Abstract
Dynamic Systems theory is a general mathematical tool that has been successfully used both in hard and in soft Sciences.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Not necessarily the total number of voters will be the same.
- 2.
The word “state” has no political implication: it simply refers to the position (state) of the system.
- 3.
In continuous time, “local” is referred to what happens at epochs close to the current date t. In discrete time, for instance, which relationship must hold between \(\mathbf {x}\left( t\right) \) and \(\mathbf {x}\left( t+1\right) \). In continuous-time systems what counts is what happens between t and \(t+h\), with h small.
- 4.
They stand at the basis of the concrete current problem of immigration.
- 5.
Assume that \(x\left( t\right) \) is differentiable:
$$\begin{aligned} x\left( t+h\right) -x\left( t\right) =x^{\prime }\left( t\right) h+o\left( h\right) \end{aligned}$$Divide both sides by h:
$$\begin{aligned} \frac{x\left( t+h\right) -x\left( t\right) }{h}=\frac{x^{\prime }\left( t\right) h+o\left( h\right) }{h} \end{aligned}$$hence:
$$\begin{aligned} \frac{x\left( t+h\right) -x\left( t\right) }{h}=x^{\prime }\left( t\right) + \frac{o\left( h\right) }{h} \end{aligned}$$When h approaches 0 the l.h.s. approaches \(x^{\prime }\left( t\right) \).
- 6.
Technically we assume that \(F\left( K, L\right) \) is a homogeneous function of the first degree:
$$\begin{aligned} F\left( aK,aL\right) =aF\left( K, L\right) \text {; }\ \ \ \ \ \ \ a>0 \end{aligned}$$Roughly speaking this means that if we increase by — say — \(10\%\) the amount of each production factor then the product amount turns out to be increased by exactly \(10\%\). For instance, a popular example of a homogeneous function of the first degree:
$$\begin{aligned} F\left( K, L\right) =b\sqrt{KL} \end{aligned}$$In fact:
$$\begin{aligned} F\left( aK, aL\right) =b\sqrt{aKaL}=ab\sqrt{KL} \end{aligned}$$It is a special case of what the reader will learn in Economics is indicated in Economics as the Cobb–Douglas production function.
- 7.
It is nothing but the vector of the time derivatives of the state variables.
- 8.
From a merely formal point of view, we could also reduce ourselves to the autonomous case. It would be sufficient to extend the state vector, in order to include time. This is mathematically correct but devastating for the applications, we will quickly see.
- 9.
Not Archimedes, but close to... It is attributed to P.S. Laplace (1749–1827), he was the the guy we already met when handling determinants or frequency/probability distributions.
- 10.
We introduce a more general symbol for the needs we will find in handling the non-homogeneous case.
- 11.
When handling scalars, the factor order is irrelevant.
- 12.
It is interesting to note that those formula holds also for \( t=1,0,-1,-2,\ldots \).
- 13.
After the name of the Italian professor of Geometry Oscar Chisini (1889–1967), who proposed this innovative perspective in 1929.
- 14.
Nothing in common with an identity matrix.
- 15.
In general, a product \(\mathop {\textstyle \prod }\limits _{p}^{q}\) with \(p>q\) is assigned to have value 1. If \(p=q\), the product has a single factor and the value of the product is that of its only factor.
- 16.
We could allow for \(A=A\left( t\right) \) and \(\mathbf {b=b}\left( t\right) \). The formulas are perfectly analogous. We will provide later the results, also in this more general case.
- 17.
Like one of the Authors.
- 18.
A nice way to say that they do not pass away.
- 19.
See Definition 4.2.3 at p. 228.
- 20.
# of male newborns per woman.
- 21.
Never forget that \(o\left( h\right) +o\left( h\right) +o\left( h\right) =o\left( h\right) \).
- 22.
This task is easily treatable with numerical methods.
- 23.
In our setting \(x_{2}\) cannot take negative values and we omit absolute values.
- 24.
We call it “gross” because, unlike the female case, here the natality rate is absent as it enters the forcing term, owing to the fact that it must interact with the other state variable \( x_{2}\left( t\right) \).
- 25.
With these notation simplifications the coefficient matrix A in (4.4.27) becomes:
$$\begin{aligned} A=\left[ \begin{array}{c@{\quad }c} V_{1} &{} \nu _{1} \\ 0 &{} V_{2} \end{array} \right] {(4.4.28)} \end{aligned}$$ - 26.
The fact that the number of male newborns slightly exceeds that of females (\(\approx \) \(+3\%\)) is usually interpreted as a compensation of the fact that \(\mu _{1}>\mu _{2}\). This natural parameter can be altered in social systems where females are not welcome.
- 27.
Remember the “hairdresser theorem”:
$$\begin{aligned} \text {D}\left[ \alpha \left( t\right) \beta \left( t\right) \right] =\alpha ^{\prime }\left( t\right) \beta \left( t\right) +\alpha \left( t\right) \beta ^{\prime }\left( t\right) \end{aligned}$$and that, thanks to the chain rule:
$$\begin{aligned} \text {D}\left[ \text {e}^{\alpha \left( t\right) }\right] =\text {e} ^{\alpha \left( t\right) }\alpha ^{\prime }\left( t\right) \end{aligned}$$ - 28.
