft = dYt ,

where d > 0 is a parameter capturing the relationship between thenational income and the perceived fundamental stock market value.

The dynamic equation of the stock market is then:

Pt+1 = Pt +γ(η(Pt−Ft )+σ(Ft−Pt )3) = Pt +η(Pt−dYt )+σ(dYt−Pt )

3.



Proposition (isolated goods and stock markets)

Suppose first that Pt = P. National income is then driven by theone-dimensional linear map Yt+1 = A + cYt + αP. Its unique steadystate Y ∗ = (A + αP)/(1− c) is positive and globally asymptoticallystable.Suppose now that Yt = Y . The stock price is then determined by theone-dimensional nonlinear map Pt+1 = Pt + η(Pt − dY ) +σ(dY −Pt )

3.There are three coexisting steady states P∗1 = dY and

P∗2,3 = P∗1 ±√

ησ . Steady state P∗1 is positive, yet unstable. Steady

states P∗2,3 are positive for dY >√

ησ and locally stable for η < 1.



Proposition (isolated goods and stock markets)

Suppose first that Pt = P. National income is then driven by theone-dimensional linear map Yt+1 = A + cYt + αP. Its unique steadystate Y ∗ = (A + αP)/(1− c) is positive and globally asymptoticallystable.Suppose now that Yt = Y . The stock price is then determined by theone-dimensional nonlinear map Pt+1 = Pt + η(Pt − dY ) +σ(dY −Pt )

3.There are three coexisting steady states P∗1 = dY and

P∗2,3 = P∗1 ±√

ησ . Steady state P∗1 is positive, yet unstable. Steady

states P∗2,3 are positive for dY >√

ησ and locally stable for η < 1.



Setting Pt = P, the goods market is decoupled by the financial market.

Setting Yt = Y , the financial market is decoupled by the real market.

In this case, the system{Yt+1 = A + cYt + αP,

Pt+1 = Pt + η(Pt − dY ) + σ(dY − Pt )3,

is composed by two independent equations.






Pt+1 = Pt + η(Pt − dY ) + σ(dY − Pt )3,







Pt+1 = Pt + η(Pt − dY ) + σ(dY − Pt )3,




However, when Yt+1 depends on Pt and Pt+1 depends on Yt , themodel in Westerhoff (2012) is 2D:{

Yt+1 = A + cYt + αPt ,

Pt+1 = Pt + η(Pt − dYt ) + σ(dYt − Pt )3.

In this case, goods and stock markets are interacting.

How do we study it? −→ Analysis of 2D discrete dynamical systems.































Introducing the interaction degree ω ∈ [0,1], we could see the twoframeworks above as the extreme cases of

{Yt+1=A + cYt + α(ωPt + (1− ω)P),

Pt+1=Pt + η(Pt − d(ωYt + (1− ω)Y )) + σ(d(ωYt + (1− ω)Y )− Pt )3.

For ω = 0 we obtain the isolated markets framework;

for ω = 1 we obtain the fully interacting markets framework;

for ω ∈ (0,1) we obtain a partial interaction between the two markets.




{Yt+1=A + cYt + α(ωPt + (1− ω)P),








{Yt+1=A + cYt + α(ωPt + (1− ω)P),








{Yt+1=A + cYt + α(ωPt + (1− ω)P),







Rmk: ω may be used as bifurcation parameter to show the role, on thesystem dynamics, of an increasing degree of interaction betweenmarkets.

Let us show the bifurcation diagrams w.r.t. ω ∈ [0,1] of the map Fωassociated to:

{Yt+1 =A + cYt + α(ωPt + (1− ω)P),

Pt+1 =Pt + η(Pt − d(ωYt + (1− ω)Y )) + σ(d(ωYt + (1− ω)Y )− Pt )3



Rmk: ω may be used as bifurcation parameter to show the role, on thesystem dynamics, of an increasing degree of interaction betweenmarkets.

Let us show the bifurcation diagrams w.r.t. ω ∈ [0,1] of the map Fωassociated to:

{Yt+1 =A + cYt + α(ωPt + (1− ω)P),

Pt+1 =Pt + η(Pt − d(ωYt + (1− ω)Y )) + σ(d(ωYt + (1− ω)Y )− Pt )3



The bifurcation diagram for Y and P w.r.t. ω ∈ [0,1] of the map Fωwhen A = 3, c = 0.95, α = 0.02, d = 1, η = 1.63, σ = 0.3



For the parameter values considered in Westerhoff (2012), raising theinterconnection between markets destabilizes Y ∗ and leads toincreasing income oscillations.

