a post-keynesian model of growth and distribution with a

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Structural Change and EconomicDynamics28 (2014) 12–24

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Structural Change and Economic Dynamics

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A post-Keynesian model of growth and distribution with aconstraint on investment

Pasquale Commendatore a,∗, Antonio Pintob, Iryna Sushko c

a Department of Economics, University of Naples ‘Federico II’, Italyb University of Naples ‘Federico II’, Italyc Institute of Mathematics, NASU, andKyiv School of Economics, Ukraine

a r t i c l e i n f o

Article history:

Received 2 April 2012

Received in revised form 28 August2013

Accepted15 September 2013

Available online 8 October 2013

JEL classification:

C62

E12

O41

Keywords:

Post-Keynesiangrowth and distribution

modelConstrained investment function

Stability

Complex dynamics

a b s t r a c t

We introduce in a post-Keynesian/Kaleckian model of growth and distribution a con-

straintonfirms’ investmentinducedby increasingadjustmentcostsand/or limitedfinancial

resources. Whereas in the short run limitingfirms’ investment reduces capacity utilization

and capital accumulation, in the long run, allowing the adjustment of the “normal” to the

actual degree of capacity utilization, the direction of the impact of the constraint goes in

the opposite direction: relaxing the constraint reduces capital utilization and accumula-

tion. Moreover, an increase in the saving propensity or a fall inwages do not always cause

a reduction in the degree of capital utilization – the so-called paradoxes of thrift and costs

– as predicted by the standard post-Keynesian/Kaleckian analysis; and growth could be

profit led. These results are not confined to long-run positions of the economy charac-

terized by convergence to a stationary equilibrium but take also into account periodic or

chaotic fluctuations. © 2013 Elsevier B.V. All rights reserved.

1. Introduction

In TheAccumulationof Capital, 1956 [1965], Joan Robin-

sonliststhe limits to long-runcapitalaccumulation.Among

such limits she includes financial and technical impedi-

ments that could affect entrepreneurs’ predisposition to

invest. In her own words: “[w]ithin the limit set by thetechnical surplus lies the inflation barrier, which opera-

tes through the mechanism of the interest rates. Within

this may lie a limit to over-cautious or clumsily oper-

ated banking policy, which keeps investment (on the

∗ Corresponding authorat:Dipartimentodi Giurisprudenza, Universitàdi Napoli ‘Federico II’, viaMezzocannone16, I-80134 Naples, Italy.

Tel.: +39 081 253 7447; fax: +39 081 253 7454.

E-mail address: [email protected]

(P. Commendatore).

average over a run of years) lower than is necessary to

avoid inflation. Within this, the limit set by the energy of

entrepreneurs involves a complex of technical, human and

financial influences” (Robinson, 1956 [1965], 243). More-

over, capacity bottle-necks and the availability of finance

represents a short-term impediment to investment aswell

(see Robinson, 1956 [1965], pp. 50–52).1

A constraint on investment can also be explained

drawing on Kalecki, 1937[1990]) principle of increasing

risk. Following Kalecki, “[i]t is reasonable to assume that

marginal risk increases with the amount invested” (1937

[1990], 287). This assumption is useful “in order to obtain

a realistic solution of the problem of limited investment

(1937 [1990], 287).

1 In Robinson (1962) are also briefly addressed the impediments to

capital accumulationandamong these the lack of finance.

0954-349X/$ – see frontmatter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.strueco.2013.09.001

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P. Commendatore et al. / Structural Change and Economic Dynamics 28 (2014) 12–24 13

The limits to the firm’s growth, especially imposed by

the financing of corporations, has been a major concern

of post-Keynesian literature dealing with managerial cap-

italists (see, e.g., Marris, 1964; Kaldor, 1966; Wood, 1975;

Eichner, 1976) or with financial crises. Indeed, follow-

ing Minsky’s (1964, 1978, 1982, 1986) “financial fragility

hypothesis” there exists a strong link between the finan-

cial structure and the expansion or contraction of the real

sector of the economy. More recently, Kaleckian mod-

els of growth and distribution, aiming at explaining the

causes and the consequences of the global financial crisis

started in 2007–2008, have been put forward stressing the

effects of “financialization” and credit constraints on capi-

tal accumulation. These analyses consider that, since early

‘80s, increasingshareholder powershifted theobjectivesof

firms from long-run accumulation toward short-term pro-

fitability. This hasdetermined a rise in dividend payments,

a reduction in the availability of internal finance and has

made firmsmore anmore reliant on bank credit and other

forms of external finance for investment in capital stock

(see, for a review, Hein and van Treeck, 2010).

