r0297.dviJournal of Computing and Information Technology - CIT 16,
2008, 4, 255–269 doi:10.2498/cit.1001393
255
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
Felix Breitenecker1, Florian Judex1, Nikolas Popper2, Katharina
Breitenecker3, Anna Mathe3 and Andreas Mathe3
1Institute for Analysis and Scientific Computing, Vienna University
of Technology, Vienna, Austria 2Die Drahtwarenhandlung’ Simulation
Services, Vienna, Austria 3ARGESIM – Vienna University of
Technology, Vienna, Austria
Laura, a very beautiful but also mysterious lady, inspired the fa-
mous poet Petrarch for poems, which express ecstatic love as well
as deep despair.
F. J. Jones – a scientist for literary work – recognized in these
changes between love and despair an os- cillating behaviour – from
1328 to 1350 – which he called Petrarch’s emotional cycle.
The mathematician S. Rinaldi investigated this cycle and
established a mathematical model based on ordinary differential
equation: two coupled nonlinear ODEs, reflecting Laura’s and
Petrarch’s emotions for each other, drive an inspiration variable,
which coincides with Petrarch’s emotional cycle.
These ODEs were the starting point for the investi- gations in two
directions: mapping the mathematical model to a suitable modelling
concept, and trying to extend the model for love dynamics in modern
times (F. Breitenecker et al.). This contribution introduces and
investigates a modelling approach for love dynamics and inspiration
by means of System Dynamics, for Laura’s and Petrarch’s emotions as
well as for a modern couple in love. In principle, emotions and
inspiration emerge from a source and are fading into a sink. But
the controlling parameters for increase and decrease of emotion
create a broad variety of emotional behaviour and of degree of
inspiration, because of the nonlinearities.
Experiments including an implementation of this model approach and
selected simulations provide interesting case studies for different
kinds of love dynamics - attrac- tion, rejection and neglect –
stable equilibria and chaotic cycles.
Keywords: love dynamics, limit cycles, dynamical sys- tems, system
dynamics
1. Introduction
Dynamic phenomena in physics, biology, eco- nomics, and all other
sciences have been ex- tensively studied with differential
equations, since Newton introduced the differential calcu- lus. But
the dynamics of love, perhaps the most important phenomenon
concerning our lives, has been tackled only very rarely by this
cal- culus. In literature, two special contributions can be
found:
• Love Affairs and Differential Equations by S. H. Strogatz ([1]),
– harmonic oscillators making reference to Romeo and Juliet,
and
• Laura and Petrach: an Intriguing Case of Cyclical Love Dynamics
by S. Rinaldi ([2]) – presenting a nonlinear ODE with cyclic
solutions.
Based on these papers, recent work has been done, and is done, by
embedding the mathe- matical concepts of Rinaldi into modelling
con- cepts like Transfer Functions (Breitenecker et al., [3]) and
System Dynamics (this paper), and by generalizing the model for
modern times.
Section 2 presents the classification of Petrarch’s poems by F.
Jones, introducing Petrarch’s emo- tional cycle. Sections 3
summarises the ODE model approach for the Laura-Petrarch
model
256 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
by S. Rinaldi. Section 4 introduces System Dy- namics as
appropriate modelling approach for the underlying dynamics, giving
insight into the principle model structure and preparing a basis
for a model extension. Section 5 shows iden- tification results for
the Laura-Petrarch model based on a MATLAB implementation with a
GUI (MATLAB graphical user interface).
Section 6 presents analytical analysis for the Laura-Petrarchmodel
aswell as numerical anal- ysis and experiments with this
model.
Section 7 extends Laura-Petrarch Model to a general model for the
dynamic love cycle – the Woman-Man Model, perhaps also applicable
for modern times and modern love affairs, Section 8 concludes with
experiments with the Woman- Man model, showing interesting case
studies with changing appeal, etc.
2. Classification of Petrarch’s poems
Francis Petrarch (1304–1374), is the author of the Canzoniere, a
collection of 366 poems (son- nets, songs, sestinas, ballads, and
madrigals). In Avignon, at the age of 23, he met Laura, a
beautiful, but married lady. He immediately fell in love with her
and, although his love was not reciprocated, he addressed more than
200 poems to her over the next 21 years. The po- ems express bouts
of ardour and despair, snubs and reconciliations, making Petrarch
the most lovesick poet of all time.
Unfortunately, only a few lyrics of the Can- zoniere are dated. The
knowledge of the cor- rect chronological order of the poems is a
pre- requisite for studying the lyrical, psychologi- cal, and
stylistical development of Petrarch and
Figure 1. Portraits of Laura and Petrarch, from Biblioteca Medicea
Laurenziana, ms. Plut. cc. VIIIv- IX, Florence, Italy (courtesy of
Ministero per i Bene
Culturali e Ambientali), from [2].
Figure 2. Portraits of Laura and Petrarch, from Internet
resources.
his work. For this reason, identification of the chronological
order of the poems in the Can- zoniere has been for centuries a
problem of ma- jor concern for scholars.
In 1995, in his book The Structure of Petrarch’s Canzoniere ([4]),
Frederic Jones presented an interesting approach and solution to
the chrono- logical ordering problem of Petrarch’s poems.
Jones concentrated on Petrarch’s poems written at lifetime of Laura
(the first, sonnet X, was written in 1330 and the last, sonnet
CCXII, in 1347). First, he analysed 23 poems with fairly secure
date. After a careful linguistic and lyrical analysis, he assigned
grades to the poems, rang- ing from -1 to +1, establishing
Petrarch’s emo- tional cycle: the maximum grade (+1) stands for
ecstatic love, while very negative grades cor- respond to deep
despair. The following exam- ples (in quotations, the English
version is taken fromanEnglish translation of theCanzoniere by
Frederic Jones) illustrate some of these grades:
• Sonnet LXXVI, great love, grade +0.6:
Amor con sue promesse lunsingando, mi ricondusse alla prigione
antica [Love’s promises so softly flattering me have led me back to
my old prison’s thrall.]
