Dynamical Systems with Applications using MATLAB

Dynamical Systems with Applications using MATLAB
اسم المؤلف
Stephen Lynch
التاريخ
6 نوفمبر 2021
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Dynamical Systems with Applications using MATLAB
Stephen Lynch
Contents
1 A Tutorial Introduction to MATLAB
1.​1 Tutorial One:​ The Basics and the Symbolic Math Toolbox (1 h)
1.​2 Tutorial Two:​ Plots and Differential Equations (1 h)
1.​3 MATLAB Program Files or M-Files
1.​4 Hints for Programming
1.​5 MATLAB Exercises
References
2 Linear Discrete Dynamical Systems
2.​1 Recurrence Relations
2.​2 The Leslie Model
2.​3 Harvesting and Culling Policies
2.​4 MATLAB Commands
2.​5 Exercises
References
3 Nonlinear Discrete Dynamical Systems
3.​1 The Tent Map and Graphical Iterations
3.​2 Fixed Points and Periodic Orbits
3.​3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number3.​4 Gaussian and Hénon Maps
3.​5 Applications
3.​6 MATLAB Program Files
3.​7 Exercises
References
4 Complex Iterative Maps
4.​1 Julia Sets and the Mandelbrot Set
4.​2 Boundaries of Periodic Orbits
4.​3 MATLAB Commands
4.​4 Exercises
References
5 Electromagnetic Waves and Optical Resonators
5.​1 Maxwell’s Equations and Electromagnetic Waves
5.​2 Historical Background
5.​3 The Nonlinear SFR Resonator
5.​4 Chaotic Attractors and Bistability
5.​5 Linear Stability Analysis
5.​6 Instabilities and Bistability
5.​7 MATLAB Commands
5.​8 ExercisesReferences
6 Fractals and Multifractals
6.​1 Construction of Simple Examples
6.​2 Calculating Fractal Dimensions
6.​3 A Multifractal Formalism
6.​4 Multifractals in the Real World and Some Simple Examples
6.​5 MATLAB Commands
6.​6 Exercises
References
7 The Image Processing Toolbox
7.​1 Image Processing and Matrices
7.​2 The Fast Fourier Transform
7.​3 The Fast Fourier Transform on Images
7.​4 Exercises
References
8 Differential Equations
8.​1 Simple Differential Equations and Applications
8.​2 Applications to Chemical Kinetics
8.​3 Applications to Electric Circuits
8.​4 Existence and Uniqueness Theorem8.​5 MATLAB Commands
8.​6 Exercises
References
9 Planar Systems
9.​1 Canonical Forms
9.​2 Eigenvectors Defining Stable and Unstable Manifolds
9.​3 Phase Portraits of Linear Systems in the Plane
9.​4 Linearization and Hartman’s Theorem
9.​5 Constructing Phase Plane Diagrams
9.​6 MATLAB Commands
9.​7 Exercises
References
10 Interacting Species
10.​1 Competing Species
10.​2 Predator-Prey Models
10.​3 Other Characteristics Affecting Interacting Species
10.​4 MATLAB Commands
10.​5 Exercises
References
11 Limit Cycles11.​1 Historical Background
11.​2 Existence and Uniqueness of Limit Cycles in the Plane
11.​3 Nonexistence of Limit Cycles in the Plane
11.​4 Perturbation Methods
11.​5 MATLAB Commands
11.​6 Exercises
References
12 Hamiltonian Systems, Lyapunov Functions, and Stability
12.​1 Hamiltonian Systems in the Plane
12.​2 Lyapunov Functions and Stability
12.​3 MATLAB Commands
12.​4 Exercises
References
13 Bifurcation Theory
13.​1 Bifurcations of Nonlinear Systems in the Plane
13.​2 Normal Forms
13.​3 Multistability and Bistability
13.​4 MATLAB Commands
13.​5 Exercises
References14 Three-Dimensional Autonomous Systems and Chaos
14.​1 Linear Systems and Canonical Forms
14.​2 Nonlinear Systems and Stability
14.​3 The Rössler System and Chaos
14.​4 The Lorenz Equations, Chua’s Circuit, and the Belousov–
Zhabotinski Reaction
14.​5 MATLAB Commands
14.​6 Exercises
References
15 Poincaré Maps and Nonautonomous Systems in the Plane
15.​1 Poincaré Maps
15.​2 Hamiltonian Systems with Two Degrees of Freedom
15.​3 Nonautonomous Systems in the Plane
15.​4 MATLAB Commands
15.​5 Exercises
References
16 Local and Global Bifurcations
16.​1 Small-Amplitude Limit Cycle Bifurcations
16.​2 Gröbner Bases
16.​3 Melnikov Integrals and Bifurcating Limit Cycles from a Center