Since η > 1 all equilibria of the isolated stock market are unstable.Raising the interconnection between markets slightly increases themodulus of oscillations.



For the parameter values considered in Westerhoff (2012), raising theinterconnection between markets destabilizes Y ∗ and leads toincreasing income oscillations.

Since η > 1 all equilibria of the isolated stock market are unstable.Raising the interconnection between markets slightly increases themodulus of oscillations.



A last 1D framework:the model in Naimzada and Pireddu (2014b)

The real and the financial sectors can be “integrated” −→ the resultingsetting is 1D.

In regard to the real side of the economy, we consider

Yt+1 = Yt + µf (Zt − Yt ),

wheref (Zt − Yt ) = a2

(a1+a2

a1e−(Zt−Yt )+a2− 1), with a1,a2 positive

parameters;

the aggregate demand in a closed economy is given byZt = Ct + It + Gt = A + cYt + αPt .

Similarly to Westerhoff (2012), we assume that private expenditureincreases with the belief about the stock price performance

Pt = (1− ω)P + ωPt , ω ∈ [0,1].






Yt+1 = Yt + µf (Zt − Yt ),


(a1+a2


parameters;



Pt = (1− ω)P + ωPt , ω ∈ [0,1].






Yt+1 = Yt + µf (Zt − Yt ),


(a1+a2


parameters;



Pt = (1− ω)P + ωPt , ω ∈ [0,1].






Yt+1 = Yt + µf (Zt − Yt ),


(a1+a2


parameters;



Pt = (1− ω)P + ωPt , ω ∈ [0,1].






Yt+1 = Yt + µf (Zt − Yt ),


(a1+a2


parameters;



Pt = (1− ω)P + ωPt , ω ∈ [0,1].






Yt+1 = Yt + µf (Zt − Yt ),


(a1+a2


parameters;



Pt = (1− ω)P + ωPt , ω ∈ [0,1].



For the financial sector, we consider the framework with chartists andfundamentalists in Tramontana et al. (2009) and in Westerhoff (2012),but with a linear demand for fundamentalists, too.

Pt+1 = Pt + γ(DCt + DF

t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).

Similarly to Westerhoff (2012), we suppose that speculators perceivethe following relation between the fundamental value and a proxy Yt ofnational income

Ft = dYt ,

where d > 0 is a parameter capturing the above described directrelationship.

We suppose that

Yt = ωYt + (1− ω)Y , ω ∈ [0,1].





t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).


Ft = dYt ,


We suppose that

Yt = ωYt + (1− ω)Y , ω ∈ [0,1].





t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).


Ft = dYt ,


We suppose that

Yt = ωYt + (1− ω)Y , ω ∈ [0,1].





t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).


Ft = dYt ,


We suppose that

Yt = ωYt + (1− ω)Y , ω ∈ [0,1].



Hence, the dynamic equation of the stock market is given by:

Pt+1 = Pt +γ(η(Pt−d(ωYt + (1− ω)Y ))+σ(d(ωYt + (1− ω)Y )−Pt )).

Since the functioning of financial markets is such that their priceadjustment mechanism is much faster than the mechanism ofadjustment of good market prices, we assume that γ → +∞.

Thus, we obtain the equilibrium condition

0 = limγ→+∞

Pt+1 − Pt

γ= η(Pt−d(ωYt +(1−ω)Y ))+σ(d(ωYt +(1−ω)Y )−Pt ),

from whichPt = d [ωYt + (1− ω)Y ].







0 = limγ→+∞

Pt+1 − Pt









0 = limγ→+∞

Pt+1 − Pt









0 = limγ→+∞

Pt+1 − Pt





We then find the integrated equation

Yt+1 = Yt +µa2

(a1 + a2

a1e−(A+cYt+α((1−ω)P+ω(d [ωYt+(1−ω)Y ]))−Yt ) + a2− 1).

We stress that for ω = 0 we obtain

Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)+αP) + a2− 1),

very similar to the model studied in Naimada and Pireddu (2014a)

Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)) + a2− 1).




Yt+1 = Yt +µa2

(a1 + a2



Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)+αP) + a2− 1),


Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)) + a2− 1).




Yt+1 = Yt +µa2

(a1 + a2



Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)+αP) + a2− 1),


Yt+1 = Yt + µa2

(a1 + a2

a1e−(A−Yt (1−c)) + a2− 1).



The system

Yt+1 = Yt + µa2

(a1 + a2

a1e−(A+cYt+α((1−ω)P+ω(d [ωYt+(1−ω)Y ]))−Yt ) + a2− 1)

has as unique steady state

Y ∗(ω) =A + α(1− ω)[P + dωY ]

1− c − αdω2 .