• Sonnet LXXIX, great despair, grade -0.6:
Cosi mancando vo di giorno in giorno, si chiusamente, ch’i’ sol me
ne accorgo et quella che guardando il cor mi strugge. [Therefore my
strength is ebbing day by day,
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 257
which I alone can secretly survey, and she whose very glance will
scourge my heart.]
• Sonnet CLXXVI, melancholy, grade -0.4:
Parme d’udirla, udendo i rami et l’ore et le frondi, et gli augei
lagnarsi, et l’acque mormorando fuggir per l’erba verde. [Her I
seem to hear, hearing bough and wind’s caress, as birds and leaves
lament, as murmuring flees the streamlet coursing through the
grasses green.]
Displaying the grades over time (Figure 3), F. Jones detected an
oscillating behaviour, which he called Petrarch’s emotional cycle
E(t), with a period of about four years.
In the second step, F. Jones analysed and ’graded’ all the other
poems of unknowndate and checked, into which part of the cycle they
could fit. Tak- ing into account additional historical informa-
tion, he could date these poems by locating them in Petrarch’s
emotional cycle E(t), see Figure 3.
Figure 3. Petrarch’s emotional cycle E(t) - dashed line, with
’graded’ poems (crosses for securely dated poems,
and circles for poems dated based on the emotional cycle).
3. Mathematical Modelling Approach
The big challenge for S. Rinaldi was to set up an ODE model for the
cyclic love dynamics and emotional dynamics of Petrarch and Laura,
which would fit the experimentally founded emotional cycle of
Petrarch.
The basis for modelling is a predator-prey – type dynamics between
the variables L(t) and P(t). L(t) represents Laura’s love for the
poet at time t; positive and high values of L mean warm
friendship, while negative values should be as- sociated with
coldness and antagonism. P(t) describes Petrarch’s love for Laura,
whereby high values of P indicate ecstatic love, while negative
values stand for despair. The general Laura-Petrarch model is given
by:
dL(t) dt
= −αPP(t) + RP(L(t)) + βPAL.
RL(P) and RP(L) are reaction functions to be specified later, and
AP and AL is the appeal (physical, as well as social and
intellectual) of Petrarch and Laura to each other.
Next, it is necessary to model in more detail the complex
personality Petrarch. A second variable has to be used to describe
his poetic in- spiration IP(t), expressing his productivity for
poems (in general modelling the inspiration for work related to the
love), and influencing the appeal AL:
dL(t) dt
1 + δPIP(t) dIP(t)
dt = −αIPIP(t) + βIPP(t).
The rate of change of the love of Laura L(t) is the sum of three
terms. The first describes the forgetting process characterizing
each individ- ual. The second term RL(P) is the reaction of Laura
to the love of Petrarch, and the third is her response to his
appeal. The rate of change of the love of Petrarch P(t) is of
similar struc- ture, with an extension: the response of Petrarch to
the appeal of Laura depends also upon his inspiration IP(t). This
takes into account the well-established fact that high moral
tensions, like those associated with artistic inspiration,
attenuate the role of the most basic instincts.
And there is no doubt that the tensions between Petrarch and Laura
are of a passionate nature, as can be read in literature.
• In Sonett XXII, Petrarch writes:
Con lei foss’io da che si parte il sole, et non ci vedess’ altri
che le stelle, sol una nocte, et mai non fosse l’alba.
258 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
[Would I were with her when first sets the sun, and no one else
could see us but the stars, one night alone, and it were never
dawn.]
• And in his Posteritati, Petrarch confesses:
Libidem me prorsus expertem dicere posse optarem quidem, sed si
dicat mentiar. [I would truly like to say absolutely that I was
without libidinousness, but if I said so I would be lying].
The equation for the inspiration IP(t) says that the love of
Petrarch sustains his inspiration which, otherwise, would
exponentially decay. The reaction functions RL(P) and RP(L) are
partly nonlinear. A linear approach would sim- ply say that
individuals love to be loved and hate to be hated.
The linearity of RP(L) is more or less obvious, since in his poems
the poet has very intense re- actions to the most relevant signs of
antagonism from Laura: RP(L(t)) = βP · L(t).
A linear reaction function is not appropriate for Laura. Only
around RL(P) = 0 it can be as- sumed to be linear, thus
interpreting the natural inclination of a beautiful high-society
lady to stimulate harmless flirtations. But Laura never goes too
far beyond gestures of pure courtesy: she smiles and glances.
However, when Petrarch becomes more de- manding and puts pressure
on her, even indi- rectly when his poems are sung in public, she
reacts very promptly and rebuffs him, as de- scribed explicitly in
a number of poems.
• In Sonnet XXI, Petrarch claims:
Mille fiate, o dolce mia guerrera, per aver co’ begli occhi vostri
pace v’aggio proferto il cor; ma voi non piace mirar si basso colla
mente altera. [A thousand times, o my sweet enemy, to come to terms
with your enchanting eyes I’ve offered you my heart, yet you
despise aiming so low with mind both proud and free.]
Consequently, the reaction functionRL(P) should, for P > 0,
first increase, and then decrease. But the behaviour of Laura’
reaction is also nonlin- ear for negative values of P. In fact,
when P < 0 (when the poet despairs), Laura feels very sorry for
him. Following her genuine Catholic ethic
she arrives at the point of overcoming her antag- onism by strong
feelings of pity, thus reversing her reaction to the passion of the
poet. This be- havioural characteristic of Laura is repeatedly
described in the Canzoniere.
