Skiadas C.H., Skiadas C. Chaotic Modelling and Simulation. Analysis of Chaotic Models, Attractors and Forms
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294 Chaotic Modelling and Simulation
(a) A pattern of a ship (b) The signature attractor
FIGURE 13.8: The ship and signature attractors
equations of the model are:
x
t+1
= (a − x
t
− by
t
)x
t
y
t+1
= (c − y
t
− dx
t
)y
t
(13.13)
(a) The Ushiki attractor with
two objects
(b) The Ushiki attractor with
four objects
(c) The Tinkerbell attractor
FIGURE 13.9: The Ushiki and Tinkerbell attractors
A four-object attractor is obtained when the parameters take the values a = 3.6, b =
3.2, c = 0.04 and d = 0.4 (Figure 13.9(b)).
The Tinkerbell model’s iterations form six chaotic attractors (Figure 13.9(c)). The
model is based on the following set of equations
x
t+1
= x
2
t
− y
2
t
+ 0.91x
t
− 0.6y
t
y
t+1
= 2x
t
y
t
+ 2.5x
t
(13.14)
Odds and Ends 295
Questions and Exercises
1. Consider the following simple three-dimensional system of Lotka-Volterra
type:
x
′
= a(x − xyz)
y
′
= b(−y + xyz)
z
′
= c(z − xyz)
Find the equilibrium points and the type of stability.
2. Find the line of symmetry of the ship attractor.
3. Consider the difference delay equation:
z
n+2
= azn + 1 + bz
n
(1 − z
n
)
Study this equation by finding equilibrium points, exploring chaotic behaviour
and drawing the related graphs and bifurcation diagrams for specific values of
the parameters a and b.
4. Find the equilibrium points of the Ushiki attractor.
5. Find the equilibrium points of the Tinkerbell attractor.
Chapter 14
Milestones
It is fascinating and very instructive for any scientific field to examine the history
and milestones that were the basis for the establishment and advancement of the
field. Some researchers were lucky to publish and disseminate their findings to a
broad audience. Others circulated their research only to a small number of friends
and colleagues. And sometimes new findings came too early to be accepted by the
scientific community.
During the preparation of any work an extensive bibliography is used, both directly
and indirectly, by influencing the approach and direction of the research. This work
was no exception. We hope the extensive bibliography, presented in the references
section, will help the reader become more acquainted with the fascinating fields of
chaotic modelling and simulation, and their historical development.
The table that follows presents an account of the ground-breaking steps in the
development of chaotic modelling and simulation, as it is perceived in our days,
through the accounts included in various books and papers. Papers written after
1980 are not included, and very few works published in the seventies appear. We
have tried to emphasise those scientific works that are considered “classical,” along
with papers and books that described chaotic phenomena or proposed theoretical or
mathematical tools that triggered the future development of the chaotic field.
Year Name Title of Publication
1798 T. Malthus An essay on the principle of population
1823 M. Faraday On a peculiar class of acoustical figures, and on cer-
tain forms assumed by groups of particles upon vi-
brating elastic surfaces
1825 B. Gompertz On the nature of the function expressing of the law
of human mortality
1859 P. Riess Das Anblasen offener Rohre durch eine Flamme
— P. L. Rijke Notiz
¨
uber eine neue Art, die in einer an beiden En-
den offenen R
¨
ohre enthaltene Luft in Schwingungen
zu versetzen
1883 G. Cantor
¨
Uber unendliche, lineare Punktmannigfaltigkeiten
— Lord Rayleigh On maintained vibrations
297
298 Chaotic Modelling and Simulation
Year Name Title of Publication
— Lord Rayleigh On the crispations of fluid resting upon a vibrating
support
1887 Lord Rayleigh On the maintenance of vibrations by forces of dou-