Another factor thatmay affect the pace of capital accu-

mulation is the occurrence of increasing adjustment costs

of productive capacity. A special case of this hypothesis,

that of smoothconvex adjustment costs, firstly introduced

by Uzawa (1969), hasbeen incorporated in post Keynesian

models by Asada (2001, 2006).2

In this paper, we present a post-Keynesian/Kaleckian

model of growth in which – taking into account all the

above considerations – a constraint on private invest-

ment expenditure is introduced. Basically, we assume that

entrepreneurs’ “propensity to invest” decreases above a

critical value of thedegree of capacity utilization. This crit-

ical value canbe interpreted asa limit imposed to investors

by the availability of credit induced by the lack of internal

financial resources and/or by the over cautious behavior

of the banking sector; or by the raising costs of adjus-

ting productive capacity above some specific threshold of

the capital stock utilization. We provide a detailed anal-

ysis of all possible scenarios that may emerge from the

possible configurations of the relevant parameters, espe-

cially those related to saving and investment decisions.

Notably, the introduction of the constraint has different

consequences in the short and in the long run.Whereas in

the short run limiting firms’ investment reduces capacity

utilization and investment demand; in the long run, allow-

ing the adjustment of the “normal” to the actual degree

of capacity utilization, the direction of the impact of the

constraint goes in the opposite direction: plant utilization

andcapital accumulationareenhanced.Wealso show that,

due to the presence of the constraint, an increase in the

propensity to save or a fall inwages do not always cause a

reductionin thedegreeofcapitalutilization– that is,theso-

called paradoxes of thrift and cost, which hold in standard

post-Keynesian models of growth and distribution, may

not occur.

2 Theexistenceof some types of inflexibilities in the adjustment of the

capital stock is acknowledged also by other heterodox authors such asSkott (1989, 2010) and Schoder (2012).

The paper is organized as follows. In Section2, we

present themodel with no constraint on firms’ investment

plans. We derive the short and long-term solutions and

identify the stability conditions. In Section3, we introduce

a constraint on the investment function.Weverify that the

long-run dynamic behavior is substantially enriched. For

some parameter combinations, thedynamic equation gov-

erning the long-run evolutionof the system is represented

by a so-called skew tentmap.We apply themathematical

results concerning its local and global stability properties

to our model. Section4 presents some final remarks.

2. The model with a linear investment function

We represent a very stylized closed economy. Pro-

duction involves fixed coefficients, infinitely elastic labor

supply and no capital depreciation. The short-run sta-

tionary equilibrium is always obtained; the link between

short and long run is represented by the revision of

entrepreneurs’ conception of thenormal degreeof capacity

utilization.

2.1. The economy in the short run

In the short run productive capacity is given, i.e. the

capital stock is fixed; moreover, firms’ conception of the

normal degree of capacity utilization is given as well.

Savings, S , are proportional to national income:

S

K = sY

K = saK u = u, (1)

where s represents the average propensity to save of the

economy, with 0< s0, dependsonthetechnology,aK ,andontheaveragepropensitytosave

of theeconomy.We donotconsiderchanges inthetechnol-

ogy. Therefore, changes in will only be the consequenceof variations of s. The latter parameter, in turn, depends

on the propensities to save out of profits and wages and

on income distribution: an increase in the average saving

propensity could be the result of a rise in the propensities

to save out of wages and/or profits or of a more favorable

distribution to profits (rather thanwages), out to which is

saved a greater proportion (than out of wages).

In the absence of constraints on productive capac-ity building, firms investment decisions are modeled via

a linear investment function according to which the

(desired) capital accumulation, g – corresponding to the

ratio between investment, I , and the stock of capital – is

determined as follows:

I

K = g = ˛+ ˇ(u−u), (2)

with˛,ˇ >0.u representsthe desired or “normal”degreeof capacity utilization,3 interpreted as the degree of capacity

3

Notethat theinvestment function(2) canbe mademoresimilar tothelinear version of Joan Robinson’s (1962) standard investment function,


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14 P. Commendatore et al. / Structural Change and Economic Dynamics 28 (2014) 12–24

utilization that firms set as their long-run objective. More-

over, we assume g (0)>0.