• In Sonnet LXIII Petrarch writes:
Volgendo gli occhi al mio novo colore che fa di morte rimembrar la
gente, pieta vi mosse; onde, benignamente salutando, teneste in
vita il core. [Casting your eyes upon my pallor new, which thoughts
of death recalls to all mankind, pity in you I’ve stirred; whence,
by your kind greetings, my heart to life’s kept true.]
A good choice for Laura’s reaction function RL(P) would be a cubic
function of the fol- lowing type, displayed in Figure 4.
RL(P) = βLP · (
.
Figure 4. Nonlinear reaction function RL(P) of Laura – grey line,
and linear reaction function RP(L) of Petrarch
– straight black line.
With these approaches, the full Laura-Petrarch model is given by
the following equations:
dL(t) dt
= −αLL(t) + βLP
1 + δPIP(t) dIP(t)
dt = −αIPIP(t) + βIPP(t).
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 259
4. Model Approach by System Dynamics
System Dynamics (SD) is a well known mod- elling approach,
introduced by J. Forrester. SD is a methodology for studying and
managing complex feedback systems, such as one finds in business
and other social systems. In fact it has been used to address
practically every sort of feedback system. Feedback refers to the
situa- tion of X affecting Y and Y in turn affecting X, perhaps
through a chain of causes and effects. One cannot study the link
between X and Y and, independently, the link between Y and X, and
predict how the system will behave. Only the study of the whole
system as a feedback system will lead to correct results.
For modelling, SD starts at qualitative level with a causal loop
diagram. A causal loop diagram (CLD) is a diagram that aids in
visualizing how interrelated variables affect one another.
The CLD consists of a set of nodes representing the variables
connected together. The relation- ships between these variables,
represented by arrows, can be labelled as positive or negative.
There are two kinds of causal links, positive and negative.
Positive causal links mean that the two nodes move in the same
direction, i.e. if the node in which the link starts decreases, the
other node also decreases. Similarly, if the node in which the link
starts increases, the other node increases. Negative causal links
are links in which the nodes change in opposite direc- tions (an
increase causes a decrease in another node, or a decrease causes an
increase in another node).
Figure 5. Causal loop diagram between Laura’s and Petrarch’s
emotions and Petrarch’s inspiration.
In case of Laura’s and Petrarch’s emotions L(t) and P(t), and
Petrarch’s inspiration IP(t), the causal relation is evident
(Figure 5), but it is not possible to identify clearly positive and
negative links.
SD continues the modelling process now at the quantitative level by
a stock and flow diagram – (SFD), sometimes also called level and
rate diagram. A stock variable is measured at one specific time. It
represents a quantity existing at a given point in time, which may
have been accumulated in the past. A flow variable is mea- sured
over an interval of time. Therefore a flow would be measured per
unit of time. The vari- ables in the CLD must be identified either
as stock (level) or flow (rate) – or as auxiliary, and each stock
(level) is connected in the SFD with its inflow – coming from a
source – and by its outflow to a sink; flows are represented by
double arrows and flow-controlling valves. The causal links from
the CLP are found in the SFD as characterizing influences from
stocks to flows (or from parameters and auxiliaries to
flows).
Figure 6. Qualitative stock and flow diagram for Laura’s and
Petrarch’s emotions and for Petrarch’s inspiration.
For the dynamics of emotion and inspiration under investigation,
all L(t), P(t), and IP(t), are considered as stocks, whereby the
adjoin flows are the changes (derivatives). A first sim- ple SFD
(Figure 6) for these dynamics shows similar structures for all
three variables L(t), P(t), and IP(t). They are emerging from a
source flow, influenced by other stocks, and they are fading to a
sink (outflow) controlled by the stock itself. The stocks’ feedback
to the outflow let them ’converge’ to zero (stabilizing
260 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
feedback), if no inflow is driving emotions and intuition.
SD’s modelling procedures now quantifies the SFD by introducing
parameters and auxiliaries for the causal links and for the
influences for the flows. Laura’s and Petrarch’s emotions and
Petrarch’s intuition are fadingwith certain celer- ity,
characterised by the parameters αL, αP, and αIP in the direct
feedbacks. And, in principle, also the feedback from Laura’s
emotion L(t) to Petrarch’s emotion P(t) and vice versa is given by
the parameters βP and βL, and the feedback from Petrarch’s emotion
P(t) to Petrarch’s in- spiration IP(t) by the parameter βIP. The
appeal parameters AL and AP influence directly the in- flow for the
stocks P(t) and L(t) – indicated by additional control inputs for
the respective inflows (summarized with the feedbacks from other
stocks).
The SFD for the dynamics of emotion and in- spiration in Figure 7
shows all basic feedbacks and direct inputs for the flows, weighted
with parameters, but neglecting the nonlinear reac- tion of Laura,
and the inspiration’s reciprocal feedback to Petrarch’s emotion
(only ’approx- imated’ by a classic feedback with parameter nature
γL, which may be seen as linearization of the reciprocal
feedback).
The model presented in the SFD in Figure 7 is still a simplified
(linear)model, but the SFD can directly generate the system
governing ODEs, which might be by used by any simulator.
This automatic generation of the ODE model is the last step in the
SD modelling procedure. In principle, the SFD balances stocks and
flows, so that for the simplified model the following ODEs are
generated:
dL(t) dt
dIP(t) dt
= −αIPIP(t) + βIPP(t).
Of course, the above equations are linear ODEs, and finally the
nonlinearities must be modelled in the FSD. For this purpose, SD
makes use of auxiliaries, which define the nonlinear nature of a
feedback. Auxiliaries may have more than one input, and the output
may be fed back into
Figure 7. Stock and flow diagram for Laura’s and Petrarch’s
emotions and for Petrarch’s inspiration with
linear influences.