ble frequency, and on the propagation of waves
through a medium endowed with a period structure
1889 S. Kowalevskaya Sur le probl
`
eme de la rotation d’un corps solide au-
tour d’un point fixe
— S. Kowalevskaya Sur une propri
´
et
´
e du syst
`
eme d’
´
equations
diff
´
erentielles qui d
´
efinit la rotation d’un corps
solide autour d’un point fixe
1890 H. Poincar
´
e M
´
emoire sur les courbes d
´
efinies par les
´
equations
diff
´
erentielles I-VI, Oeuvre I
— H. Poincar
´
e Sur les
´
equations de la dynamique et le probl
`
eme de
trois corps
1891 Lord Rayleigh On the problem of random vibrations, and of random
flights in one, two, or three dimensions
1892 M. A. Lyapunov Probl
`
eme g
´
en
´
eral de la stabilit
´
e du mouvement,
(1907 translation from the 1892 Russian original)
— H. Poincar
´
e Les m
´
ethodes nouvelles de la m
´
ecanique c
´
eleste
1898 J. Hadamard Les surfaces
`
a courbures oppos
´
ees et leurs lignes
g
´
eod
´
esiques
1899 H. Poincar
´
e Les m
´
ethodes nouvelles de la m
´
ecanique c
´
eleste
1905 H. Poincar
´
e Lecons de m
´
ecanique
1908 H. Poincar
´
e Science et methode
1910 A. Lotka Zur Theorie der periodischen Raktionen
1918 G. Duffing Erzwungene Schwingungen bei ver
¨
anderlicher
Eigenfrequenz
— G. Julia M
´
emoire sur l’it
´
eration des fonctions rationelles
1919 F. Hausdorff Dimension und
¨
außeres Maß
— P. Fatou Sur les
´
equations fonctionelles
1920 P. Fatou Sur les
´
equations fonctionelles
1923 M. H. Dulac Sur les cycles limites
— F. C. Ritt Permutable rational functions
1926 P. Fatou Sur l’it
´
eration des fonctions transcendentes enti
`
eres
Milestones 299
Year Name Title of Publication
1927 B. van der Pol Forced oscillations in a circuit with non-linear
resistance
— B. van der Pol and
J. van der Mark
Frequency demultiplication
1927 G. D. Birkhoff Sur le probl
`
eme restreint des trois corps
1932 G. D. Birkhoff Sur l’existence de r
´
egions d’instabilit
´
ee dynamique
— A. Denjoy Sur les courbes d
´
efinies par les
´
equations
diff
´
erentielles
`
a la surface du tore
1934 G. F. Gause The struggle for existence
1935 G. D. Birkhoff Sur le probl
`
eme restreint des trois corps
— P. O. Pedersen Subharmonics in forced oscillations in dissipative
systems
1936 A. N. Kolmogorov Sulla teoria di Volterra della lotta per l’esistenza
1937 R. A. Fisher The wave of advance of advantageous genes
— A. Kolmogorov et
al.
Etude de l’
´
equation de la diffusion avec croissance
de la quantit
´
ede mati
`
ere et son application
`
a un
probl
`
eme biologique
1942 J. L. Doob The Brownian movement and stochastic equations
1942 E. Hopf Abzweigung einer periodischen L
¨
osung von einer
station
¨
aren L
¨
osung eines Differentialsystems
1942 A. Kolmogorov Interpolation and extrapolation of stationary series
1943 S. Chandrasekhar Stochastic problems in physics and astronomy
1944 L. D. Landau On the problem of turbulence
1945 M. L. Cartwright
and J. E. Little-
wood
On nonlinear differential equations of the second
order
— S. O. Rice Mathematical analysis of random noise
1947 M. Kac Random walk and the theory of Brownian motion
1948 E. Hopf A mathematical example displaying features of
turbulence
— M. L. Cartwright Forced oscillations in nearly sinusoidal systems
1949 C. E. Shannon and
W. Weaver
The mathematical theory of information
1951 G. H. Markstein Experimental and theoretical studies of flame-front
stability
300 Chaotic Modelling and Simulation
Year Name Title of Publication
1952 A. L. Hodgkin and
A. F. Huxley
A quantitative description of membrane current and
its application to conduction and excitation in nerve
— A. L. Hodgkin and
A. F. Huxley
Current carried by sodium and potassium ions
through the membrane of the giant axon of loligo
1953 N. Metropolis et al. Equations of state calculations by fast computing
machine
1954 A. N. Kolmogorov Preservation of conditionally periodic movements