The short-run equilibrium condition is

I

K = S

K . (3)

The solutions are:

u∗ = ˛− ˇu − ˇ , g

∗ = ˛− ˇu − ˇ . (4)

We have that, 0


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Fig. 1. Region A is related to short and long-run instability; region B to

short-run stability but long-run instability; region C 1 to short and long-

runstabilitywith(long-run)fluctuations;and regionC 2 toshort andlong-

run stability withmonotonic convergence.

and short-run stationary equilibrium values of the degree

of capacity utilization. It is unique and corresponds to

u∗lr =˛

, g ∗lr = ˛. (7)

If the equilibrium state represented by expression (7)

prevails, the economy is characterized by a stagnation-

ist regime (see Bhaduri and Marglin, 1990), according to

which capacity utilization is inversely related to the sav-

ing propensity of the economy (and to the profit share).

Instead, the rate of growth is exogenously determined by

the autonomous component of the investment function.

Therefore, the so-called paradoxes of thrift and costs onlyapply to the degree of capacity utilization.

For >ˇ, the stability of u∗lr requires

ˇ < ˇcrit ≡ (2− )

2 , (8)

where theconvergence ismonotone forˇ


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Fig. 2. Impactof theconstraint on the investment function g given in (9)

as is varied. Here ˛=0.15,ˇ=0.24, ı=0.05, =0.25 andu = 0.6.could reduce the pace of capital accumulation. The

possibility of increasing adjustment costs of pro-

ductive capacity has been firstly introduced in a

neoclassical model of growth by Uzawa (1969), even

though in the special case of smooth convex adjust-

ment costs; whereas Asada (2001, 2006) applies this

concept within post Keynesian models of the growth

cycle.8 According to the theoretical and empirical lit-erature – starting from the seminal contribution by

Abel and Eberly (1994) – which explores the rela-

tionship between adjustments costs and the shape

of the investment function (see Honda and Suzuki,

2008, for a review), different types of nonlinearities

could follow from the existence of thresholds that

modify abruptly – that is, non-smoothly – the way

in which firms adjust their productive capacity. These

thresholds can be justified by the existence of a finan-

cial constraint (see Fazzari et al., 1988; Hoshi et al.,

1991; Honda and Suzuki, 2006) or by the presence

of increasing adjustment costs. Basically, this follows

from the fact that fixed investment is characterized bylumpiness,irreversibilityandmoregenerallybyasym-

metric adjustment costs (which involve additional,

sometimes even prohibitive, costs above a certain

threshold).9

Fig. 2 shows the impact on investment decisions of the

constraint as is varied.10

Given theshape of the investment function (9), twodif-

ferent types of stationary equilibrium solution can emerge

depending on parameters (see Fig. 3, where we have

8 SeealsoSkott (1989, 2010)andSchoder(2012)whoalsoacknowledge

the possibility of asymmetries in the adjustment process of the capital

stock in order to justify a nonlinear investment functionin thecontext of

post-Keynesiangrowthmodels.9 The relationship explored by these authors is between the rate of

investmentand Tobin’sq. However,bothcapacityutilizationandq contain

thesame informationrelativelyto firm’s investmentdemandandtechnol-

ogy, thus theymay be considered as alternative indicators of investment

behavior (seeKim, 2005).10 Referring back to footnote 3, the investment function (9) can also

be interpreted as a piecewise linear approximation of Joan Robinson’s

(1962) standard concave investment function, represented in the well

known “banana diagram”. Indeed, we could have assumed for our analy-

sis a smooth convex nonlinear function alternative to (9) – that can also

generate complex dynamic behavior. However, such a function does not

allow,except forspecialcases, forexplicitshort-runequilibriumsolutionsand therefore ismuch less amenable to analytical treatment.