Figure 8. Stock and flow diagram for Laura’s and Petrarch’s
emotions and for Petrarch’s inspiration with
nonlinear reactions.
many different flows, or – if necessary, into fur- ther
auxiliaries. Figure 8 presents the full SD model with all nonlinear
reactions. The non- linearities are of different quality. Choosing
in the nonlinear cubic-like gain for Laura’s reac- tion RL(P) a big
value for the parameter γL, the nonlinear auxiliary becomes almost
linear (the nominator is bounded, usually less than 1). The
nonlinear auxiliary for Laura’s appeal AL be- comes linear if the
parameter δP is set to zero, letting the influence of Petrarch’s
poetic inspi- ration vanish.
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 261
As with the simplified model in Figure 7, the ODE system is
generated automatically from the FSD in Figure 8, to be used by a
simula- tor, to be used directly in the SD modelling and simulation
environment.
It can be concluded, that System Dynamics is a very appropriate
tool for modelling the emo- tional dynamics under investigation –
again it is underlined, that SD offers big benefits for modelling
social systems, because relations and influences can be modelled
step by step, from qualitative causal links to quantitative stock
and flow diagrams, which can be simulated directly.
Finally it should be remarked, that SFDs can be directly mapped to
block diagrams with trans- fer functions: stock and output flow
with direct feedback of stock correspond to a first-order transfer
function, with input represented by the input flow (mapped on a sum
block); parame- ters in feedback links are mapped to gain blocks,
auxiliaries nonlinear blocks (for details on ap- proach by transfer
functions see [3]).
5. Identification of the Laura-Petrarch Model
The big challenge is to identify the model pa- rameters in the
nonlinear Laura-Petrarch model, with two appeal parameters, with
three gains, with three time constants, and with two param- eters
for the nonlinearity – in sum ten parame- ters. A brute-force
identification starting with arbitrary values for these parameters
is not suc- cessful, especially as the appeals may also be
negative.
Consequently, first the size of the parameters and relations
between them should be qualita- tively analysed, following S.
Rinaldi ([2]). The time constants αL, αP, and αIP describe the for-
getting processes. For Laura and Petrarch, obvi- ously αL > αP
holds, because Laura never ap- pears to be strongly involved,while
the poet def- initely has a tenacious attachment, documented by
poems:
• In sonnet XXXV Petrarch claims:
Solo et pensoso i piu deserti campi vo mesurando a passi tardi e
lenti, ....... Ma pur si aspre vie ne’ si selvage cercar non so ch’
Amor non venga sempre ragionando con meco, et io col’lui. [Alone
and lost in thought, each lonely strand
I measure out with slow and laggard step, ......... Yet I cannot
find such harsh and savage trails where love does not pursue me as
I go, with me communing, as with him do I.]
The inspiration of the poet wanes very slowly, because Petrarch
continues to write (over one hundred poems) for more than ten years
after the death of Laura. The main theme of these lyrics is not his
passion for Laura, which has long since faded, but the memory for
her and the invocation of death:
• In Sonett CCLXVIII,written about two years after Laura’s demise,
Petrarch remembers:
Tempo e ben di morire, et o tardato piu ch’i non vorrei. Madonna e
morta, et a seco il mio core; e volendol seguire, interromper
conven quest’anni rei, perche mai veder lei di qua non spero, et
l’aspettar m’e noia. [It’s time indeed to die, and I have lingered
more than I desire. My lady’s dead, and with her my heart lies;
and, keen with her to fly, I now would from this wicked world
retire, since I can no more aspire on earth to see her, and delay
will me de- stroy.]
Consequently, between the time constants αIP and αp the relation
αIP < αp must hold.
As Petrarch’s inspiration holds about ten years, whereas Laura
forgets Petrarch in about four months, and Petrarch’s passion fades
in one year, suitable relations and values are
αL ∼ 3 · αP,αP ∼ 10 · αIP,αP ∼ 1.
The gains or reaction parameters βL, βp, and βIP also can be
estimated qualitatively, with respect to the time constants:
βL ∼ αP, βP ∼ 5 · αP, βIP ∼ 10 · αP.
Here the assumption is that Laura’s reaction equals the forgetting
time of Petrarch, and Pe- trarch reacts five times stronger. For
simplicity, the parameters γL and δP are normalized to one, since
it is always possible to scale P(t) and IP(t) suitably.
262 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
The choice of the appeal parameters AL and AP is crucial, because
these parameters determine the qualitative behaviour of the love
dynamics – cyclic nonlinear behaviour, or damped oscilla- tion
toward an equilibrium. In case of Laura and Petrarch, cyclic love
dynamics are expected in order to meet the experimentally founded
emo- tional cycle E(t) of Petrarch.
Clearly, Petrarch loves Laura, so AL > 0 must hold. By contrast,
Petrarch is a cold scholar in- terested in history and letters. He
is appointed a cappellanus continuus commensalis by Car- dinal
Giovanni Colonna, and this ecclesiastic appointment brings him
frequently to Avignon, where Laura lives. Consequently, Petrarch’s
ap- peal AP is assumed to be negative. Appropriate choices for the
appeals AL and AP are:
AL ∼ 2, AP ∼ −1.
The negativity of the appeal of Petrarch for Laura is somehow
recognized by the poet him- self:
• In sonnet XLV, while Petrarch is talking about Laura’s mirror, he
says:
Il mio adversario in cui veder solete gli occhi vostri ch’Amore e’l
ciel honora, ... [My rival in whose depths you’re wont to see your
own dear eyes which Love and heaven apprize, ...]
The above estimated ten parameter values, to- gether with zero
initial values for the love dy- namics and for the poetic
inspiration, are a good choice for identification. The ODE model
has been implemented in MATLAB (for compari- son, the transfer
function model has been imple- mented in Simulink); for
identification, a least squares method can be used:∑
(P(tk) − Ek)2 → min
There, it makes sense to use relations between parameters, so that
the number of parameters to be identified is reduced. As the
quality of data is relatively poor, also different sets of parame-
ter values may be seen as good approximation. Figure 9 shows an
identification result for P(t), with data Ek (’graded’
poems).