16.​4 Bifurcations Involving Homoclinic Loops16.​5 MATLAB and MuPAD Commands
16.​6 Exercises
References
17 The Second Part of Hilbert’s Sixteenth Problem
17.​1 Statement of Problem and Main Results
17.​2 Poincaré Compactification​
17.​3 Global Results for Liénard Systems
17.​4 Local Results for Liénard Systems
17.​5 Exercises
References
18 Neural Networks
18.​1 Introduction
18.​2 The Delta Learning Rule and Backpropagation
18.​3 The Hopfield Network and Lyapunov Stability
18.​4 Neurodynamics
18.​5 MATLAB Commands
18.​6 Exercises
References
19 Chaos Control and Synchronization
19.​1 Historical Background19.​2 Controlling Chaos in the Logistic Map
19.​3 Controlling Chaos in the Hénon Map
19.​4 Chaos Synchronization
19.​5 MATLAB Commands
19.​6 Exercises
References
20 Binary Oscillator Computing
20.​1 Brain-Inspired Computing
20.​2 Oscillatory Threshold Logic
20.​3 Applications and Future Work
20.​4 MATLAB Commands
20.​5 Exercises
References
21 Simulink
21.​1 Introduction
21.​2 Electric Circuits
21.​3 A Mechanical System
21.​4 Nonlinear Optics
21.​5 The Lorenz Equations and Chaos Synchronization
21.​6 ExercisesReferences
22 Examination-Type Questions
22.​1 Examination 1
22.​2 Examination 2
22.​3 Examination 3
23 Solutions to Exercises
23.​1 Chapter 1
23.​2 Chapter 2
23.​3 Chapter 3
23.​4 Chapter 4
23.​5 Chapter 5
23.​6 Chapter 6
23.​7 Chapter 7
23.​8 Chapter 8
23.​9 Chapter 9
23.​10 Chapter 10
23.​11 Chapter 11
23.​12 Chapter 12
23.​13 Chapter 13
23.​14 Chapter 14
23.​15 Chapter 15
23.​16 Chapter 16
23.​17 Chapter 17
23.​18 Chapter 18
23.​19 Chapter 19
23.​20 Chapter 20
23.​21 Chapter 21
23.​22 Chapter 22
Index
Aabs
orptive nonlinearity
action potential
activation
function
level
potential
ADALINE network
affine linear transformation
age class
Airy equation
algebraicity of limit cycles
All or None principle
ampere
Ampere’s law
anemia
angular frequency of the wave
ants and termites
aperiodic
aperiodic behavior
applying a damper
arrhythmic
Artificial Intelligence Group
artificial neural networks
associative memory
asymptotic expansion
asymptotically stable
critical point
asynchronous updating
attractor
attributes
autocatalysisautonomous differential equation
autonomous system
auxiliary system
auxiliary system approach
average Lyapunov exponent
axial flow compressors
axon
axon terminals
Bba
ckpropagation
backpropagation algorithm
backward training
ballistic propagation
bandwidth
Barnsley’s fern
basin of attraction
basis
batch data
Belousov-Zhabotinski reaction
Bendixson’s criteria
bias
bifurcating limit cycles from a center
bifurcation
at infinity
curve
diagram
point
value
bifurcation diagram
CR resonator
Duffing equation
Gaussian map
logistic map
neuromodule
SFR resonator
binary half adderbiology
bipolar activation function
bistability
bistable
cycle
device
optical resonator
region
solution
bistable neuromodule
bistable region
block
block icon
blowflies
bluegill sunfish
Boston housing data
boundaries of periodic orbits
box-counting dimension
brain functions
butterfly effect
Cca
nonical form
Cantor
multifractal
Cantor set
capacitance
cardioid
cardiology
carrying capacity
cavity ring (CR) resonator
cavity round-trip time
cell body
center
manifold
theorem
center eigenspacecenter manifold
changing the system parameters
chaologist
chaos
control
control in an SFR
synchronization
chaos control
OGY method
periodic proportional pulses
chaos game
chaotic
attractor
dynamics
phenomena
chaotic attractor
Hénon map
neuromodule
Sierpiński
Chapman cycle
characteristic
equation
exponent
characteristic multiplier
charge density
chemical
kinetics
signals
chemical kinetics
chemical law of mass action
chemical reaction
chemical substance
Chua’s circuit
circle map
circular frequency of light
classical symmetry argument