Fig. 3. The investment function g given in (9) at =0.05 and examples

of two types of equilibrium for twodifferent values of : unconstrained

equilibrium at =0.22 and constrained oneat =0.3 (the other parame-

tersareas in Fig. 2).

considered two different values of ): (i) an unconstrainedequilibrium,

ua = ˛− ˇu − ˇ , (10)

which is identical to that identified in theprevious section;

and (ii) a constrained equilibrium,

ub = ˛− u+ ı(1− /ˇ) − , (11)

that also depends on ı and .More specifically, let g 1(0)>0and ≤ˇ ˇ and g 1(0) = ˛−ˇu > 0 are a sufficient condition for a positive short-runequilibrium,u∗ >0;whereas a sufficientcondition foru∗ ˇ, whereas a necessarycondition is > . Put it in other terms, when the equilib-rium ua prevails the stability condition is >ˇ, whereaswhen the equilibrium ub prevails it is > (this allowingfora quantity adjustmentmechanismthat goes in theright

directionwitha sufficiently lowspeed; seeCommendatore,

2006).

11 The condition ˇ


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Finally, we compute the following partial derivatives:

∂u∗

∂ = − u

b

− < 0 for 0 ≤ u ≤ u,∂u∗

∂ = − u

a

− ˇ < 0 for u <

u ≤ 1,

∂ g ∗∂

= − ub

− < 0 for 0 ≤ u ≤ u,∂ g ∗

∂ = − ˇu

a

− ˇ < 0 for u < u ≤ 1,that confirm the paradoxes of thrift and costs.

It is also interesting to verify the effects of ̌ , and ı onthe degree of capacity utilization and on the rate of capital

accumulation:

∂ua

∂ˇ= ˛− u( − ˇ)2

≥()˛ ;

∂ g a

∂ˇ= ∂u

a

∂ˇ≥() ̨ ;

∂ua

∂ = 0, ∂ g

a

∂ = 0, ∂u

a

∂ı= 0, ∂ g

a

∂ı= 0.

Moreover, letting /= ˇ, we have that

∂ub

∂ˇ= ı

ˇ2( − ) > 0, ∂ g b

∂ˇ= ∂u

b

∂ˇ> 0,

∂ub

∂ =

˛+ ı1−

ˇ

−

u

( − )2

≥0, ∂ g b

∂ = ∂u

b

∂ ≥0,

∂ub

∂ı= ˇˇ−

− ≥0, ∂ g b

∂ı= ∂u

b

∂ı≥0.

In general, therefore, less stringent financial and tech-

nological conditions have a positive (or no) effect on the

short-run equilibrium values of the degree of capacity uti-

lization and of investment demand. The effect of ˇ on u∗

and g ∗ is not univocal when the constraint is not binding(confirming the result in the previous section); whereas it

is positive when firms’ investment demand is limited by

financial, technical or behavioral factors.

3.2. The economy in the long run

3.2.1. The basic dynamic equation

The dynamic law that governs the evolution of the nor-

mal degree of capacity utilization in the presence of a

constraint is obtained by substituting (12) into (5):

F (u) = (1− )− − u+ [˛+ ı(1− /ˇ)] − for 0 ≤ u ≤ u,

F (

u) = (1− )− ˇ

− ˇ

u+ ˛

− ˇ for u <

u ≤ 1,

(14)

where u is given in (13).

Fig. 4. The function F (u) defining the skew tent map given in (14). Here˛=0.15,ˇ=0.22, =0.1, ı=0.05,=0.4, =0.25.

Inorder to simplify theanalysis,wereformulateexpres-

sion (14) as follows:

f ( x) = f L( x) = ax+ ε1 for 0 ≤ x ≤ xc f R( x) = bx + ε2 for xc < x ≤ 1 (15)where

a = (1− )− − , b =

(1− )− ˇ − ˇ , xc = u, (16)

ε1 =[˛+ ı(1− /ˇ)]

− , ε2 =˛

− ˇ . (17)

When the condition

1− < < ˇ

1− (18)

is applied,12wehavea>0andb the inequality a (ˇ(˛− ))/( −ˇ), xc 1when ı< (ˇ(˛− ))/( −ˇ). Then the constraint is always binding and no

long run stationary solution exists. Note also that the latter inequality isnever satisfied as long as ˛


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3.2.2. Stationary equilibrium and local stability analysis

Sinceˇ


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Fig. 6. 1Dbifurcationdiagramof themap(14) asˇ varies in interval [0.22,

0.25] ; the other parameters are fixed: ˛=0.15, =0.15, ı=0.05, =0.2,

=0.25. SeeAppendix f or the description of particular bifurcations and

related values of ̌ .