In particular Y ∗(0) = Y ∗ = A+αP1−c and Y ∗(1) = Y1 = A

1−c−dα , with Y ∗

and Y1 found in Westerhoff (2012).



The system

Yt+1 = Yt + µa2

(a1 + a2

a1e−(A+cYt+α((1−ω)P+ω(d [ωYt+(1−ω)Y ]))−Yt ) + a2− 1)

has as unique steady state

Y ∗(ω) =A + α(1− ω)[P + dωY ]

1− c − αdω2 .

In particular Y ∗(0) = Y ∗ = A+αP1−c and Y ∗(1) = Y1 = A

1−c−dα , with Y ∗

and Y1 found in Westerhoff (2012).



Proposition

Setting µ = 2(a1+a2)(1−c)a1a2

and µ = 2(a1+a2)(1−c−dα)a1a2

, the steady state Y ∗(ω) isstable in the following cases:

for every ω ∈ [0,1], if µ < µ ;

for ω ∈ [R,1], where R =

√1

dα

(1− c − 2(a1+a2)

µa1a2

), if µ < µ < µ .

Indeed, for

ξ(Y ) = Y + µa2

(a1 + a2

a1e−(A+cY+α((1−ω)P+ω(d [ωY+(1−ω)Y ]))−Y ) + a2− 1)

it holds thatξ′(Y ∗) = 1− µa1a2(1− c − αdω2)

a1 + a2∈ (−1,1)

for the above values of µ, recalling that ω ∈ [0,1].



Proposition


and µ = 2(a1+a2)(1−c−dα)a1a2




√1

dα

(1− c − 2(a1+a2)

µa1a2

), if µ < µ < µ .

Indeed, for

ξ(Y ) = Y + µa2

(a1 + a2



a1 + a2∈ (−1,1)




Proposition


and µ = 2(a1+a2)(1−c−dα)a1a2




√1

dα

(1− c − 2(a1+a2)

µa1a2

), if µ < µ < µ .

Indeed, for

ξ(Y ) = Y + µa2

(a1 + a2



a1 + a2∈ (−1,1)




In orange the stability region of Y ∗(ω) in the (µ, ω)-plane



Hence, for intermediate values of µ, an increasing degree of interactionbetween the real and financial sectors has a stabilizing effect.

This is probably due to the fact that, in our formalization, we assumethat the stock market is an equilibrium market.

Indeed, in Naimzada and Pireddu (2015b), where both the real and thefinancial markets are not always in equilibrium, different scenarios mayoccur.













The bifurcation diagram w.r.t. ω for the map ξ with A = 5, c = 0.2,α = 1, d = 0.6, a1 = 3, a2 = 2, µ = 6, Y = P = 1

The stabilization occurs via a sequence of period-halving bifurcations.

In particular, for ω = R =

√1

dα

(1− c − 2(a1+a2)

µa1a2

), a flip bifurcation

occurs.






√1

dα

(1− c − 2(a1+a2)

µa1a2


occurs.






√1

dα

(1− c − 2(a1+a2)

µa1a2


occurs.




We highlight a multistability phenomenon characterized by thecoexistence of chaotic and periodic attractors (in red) with aperiod-eight orbit (in blue).




We highlight a multistability phenomenon characterized by thecoexistence of chaotic and periodic attractors (in red) with aperiod-eight orbit (in blue).



References on the first applications of HAMs:

– Day RH, Huang W (1990) Bulls, bears and market sheep. Journal ofEconomic Behavior and Organization 14, 299–329– Naimzada A, Pireddu M (2014a) Dynamics in a nonlinear Keynesiangood market model. Chaos 24, 013142. DOI: 10.1063/1.4870015– Naimzada A, Pireddu M (2014b) Dynamic behavior of product andstock markets with a varying degree of interaction. EconomicModelling 41, 191–197– Naimzada A, Pireddu M (2015a) Introducing a price variation limitermechanism into a behavioral financial market model. Chaos 25,083112. DOI: 10.1063/1.4927831– Naimzada A, Pireddu M (2015b) Real and financial interactingmarkets: A behavioral macro-model. Chaos Solitons Fractals 77,111–131



– Tramontana F, Gardini L, Dieci R, Westerhoff F (2009) Theemergence of bull and bear dynamics in a nonlinear model ofinteracting markets. Discrete Dynamics in Nature and Society 2009,310471– Westerhoff F (2012) Interactions between the real economy and thestock market: A simple agent-based approach. Discrete Dynamics inNature and Society 2012, Article ID 504840


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