Recent facts (end of the Twenties) in Southern Europe should make this variant of some interest.
- 29.
The concrete interpretation of this equality is evident: at the equilibrium, the net number of females leaving exogenously the population must balance the net endogenous number of new females.
- 30.
The case of singularity of A is mathematically relevant, but, empirically, irrelevant.
- 31.
The case of singularity of \(I-A\) is mathematically relevant. Empirically, it is irrelevant.
- 32.
The first entry in this row is the coefficient \(\alpha _{0}\) of \(\lambda ^{n} \), which is always unitary.
- 33.
We inform our readers that for non-autonomous systems there is a parallel notion we will not take into consideration.
- 34.
Suggestion for the reader: using (4.6.6), try to check this fact!
- 35.
Suggestion for the reader: try to check that both:
$$\begin{aligned} \left\{ \begin{array}{c} x\left( t\right) \equiv 0 \\ x\left( t\right) \equiv 1 \end{array} \right. \end{aligned}$$are constant solutions of the differential equation: they obey the motion law.
- 36.
Nothing to do with the prestigious London School of Economics and Political Science, usually labelled LSE.
- 37.
This situation is common in Social Sciences and makes them different from Hard Sciences like Physics. When studying Solow’s model, in the Example 4.2.6 on p. 227, we have handled a production function F(K, L) asking that this function is homogeneous of degree 1 (constant returns to scale), that it is concave (decreasing marginal productivity of the factor) and we have examined a small piece of the neoclassical Economic Theory. If we specify F:
$$\begin{aligned} F\left( K, L\right) =aK^{\alpha }L^{1-\alpha } \end{aligned}$$according to the well known scheme by Cobb–Douglas, the quality of our analysis goes down, moving from theory to an exercise.
- 38.
The reason for this label will appear clear very soon.
- 39.
Recall that:
$$\begin{aligned} x^{\prime }\left( t\right) h=\text {d}x \end{aligned}$$is the (time) differential of x, or an approximation of its increment \( x\left( t+h\right) -x\left( t\right) \).
- 40.
In this case some iterations gave us a rather clear perspective, but examples could be easily constructed, which require millions of iterations to reveal the end of the story.
- 41.
It is one of the most important numbers of Mathematics and Art. It is known as Golden Ratio (or Aurea Sectio in Latin). For instance, it governs proportions in Ancient Greece architecture. See, for instance, the masterpiece [8].
- 42.
A vivid alternative name for this method is cobweb method.
- 43.
Such an equilibrium is technically said to be a Nash-equilibrium, from the name of the Nobel Prize for Economics John Nash, made popular by the film A beautiful mind. A Nash-equilibrium is a point at which both the players are at an optimum. If one of them moves from that point looses something. If they are there and act rationally they do not move. A delicate question is “What happens if the two players are not at a Nash-equilibrium?” This is exactly what we are exploring here. This example is brand new.
- 44.
For obvious reasons we do not discuss the very, very... special case \( k+\beta =1\). It is of interest only for mathematicians.
- 45.
Realpolitik suggests us to exclude 0-probability cases in which the starting point is an equilibrium one.
- 46.
Relevant for politics.
- 47.
Think of whitebait, to have an idea.
- 48.
It solves the equation:
$$\begin{aligned} 0=-\alpha x^{*}+r. \end{aligned}$$ - 49.
The mathematical analysis of the model is not due to the Author of the paper, who is a sociologist, but to Christopher Winship, who has had quite a different career.
- 50.
The display of scenarios excludes the two unstable initial conditions:
$$\begin{aligned} x\left( 0\right) =x_{1}^{*}\text { and }x\left( 0\right) =x_{3}^{*}. \end{aligned}$$ - 51.
The phase curve starts at 0 with positive ordinate, it is a continuous curve and it ends at 1 with a negative ordinate. Therefore there must be \( x^{*}\), between o and 1 at which it crosses the abscissae axis.
- 52.
It suffices that f satisfies the Inada conditions:
-
\(f^{\prime }\left( 0\right) =+\infty \);
-
\(f^{\prime }\left( +\infty \right) =0\);
-
f increasing and concave.
-
- 53.
We’re Italian.
- 54.
This fact holds generally. If T is a square stochastic n-matrix, then \( I-T\) is singular, being I an identity n-matrix. At least one of the equilibrium equations turns out to be redundant.
- 55.
The quantity\( \displaystyle \) \( \frac{\beta i\left( t\right) \left[ N-i\left( t\right) \right] -\gamma i\left( t\right) }{N}\) is usually called the incidence rate of the disease.
- 56.
With attraction basin \(\left( 0,+\infty \right) \).
- 57.
The computations, which are quite analogous to the ones of the basic model are left to the reader as a useful exercise.
- 58.
% of non-detected affected \(+\) % of isolated affected \(=\) 0.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Peccati, L., D’Amico, M., Cigola, M. (2018). Dynamic Systems. In: Maths for Social Sciences. UNITEXT(), vol 113. Springer, Cham. https://doi.org/10.1007/978-3-030-02336-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-02336-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02335-5
Online ISBN: 978-3-030-02336-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)