Fig. 6 presents a one-dimensional (1D for short) bifur-

cation diagram illustrating how the long-term behavior of u (or x) changes asˇ varies.In particular, concerning the local stability of the sta-

tionary equilibrium, notice that at

ˇ = ˇcrit DFB1a so-called degenerate flip bifurcation16 (DFB for short)

takes place characterized by the condition b=−1. Imme-diately after theDFBa period 2-cycle originates, that is, the

long-term behavior of the economy is characterized by a

fluctuation between two values obtained from f 2( x) = x :

x2,1 =aε2

+ε1

1− ab and x2,2 =bε1

+ε2

1− ab . (23)Moreover, as ˇ is increased further more complex

behavior occurs. Fig. 7 presents, instead, a 2D bifurcation

diagram showing the long-term behavior of u as ˇ and varies simultaneously (Fig. 6 representing a movementalong thevertical arrowdrawn inFig. 7 starting at =0.15).As shown in Fig. 7, does not affect the local stabilitypropertiesof thestationary equilibrium.Indeed, theborder

ˇ =ˇcrit betweentheorangeandgreen region, whichcorre-sponds to the condition (20), is a horizontal line. However,

abovesuchalinealso playsa crucial role forthe long-termdynamics.

The 2-cycle which emerges at ˇ =ˇcrit is stable as long

as its eigenvalue satisfies the condition |ab|0 are

observable in the ( , ˇ)-parameter plane shown in Fig. 7.One can compare itwith a more standard (a, b)-parameter

plane, where a and b are slopes of the linear branches of

the skew tent map: in such a case one has to introduce

some artificial scaling to study the complete bifurcation

structure. This is not necessary in our analysis due to the

particular constellation of the parameters of the model

which are naturally scaled.

3.2.4. Some comparative dynamics results

As mentioned before, the stationary long-run equilib-

riuminvolves a non-binding constraint: u∗lr =˛/ and g ∗

lr =˛.

In such an equilibrium state the degree of capacity utiliza-

tion is inversely related to the average saving propensity;

whereas the rate of capital accumulation corresponds to

the autonomous component of the investment function.

Moreover, this equilibrium is stable forˇ ˇcrit , the economy may not be traveling along a

stationary growth path, but it could settle down on a peri-

odic or chaotic motion. In order to evaluate the impact of

parameter changes along a non stationary long-term evo-

lution, it ismore suitable toconsideraverage values. Notice

that following from theadaptive adjustment hypothesis in


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Fig. 8. The function f ( x) at ˛=0.15, ˇ=0.232, ı=0.05, =0.25, =0.2. To compare: =0.15 in(a)and =0.19 in (b).

(5) and the properties of means, the average values of the

normal andof the short-run stationary equilibrium degree

of capacity utilization are equal.

To begin with, we study more in detail the effects of

parameter changes on the stable 2-period cycle, then wepresent somesimulations inorder togeneralize ourresults.

Let x2,1 and x2,217(see (23)) the periodic points of the sta-

ble 2-cycle for the degree of capacity utilization. In order

to assess the impact of the constraint, we evaluate first

the effect of a change of on the average values of thenormal degree capacity utilization, uM , and rate of capital

accumulation, g M , along the cycle, that is:

uM (2) =1

2

2i=1

x2,i =(1+ a)ε2 + (1+ b)ε1

2(1− ab) and

g M (2) = uM (2)

Wehavethat∂uM (2)/∂ =−ı/ˇ ((ˇ−ˇcrit )/(ˇ−ˇdiv))2


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Fig. 9. Theaverage values of u∗ (a) and g ∗ (b) shownby solid lines, and their stationary equilibrium values (dashed lines). Here varies in [0, 0.2] and theother parameters are fixed:˛=0.15,ˇ =0.23,ı=0.05, =0.25,=0.2.

Fig. 10. The average values of u∗ (a) and g ∗ (b) shownby solid lines, and their stationary equilibrium values (dashed lines). Here varies in [0, 0.2, 0.28]and theother parameters arefixed: ˛=0.15,ˇ=0.23, ı=0.05, =0.15,=0.2.