Figure 9. Result of model identification: love dynamics P(t) for
Petrarch coinciding with data from Petrarch’s
emotional cycle E(tk), with data Ek (crosses and circles).
Of interest are, of course, also the love dynamics L(t) for Laura
and Petrarch’s poetic inspiration IP(t). All variables are
presented in a MAT- LAB GUI, which drives experiments with the
Laura-Petrarch model. The GUI (Figure 10) of- fers parameter input
(sliders) and displays time
Figure 10. MATLAB GUI for experimenting with the Laura-Petrarch
model:
i) above left: sliders for gain βL, time constant αL, gain βP, time
constant αP, and appeals AL and AP;
ii) upper right: love dynamics for Laura – L(t), green – and
Petrarch – P(t), red – over time period 1130 – 1360; iii) lower
left: poetic inspiration of Petrarch – IP(t), blue
– over time period 1130 – 1360; iv) lower right: phase portrait
P(L) of love dynamics of
Petrarch and Laura – P over L.
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 263
courses for P(t) and L(t) (together), the time course for IP(t),
and a phase portrait P(L).
Figure 10 shows all results for the identified pa- rameters. The
results of the numerical solution are qualitatively in full
agreement with the Can- zoniere and with the analysis of Frederic
Jones. After a first high peak, Petrarch’s loveP(t) tends toward a
regular cycle characterized by alternate positive and negative
peaks. Also, Laura’s love L(t) and Petrarch’s poetic inspiration
IP(t) tend towards a cyclic pattern.
At the beginning, Petrarch’s inspiration IP(t) rises much more
slowly than his love and then remains positive during the entire
period. This might explainwhy Petrarchwrites his first poem more
than three years after he has met Laura, but then continues to
produce lyrics without any significant interruption.
By contrast, Laura’s love is always negative. This is in perfect
agreement with the Can- zoniere, where Laura is repeatedly
described as adverse.
• In sonnet XXI, Petrarch calls Laura
dolce mia guerrera [my sweet enemy].
• But in sonnet XLIV Petrarch says:
ne lagrima pero discese anchora da’ be’ vostr’occhi, ma disdegno et
ira. [and still no tears your lovely eyes assail, nothing as yet,
but anger and disdain.]
The fit between P(t) for Petrarch’s love and E(tk) for Petrarch’s
emotional cycle is actually very good. It is of similar quality
which is usu- ally obtained when calibrating models of elec- trical
and mechanical systems. Moreover, the fit could be further improved
by slightly modi- fying the parameter values and by loosing some
parameter relations.
But, improvement might be skipped, citing and agreeing with
Rinaldi: I do not want to give the impression that I believe that
Petrarch had been producing his lyrics like a rigid, determin-
istic machine. Nevertheless, one can conclude that the
Laura-Petrarchmodel strongly supports Frederic Jones’s conjecture
on Petrarch’s emo- tional cycle.
6. Experiments and Analysis with the Laura-Petrarch model
Experiments with the parameters show that the cyclic love dynamics
may change to a damped oscillation converging to equilibrium. It is
diffi- cult to find out which parameter quality causes a cyclic
behaviour, and which the damped oscilla- tions. Starting with the
classic Laura-Petrarch parameters, for instance, an increase of
only one parameter αL by a factor of 2.5 changes the qualitative
behaviour essentially (Figure 11) – this parameter change means
that Laura forgets Petrarch in about half time than before.
In this case all variables for P(t), L(t), and IP(t), show strongly
damped oscillations, reach- ing equilibrium almost in ten years.
The phase portrait P(L) underlines this convergence in rel- atively
fast time to an equilibrium of about P = 0.05 and L = −0.0105. This
equilib- rium still has a negative value for Laura’s love emotion,
but it is very small – almost as small as the positive value for
Petrarch’s emotion (see Figure 11).
Figure 11. Experiment with Laura-Petrarch parameters and changed
parameter αL by a factor of 2.5: strongly damped variables
converging to equilibrium with very small positive and negative
values for P and L, resp.
In principle, the existence of an equilibrium is tied to the
existence of steady state solutions for the ODE. Setting all
derivatives to zero, and substituting IP by the corresponding
steady state for P, i.e.
264 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
IP(t → ∞) = βIP
αIP P(t → ∞),
results in two coupled nonlinear algebraic equa- tions for P and L
at steady state:
0 = −αLL + βLP − βLP 3 + βLAP
0 = −αP(αIP + βIPP)P + βP(αIP + βIPP)L + αIPβPAL.
Rinaldi investigated in detail, by means of ana- lytical methods,
the case AL < 0 and AP > 0. First he tested the robustness of
the Laura – Petrarch cyclic love dynamics, with respect to
perturbations of the parameters.
For this purpose, the package LOCBIF, a pro- fessional software
package for the analysis of the bifurcations of continuous-time
dynamical systems, has been used. By varying only one parameter at
a time, Rinaldi detected a super- critical Hopf bifurcation, by
which the cycle eventually disappears.
Rinaldi continued with a detailed analysis of the limit cycle using
substitution of the variable L(t), transformation of parameters,
and singular perturbation for P(t) and IP(t). Results derive that,
indeed, a supercritical Hopf bifurcation causes a change of the
qualitative behaviour. One detailed result is a stability chart
(Figure 12) for new parameters ε and μ:
ε = αIP, μ = βIP
αIP , βPβL > αPαL.
Figure 12. Stability chart μ(ε) with border (red line; Hopf
bifurcation) between cyclic solutions (left) and solutions
converging to equilibrium (right); circle
indicates the classical cyclic Laura-Petrarch solution.