with small change in the Hamiltonian function
1956 W. Feller An introduction to probability theory and its
applications
1956 R. E. Kalman Nonlinear aspects of sample data control systems
— P. Landberg Vibrations caused by chip formation
— E. N. Lorenz Empirical orthonogal functions and statistical
weather prediction
— G. Contopoulos On the isophotes of ellipsoidal nebulae
1958 A. N. Kolmogorov A new invariant of transitive dynamical systems
— P. J. Myrberg Iteration der reellen Polynome zweiten Grades
— G. Contopoulos On the vertical motions of stars in a galaxy
1959 N. N. Leonov Transformations d’une droite en elle-m
`
eme
— A. R
´
enyi On the dimension and entropy of probability
distributions
1960 R. E. Kalman A new approach to linear filtering and prediction
problems
— N. N. Leonov Transformation ponctuelle d’une droite en elle-
m
`
eme discontinue, lin
´
eaire par morceaux
— N. N. Leonov Th
´
eorie des transformations discontinues d’une
droite en elle-m
`
eme
— G. Contopoulos A third integral of motion in a galaxy
1961 R. A. FitzHugh Impulses and physiological states in theoretical
models of nerve membrane
1962 P. J. Myrberg Sur l’it
´
eration des polynomes r
´
eels quadratiques
— J. Nagumo et al. An active pulse transmission line simulating nerve
axon
1963 E. N. Lorenz Deterministic nonperiodic flow
— B. Mandelbrot The variation of certain speculative prices
Milestones 301
Year Name Title of Publication
1964 E. N. Lorenz The problem of deducing the climate from the gov-
erning equations
— M. H
´
enon and C.
Heiles
The applicability of the third integral of motion:
Some numerical experiments
— A. N. Sarkovskii Coexistence of cycles of a continuous map of a line
into itself
— G. Contopoulos
and G. Bozis
Escape of stars during the collision of two galaxies
1965 D. Coles Transition in circular couette flow
— A. N. Kolmogorov Three approaches to the quantitative definition of
information
— J. R. Pasta et al. Studies on nonlinear problems
— L. P. Shilnikov A case of the existence of a countable number of
periodic motions
— G. Contopoulos Periodic and tube orbits
1967 H. Degn Effect of bromine derivatives of malonic acid on the
oscillating reaction of malonic acid, cerium ions and
bromate
1967 B. Mandelbrot The variation of some other speculative prices
— L. P. Shilnikov On the Poincar
´
e-Birkhoff problem
— L. P. Shilnikov The existence of a denumerable set of periodic
motions in four-dimensional space in an extended
neighbourhood of a saddle-focus
1968 M. Kuczma Functional equations in a single variable
— I. Prigogine and R.
Lefever
Symmetry breaking instabilities in dissipative sys-
tems. II
1969 R. FitzHugh Mathematical models of excitation and propagation
in nerve and muscle
— M. H
´
enon Numerical study of quadratic area-preserving
mappings
– E. N. Lorenz Atmospheric predictability as revealed by naturally
occurring analogies
— I. Prigogine et al. Symmetry breaking instabilities in biological
systems
1976 O. E. R
¨
ossler Chaotic behavior in simple reaction systems
302 Chaotic Modelling and Simulation
Year Name Title of Publication
— O. E. R
¨
ossler Chemical turbulence: Chaos in a simple reaction-
diffusion system
— O. E. R
¨
ossler Different types of chaos in two simple differential
equations
1978 M. J. Feigenbaum Quantitative universality for a class of nonlinear
transformations
1979 M. J. Feigenbaum The universal metric properties of nonlinear
transformations
— O. E. R
¨
ossler An equation for hyperchaos
— O. E. R
¨
ossler Continuous chaos — four prototype equations
— K. Ikeda Multiple-valued stationary state and its instability of
the transmitted light by a ring cavity system
1980 K. Ikeda et al. Optical turbulence: Chaotic behaviour of transmit-
ted light from a ring cavity
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