Similarly, other simulations (not presented here) con-firm, for a range of values beyond the stable 2-cycle, the

negative impact of an increase in ˇ on capacity utilizationandaccumulation.Further,alsofor thisparameterto higher

values correspond wider fluctuations of u∗ and g ∗.Fig. 10, instead, shows that the paradoxes of thrift

and costs may or may not hold in the long-run. Indeed,

for sufficiently small, the degree of capacity utilizationincreases with this parameter; whereas above a certain

value the degree of capacity utilization begins to decrease

with . On the other hand, capital accumulation alwaysincreaseswhenever the stationary equilibrium is unstable.

Moreover, increasing has also a stabilizing effect on the

dynamics.These pictures confirm the results obtained for the sta-

ble 2-cycle.

4. Final remarks

Inthispaperwehaveseenthattheintroductioninasim-

ple and quite typical post-Keynesian/Kaleckianmodel of a

constraint on investment decisions– that couldbe induced

byfinancial,technical andbehavioral factors –hassubstan-

tially enriched the long-run behavior. We presented both

analytical and simulation results which provide a com-

plete taxonomy of the possible long-run growth scenarios

depending on parameter values.

Concerning theconstraint: in theshort-runit canhave anegative impacton thestationary equilibriumvaluesof the

degree of capacity utilization and of investment demand.

In the long run, if the system converges to the stationary

growth path the constraint has no effect and the standard

results prevail (see Amadeo, 1986; and Lavoie, 1995). The

normaland theeffectivedegree of capacity utilization (the

latteremergingas a short-term stationary equilibrium) are

alwaysequalandinverselyrelated totheoverall propensity

to save of the economy and the rate of capital accumula-

tion isequalto theexogenouscomponentof theinvestment

function;however, if the economy is characterized by per-

sistent regular or chaotic fluctuations, the normal and the

effective degree of capacity utilization do not necessarilycoincide, even though their average values are equal, that

is,onaverage, firms’ investment plansarerealized;andthe

rateof capital accumulation becomes endogenousdepend-

ing on the degree of capacity utilization and, with a given

income distribution, on profitability.19 However, both the

average degree of capacity utilization and rate of capital

accumulation lie below their stationary equilibrium value.

Relaxing the constraint, moreover, worsens the economic

conditions by widening the fluctuations of the economy

19 Giventheprofit share,, r M =aK uM is therate ofprofit corresponding

to average capital accumulation. Thus, the average rate of growth canbeexpressed as g M = saK uM = r M /.


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and reducing capital utilization ad accumulation. An intu-

ition of these results can be grasped considering the effect

of a relaxation of the constraint along a two-period cycle,

referringto theanalysispresentedinSection3.2.4, by com-

paring the short-run equilibrium values of the relevant

variables for a lower and a higher constraint. We denote

by (u1, u2) and (u1, u

2) the short-run equilibrium values

of the degree of capacity utilization along the two 2-cycles

with a relatively lowandhigh constraint, respectively, and

by (u1,u2) and (u1,u2) the corresponding normal values:(i) looking first at the 2-cycle with a lower constraint,

we choose as initial state the end of period 2, when

the economy is not constrained: the current and nor-

mal degree of capacity utilization are equal to u2 andu2, respectively. During period 1, when the economyis constrained, since the realized degree of capacity

utilization in the previous period is below the previ-

ouslyanticipated normal one,u2 <

u2, according tothe

mechanism in (5), firms revise downwards their con-

ception of the normal degree of capacity utilization,u1 < u2; this determines a higher degree capacity uti-lization(andhigher investmentdemand)thatis limited

by thepresenceof theconstraint,u1 >u2; duringperiod

2, the current degree of capacity utilization is now

abovethenormaloneu1 > u1 andfirmsreviseupwardsthenormal degreeof capacityutilization forthefollow-

ingperiod,u2 > u1. It followsa lowerdegreeof capacityutilization (andlowerinvestmentdemand),u2 ˇcrit (see Figs. 6 and 7). As we have shown inSection3.2.3, at ˇ =ˇ4− A given in (24) the DFB of the 2-cycle occurs. As soon as ˇ crosses ˇ4- A, the period 2-cycleloses stability and a period 4-cycle is originated. How-

ever, the eigenvalue of the 4-cycle is (ab)2 >1 (because

ab


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correspondsto afirsthomoclinicbifurcation20 of theunsta-

ble 2-cycle which occurs if the following implicit equation

is satisfied:

a2b3 − b+ a = 0 H 2

The next bifurcation occurring by increasing ˇ, whentwopieces of theattractormergesintoaonebandattractor,is due to the first homoclinic bifurcation of the fixed point.