In Figure 12, the border between cyclic solu- tions and solutions
converging to equilibrium can be seen, represented as graph
μ(ε).
In general, cyclic solutions exist only in cases of nonsymmetric
reactions, or in simple cases for appeal parameters with different
signs. More details about these analytical investigations can be
found in Rinaldi’s work ([2]).
Usually, if people fall in love, both are attractive for each
other, which means that AL and AP must be positive. In this case,
we meet only damped oscillations converging to a stable steady
state:
AL > 0, AP > 0 → ∃(P > 0, L > 0).
An interesting experiment is the case of an at- tractive Petrarch.
Assuming e.g. that Petrarch is a young beautiful man, almost like
Apollo, he may have the appeal AL ∼ 6 to Laura, three times the
appeal of Laura to him (all other pa- rameters unchanged).
Figure 13. Experiment with Laura-Petrarch parameters and positive
appeal AL ∼ 6 of Petrarch to Laura: very
strongly damped variables converging to an equilibrium with
positive steady states for P and L, resp.
Figure 13 shows the MATLAB GUI with the results of this experiment.
All variables P(t), L(t), and IP(t), are very strongly damped and
converge to steady states with relative high pos- itive values. The
poetic inspiration IP(t) shows almost (negative) exponential
behaviour. Nev- ertheless, L(t) becomes negative for a short time
period in the first three years. The phase dia- gram shows two
crossings and it seems that the love dynamics first cycles two
times, before it decides to converge to a stable positive feeling
of both lovers.
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 265
The MATLAB GUI allows experiments with a broad variety of parameter
sets. Of interest are, for instance, also cases which cycle more
than hundred times before they decide to converge to an
equilibrium. For ’extreme’ parameter values the numerical solution
may cause problems, or at least, take a long time; as ODE solver, a
stiff solver has been chosen, having the best relative
performance.
7. From Laura-Petrarch Model to Woman-Man Model
In times of gender equality women, as well as men, may play an
active part in a love affair. Consequently, women also express
their love by poems or other media, and they confess their love to
public. By this, an additional stock with flow for the women’s
inspiration can be introduced easily. For Laura and Petrarch, this
would mean that also Laura writes poems, that Petrarch’s appeal is
influenced by Laura’s po- etic inspiration, and that Petrarch shows
more sensibility in his reactions to Laura. Conse- quently, the
structure of the System Dynamics model (Figure 7) suggests a
genuine and nat- ural extension: symmetric stocks, flows, and
feedbacks gains as well for ’Petrarch’ and for ’Laura’, which
should now generally represent a man and a woman who fall in
love.
The Woman-Man Model presented in Figure 14 describes the love
dynamics W(t) for a woman, andM(t) for a man both falling in
lovewith each other; love inspires both to communicate their love
to public, in letters, in videos, with CDs and DVDs, etc. –
represented by the inspiration variables IW(t) and IM(t).
A quantification of the qualitative SD model (Figure 14) with
linear causal links and in- fluences yields a SD – representation
of Man- Woman Model with full symmetry (Figure 15).
As with the simplified Laura-Petrarch Model, the ODEs corresponding
to the SFD in Figure 15 are linear, and they show the full symme-
try between woman’s and man’s emotions and inspirations:
dW(t) dt
dIW(t) dt
= −αIWIW(t) + βIWW(t)
Figure 14. Qualitative System Dynamics Model for Woman’s and Man’s
emotions and inspirations.
Figure 15. Simplified System Dynamics Model for Woman-Man Model
with linear influences.
dM(t) dt
dIM(t) dt
= −αIMIM(t) + βMM(t).
Finally, the nonlinearities must be modelled in the FSD by
auxiliaries, replacing the weight- ing parameters in the feedback.
Because of the symmetry in emotions and inspirations, the model
makes use of two nonlinear cubic-like reaction functions for
woman’s and man’s re-
266 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
actions to each other, RW(M) and RM(W), and of two nonlinear
relations between inspiration, appeal, and emotion. The
nonlinearities for the reactions become almost linear for big
values for the parameters γW and γM, resp; the nonlin- earities for
the inspiration-appeal relation be- come linear, if the parameters
δW and δM tend towards zero. Figure 16 presents this complete
nonlinear Woman-Man Model in SD notation (SFD).
Compared with the Laura-Petrarch Model, the Woman-Man Model must
make use of an in- creased number of parameters: four fading pa-
rameters (instead of three), four (linear)weight- ing factors for
the cross-feedbacks (instead of three), two appeal parameters
(instead of one), and four parameters in the nonlinear functions
(instead of two) – in sum 14 parameters. An analysis of Man-Woman
Model is almost im- possible, but numerical experiments may give
interesting insight into love dynamics.
Figure 16. System Dynamics Model Stock and flow diagram for
Woman-Man Model with nonlinear
reactions.
Figure 17. Experiment with Woman-Man Models: case studies ’Everyday
Boring’ and ’Pretty and Ugly’
(red/blue – woman’s/man’s love emotion; red/blue dashed –
woman’s/man’s love inspiration).
The ODE model equivalent to the SFD in Figure 16 shows now typical
symmetric structure:
dW(t) dt
= −αLW(t) + βWM
1 + δMIM(t) dIM(t)
dt = −αIMIM(t) + βIMM(t).
As the Laura-Petrarch Model, the Woman-Man Model has been
implemented inMATLAB (ODE model) and in Simulink (transfer
functionsmodel).
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 267
Experiments can be controlled by a simplified MATLAB GUI, as
suggested by case studies. Figure 17 shows results of two case
studies in a GUI, with predefined parameters character- izing
typical parameter configurations (’Every- day Boring’ and ’Pretty
and Ugly’).
8. Woman-Man Model with Dynamic Appeal Parameters
Does the Woman-Man model reflect reality? The model is able to
mimicry different situa- tions, but with one assumption: the appeal
pa- rameters AM and AW are constant. This assump- tion may not meet
reality; the appeal for each other may change and may be
controlled.
A dynamic appeal can be easily modelled by time-dependent appeal
variables AM(t) and AW(t), resulting in a small change in the ODE
model:
dW(t) dt
= −αLW(t) + βWM
1 + δMIM(t) dIM(t)
dt = −αIMIM(t) + βIMM(t).
Case studies may become now very compli- cated, because not only 14
parameters have to be chosen appropriately, but also the function
AM(t) and AW(t) have to be provided meaning- fully. An extended
version of the MATLAB GUI presented in Figure 17 allows
additionally providing predefined appeal functions. Figure 18 and
Figure 19 show results for perhaps inter- esting cases: the appeal
decreases exponentially – should happen, and second, the appeal
AM(t) is an increasing step function – could model a plastic
surgery.
Figure 18. Experiment with Woman-Man Model: exponentially
decreasing appeal functions (red/blue –
woman’s/man’s love emotion; red/blue dashed – woman’s/man’s love
inspiration).
Figure 19. Experiment with Woman-Man Model – positive step in
appeal at ten years (time courses as in
Figure 18).
9. Conclusion
In principle, the contribution could show,
• that it is possible to model in some detail the love dynamics
between two persons by Sys- tem Dynamics, resulting in nonlinear
ODE systems by ODEs,
• that, additionally, a poetic inspiration caused by love emotions
can be modelled in System Dynamics, extending the ODE system for
the love dynamics,
• and that the model can be validated with data from history,
reflecting the emotions between Petrarch and Laura.
268 Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics
Of course, this contribution presents serious in- vestigations. But
is it possible to investigate the dynamics of love, perhaps the
most important phenomenon concerning our lives, seriously, by
methods of mathematics and engineering? One could also conclude, it
might be better not to tackle the secrets of love, because
described and controlled by formula, it is not love any longer. In
this view, the contribution might be seen as reference to Petrarch
and the most beautiful love poems the author has ever read.
References
[1] S. H. STROGATZ, Love affairs and differential equa- tions.
Math. Magazine, 61 (1988), pp. 35.
[2] S. RINALDI, Laura and Petrarch: an intriguing case of cyclical
love dynamics. SIAM J. App. Math. Vol. 58 (1998), No. 4, pp.
1205–1221.
[3] F. BREITENECKER, K. BREITENECKER, F. JUDEX, N. POPPER, S.
WASSERTHEURER, An ODE for the Re- naissance. InB. Zupancic,R.Karba,
S. Blazic (Eds.) Proc. 6th EUROSIM Congress on Modelling and
Simulation; ISBN-13: 978-3-901608-32-2; ARGESIM – ARGE Simulation
News, Vienna; www.argesim.org; CD, 11 p.
[4] F. J. JONES. The Structure of Petrarch’s Canzoniere. Brewer,
Cambridge, UK, 1995.
Received: June, 2008 Accepted: September, 2008
Contact addresses:
Vienna University of Technology Wiedner Hauptstrasse 8-10
1040 Vienna, Austria e-mail:
[email protected]
Florian Judex Institute for Analysis and Scientific Computing
Vienna University of Technology Wiedner Hauptstrasse 8-10
1040 Vienna, Austria
Wiedner Hauptstrasse 8-10 1040 Vienna, Austria
Anna Mathe ARGESIM – Vienna Univ. of Technology
Wiedner Hauptstrasse 8-10 1040 Vienna, Austria
Andreas Mathe ARGESIM – Vienna Univ. of Technology
Wiedner Hauptstrasse 8-10 1040 Vienna, Austria
FELIX BREITENECKER studied “Applied Mathematics” and, after guest
professorships at University Glasgow, at University Budapest, at
Uni- versity Ljubljana and at other universities, since 1992 he has
acted as professor of Mathematical Modelling and Simulation at
Vienna Uni- versity of Technology. He covers a broad research area,
from mathe- matical modelling to simulator development, from DES
via numerical mathematics to symbolic computation, from biomedical
and mechani- cal simulation to process simulation. In teaching
area, he is organizing E-learning courses on basic mathematics and
basic simulation.
Felix Breitenecker is active in various simulation societies:
president and ex president of EUROSIM since 1992, board member and
presi- dent of the German Simulation Society ASIM, member of
INFORMS, SCS, etc. He is engaged in many industrial projects in the
area of modelling and simulation, industrial partners being Airbus,
Daimler, Austrian Social Insurance, and Fraunhofer Foundation.
Felix Breite- necker has published about 300 scientific
publications, and he is author of two 3 books and editor of 22
books. Since 1995 he has acted as Editor in Chief of the journal
Simulation News Europe.
In 2006 he initiated the LAURA Group at Vienna University of Tech-
nology, gathering people interested in modelling and simulation of
emotions, starting the group with the Laura – Petrarch model. Fe-
lix Breitenecker is head of LAURA Group; he develops different
model approaches, and acts as narrator in multimedia speeches on
love emo- tions.
FLORIAN JUDEX currently works for the Vienna University of Technol-
ogy on a post-doctoral position. He studied “Applied Mathematics
and Applications in Computer Science” and specialized in parameter
iden- tification in modelling and simulation with his Master Thesis
(2006). His PhD Thesis (2009) dealt with groundwater modelling and
model comparison with respect to identification aspects.
Currently he is concentrating on E-learning, especially enhancing
the mathematical education of engineering students through
E-learning us- ing MATLAB and Maple T.A by developing web courses
for blended learning. Up to now, Florian Judex has published 10
papers in the area of modelling and simulation. He is member of
ASIM, the Ger- man Simulation Society, and winner of the TU Vienna
E-learning Prize 2008.