The related curve is obtained from the condition f 3( xc ) = x∗

that holds at

b = (−1− 1+ 4a2)/2a H 1

It has been shown for skew tent maps (see, for exam-

ple, Maistrenko et al., 1993) that, by increasing further ˇ,cycles having period k≥3 come in pair, denoted qk and qk ,one stable and one unstable. In Fig. 6 windows originated

at ˇ=ˇ3 and ˇ =ˇ4 are clearly visible, these are related tostable 3-cycle q3 and 4-cycle q4, respectively (the related

unstable cycles q3 and q4 are not shown). A periodic point

denoted xk,s1 of a stable k-cycle is obtained by solving the

equation f k−1L ◦ f R( xk,s1 ) = xk,s1 , fromwhich

xk,s1 =(1− ak−1)/(1− a)ε1 + ak−1ε2

1− ak−1b.

For example, a periodic point belonging to the stable 3-

cycle is given by

x3,s1 =(1+ a)ε1 + a2ε2

1− a2b .

A periodic point denoted xk,u1 of an unstable k-cycle is

obtained by solving the equation f k−2L ◦ f 2R ( xk,u1 ) = x

k,u1 from

which

xk,u1 =(1− ak−2)/(1− a)ε1 + ak−2ε2

1− ak−2b2 .

For the case k=3 we have that

x3,u1 =ε1 + a(1+ b)ε2

1− ab2 .

20 A first homoclinic bifurcation of an unstable equilibrium (or cycle)

occurswhenitsunstable setcontactsits stableset(which isa setof preim-

ages of the fixed point). The equilibrium (or cycle) becomes a so-called

snap-back repellor. It is a global bifurcation of the map f since it involves

general properties and it is not confined to a neighborhood around to a

stationary equilibrium or to a periodic equilibrium (for details see, e.g.,Gardini et al., 2011).

A so-calledbordercollisionbifurcation21 (BCBfor short)

occurswhen xk,u1 = xk,s1 = xc . For the casek=3wehave that

the related curve denoted BCB3 is given by

ˇ = ˇ3 =3( − )− ( (3− )+ )

3( − )− 2 BCB3

Therefore as theBCB3 curve is crossed a pair of 3-cycles

emerges, onestableandoneunstable,q3 and q3. More gen-erally, a pairof k-cycles,withk≥3, qk and qk, emerge as thecurve BCBk is crossed (note that in Fig. 7 all the curves have

been derived analytically).

The stable k-cycle qk loses stabilitywhen its eigenvalue

becomes smaller than −1. The eigenvalue of qk is

z k = ak−1b.

A DFB of qk occurs when z k =−1, that is, at b=−1/ak−1 . Thecorresponding curve denoted DFBk is given as

ˇ=ˇ2k

− A

= 1 + (1− )(( (1− )− )/( − ))k−1

1+ (( (1− )− )/( − ))k−1 DFBk

The DFB of qk results in the appearance of a 2k-band

chaotic attractor (in other terminology, 2k-cyclic chaotic

intervals). For k=3we have that b=−1/a2 and therefore

ˇ = ˇ6− A =1+ (1− )[( (1− )− )/( − )]2

1+ [( (1− )− )/( − )]2 DFB3

So, the3-cycle is stablewithintheintervalˇ3


13/13


of the cycle qk. This bifurcation occurswhen the following

equation is satisfied (the related curve is denoted H k):

ak−1b2 + b− a = 0 H kTo summarize, for the parameter range chosen to plot

Fig. 6 (see also the vertical line starting from =0.15 inFig. 7) in the meaningful parameter regionwe observe the

following cascade of bifurcations for increasing ˇ:

x∗DFB1⇒ q2

DFB2⇒ 4− AH 2⇒2− AH 1⇒1− ABCB3⇒

q3DFB3⇒ 6− AH 3⇒3− A

H 3⇒1− ABCB4⇒

q4DFB4⇒ 8− AH 4⇒4− A

H 4⇒1− A.

Note that changing other parameters of the model, it may

happen that higher periodicity regions fall into the feasi-

ble region. Then a longer chain of the bifurcations can be

observed.

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