In LAURA Group, Florian Judex programs and implements the MAT- LAB
and MAPLE models and GUIs, and presents them in multimedia
speeches.
NIKOLAS POPPER has earned a degree in technical mathematics at the
Vienna University of Technology. His thesis title is “Simulation of
the Respiratory System – Compartment Modelling and Modelling of
Perfusion”. After research stays abroad in Barcelona, Universidad
PolitU¤cnica de Catalunya and Moscow, Idaho, University of Idaho,
he has gained experience in industry projects as well as research
and devel- opment knowledge. Currently he is working in the area of
visualization in computer graphics, modelling and simulation of
health economy and theory of modelling & simulation. Nikolas
Popper is co-proprietor of the company “Die Drahtwarenhandlung”
Simulation Services. The company also offers technical solutions
(defect detection on pictures, modelling & simulation, . . .)
as well as animations and films in the area of science journalism,
and E-learning. Furthermore, he is doing a PhD thesis in the area
of alternative and structural dynamic models at the Vienna
University of Technology.
For LAURA Group, he is preparing multimedia content for multime-
dia speeches, based on historical research on Petrarch, Laura, and
on research on emotions modelling.
Love Emotions between Laura and Petrarch – an Approach by
Mathematics and System Dynamics 269
KATHARINA BREITENECKER studied “Technical Physics” at the Vienna
University of Technology with an emphasis on nuclear physics.
During her last terms of her master’s she was also involved in
modelling and simulation, doing some projects at the department of
atomic physics. In 2005 she got her master degree with a master
thesis about the anal- yses of trace elements by nuclear activation
analysis in Greek pumice within an UNESCO project. She continued
working on her PhD on the migration behaviour of transuranic
elements in the soil by cellular automata modelling, finished in
2008. Currently she is working for the International Atomic Energy
Agency in the Safeguards Analytical Laboratory.
She is an active member of the Austrian Society for Nuclear Tech-
niques, of the German Society for Liquid Scintillation Counting,
and of the German Simulation Society ASIM. She is also presenting
E-learning courses organized by ARGESIM. She has made publications
in both, nuclear analytics (radiometry, radiochemistry), and
mathematical mod- elling.
In LAURA Group, Katharina Breitenecker takes care of selection and
classification of Petrarch’s sonnets, and in multimedia speeches
she presents the poems in Italian and/or English; for future work,
she is trying a cellular automata approach for emotionszz
behaviour.
ANNA MATHE studied Media Design in Linz and now works as a free-
lancer for web design, copywriting and graphic design. Since 2003,
she has managed organisational affairs for ARGESIM – ARGE
Simulation News TU Vienna – in supporting organization of ARGESIM
seminars, in co-ordination and supporting editing and production of
SNE (Simu- lation News Europe). Since 2008, she is engaged in
administration and in graphical design for ASIM, the German
Simulation Society.
In LAURA Group, AnnaMathe takes care of graphical design for
contri- butions, and she presents poems in Italian and/or English
at multimedia speeches.
ANDREAS MATHE got the education of a civil engineer in Linz, and he
is working as reviewer in the area of building construction for an
insurance company. He is active in documentation, web research and
multimedia design for ARGESIM, and travelling organization and
accommodation arrangements for ARGESIM seminars abroad (Siena
Seminars).
In LAURA Group, Andreas Mathe takes care of quality management. In
multimedia speeches, he co-ordinates voice, sound and multimedia
presentations, and he presents sonnets in English.
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<FEFF9ad854c18cea306a30d730ea30d730ec30b951fa529b7528002000410064006f0062006500200050004400460020658766f8306e4f5c6210306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103055308c305f0020005000440046002030d530a130a430eb306f3001004100630072006f0062006100740020304a30883073002000410064006f00620065002000520065006100640065007200200035002e003000204ee5964d3067958b304f30533068304c3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
/KOR
<FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken
die zijn geoptimaliseerd voor prepress-afdrukken van hoge
kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met
Acrobat en Adobe Reader 5.0 en hoger.) /NOR
<FEFF004200720075006b00200064006900730073006500200069006e006e007300740069006c006c0069006e00670065006e0065002000740069006c002000e50020006f0070007000720065007400740065002000410064006f006200650020005000440046002d0064006f006b0075006d0065006e00740065007200200073006f006d00200065007200200062006500730074002000650067006e0065007400200066006f00720020006600f80072007400720079006b006b0073007500740073006b00720069006600740020006100760020006800f800790020006b00760061006c0069007400650074002e0020005000440046002d0064006f006b0075006d0065006e00740065006e00650020006b0061006e002000e50070006e00650073002000690020004100630072006f00620061007400200065006c006c00650072002000410064006f00620065002000520065006100640065007200200035002e003000200065006c006c00650072002000730065006e006500720065002e>
/PTB
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/SUO
<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>
/SVE
<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>
/ENU (Use these settings to create Adobe PDF documents best suited
for high-quality prepress printing. Created PDF documents can be
opened with Acrobat and Adobe Reader 5.0 and later.) >>
/Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ <<
/AsReaderSpreads false /CropImagesToFrames true /ErrorControl
/WarnAndContinue /FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug
false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps
false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint
/Legacy >> << /AddBleedMarks false /AddColorBars false
/AddCropMarks false /AddPageInfo false /AddRegMarks false
/ConvertColors /ConvertToCMYK /DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK /Downsample16BitImages
true /FlattenerPreset << /PresetSelector /MediumResolution
>> /FormElements false /GenerateStructure false
/IncludeBookmarks false /IncludeHyperlinks false
/IncludeInteractive false /IncludeLayers false /IncludeProfiles
false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe)
(CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector
/DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling
/LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false >> ] >> setdistillerparams
<< /HWResolution [2400 2400] /PageSize [595.276 841.890]
>> setpagedevice