classification of critical pointsclipping problem
clockwise bistable cycle
clockwise hysteresis
clockwise hysteresis loop
cluster
coarse Hölder exponent
codimension-1 bifurcation
codimension-2 bifurcation
Cody Coursework
coexistence
coexisting chaotic attractors
col
Command Window
commutative ring
competing species
complete synchronization
completely
reduced
completely integrable
complex eigenvalues
complex iterative equation
compound interest
computer algebra
concentrations
conditional Lyapunov exponents
conductivity
conformal mapping
conservation of energy
conservation of mass
conservative
contact rate
content-addressable memory
continuous Hopfield model
control curves
control engineering
control parameter
control regioncontrolling chaos
Hénon map
logistic map
conversational agents
convex closed curve
convoluted surfaces
core area of the fiber
corollary to Poincaré-Bendixson theorem
correlation dimension
coulomb
Coulomb’s law
counterclockwise hysteresis
coupler
critical point
at infinity
culling
policy
current
density
cusp
cylindrical polar coordinates
Ddam
ping
damping coefficient
dangerous bifurcation
Daphnia dentifera
data mining
databases
defibrillator
defraction
degenerate
critical point
degenerate node
degree
degree lexicographical order
deleted neighborhooddelta learning rule
dendrites
depolarization
depolarize
the Poincaré map test
desired vector
deterministic
chaos
deterministic chaos
deterministic system
D
f
dielectric
difference equation
differential amplifier
diffusion limited aggregates (DLA)
dimension
direction
field
vector
discrete Fourier transform
discrete Hopfield model
dispersive nonlinearity
displacement function
distributive laws
divergence test
domain of stability
double-coupler fiber ring resonator
double-scroll attractor
double-well potential
D
q
driver system
Duffing
system
Duffing equation
Dulac’s criteria
Dulac’s theoremEE
C
economic model
economics
eigenvector
electric
circuit
displacement
displacement vector
field
field strength
flux density
electric circuit
electromotive force (EMF)
elliptic integral
energy level
enrichment of prey
environmental effects
epidemic
epoch
point
ergodicity
error backpropagation rule
error function
erythrocytes
E S
E
U
Euclidean dimension
exact
exact differential equation
Examination-type questions
excitatory
excitory
existence and uniqueness
limit cycle
existence theoremextinct
Ff(α
)spectrum
Fabry-Perot
interferometer
resonator
farad
Faraday’s law
of induction
fast Fourier transform
feedabck mechanism
feedback
feedforward single layer network
Feigenbaum constant
FFT
fiber parameters
Fibonacci sequence
field
fine focus
first integral
first iterative method
first return map
first-order difference equation
fish population
Fitzhugh-Nagumo
equations
oscillator
fixed point
period m
period N
period one
period two
fixed size box-counting algorithm
fixed weight box-counting algorithm
flow
focal valuesfold bifurcation
forced system
forward rate constant
fossil dating
Fourier spectrum
Fourier transform
fractal
attractor
dimension
Cantor set
Koch curve
Koch square
Sierpiński triangle
geometry
structure
fractal structure
fragmentation ratios
function approximators
fundamental memory
fuzzy discs
Ggai
n
Gauss’s law
electricity
magnetism
Gauss-Newton method
Gaussian input pulse
Gaussian map
Gaussian pulse
generalized delta rule
generalized fractal dimension
generalized fractal dimensions
generalized mixed Rayleigh Liénard equations
generalized synchronization
global bifurcation
globally asymptotically stableglucose in blood
Gröbner bases
gradient
gradient vector
graph block
graphic
graphical method
gray scale
Green’s theorem
grey scale
Gross National Product (GNP)
HHé
non map
Hénon-Heiles Hamiltonian
Hamiltonian
Hamiltonian systems
with two degrees of freedom
handcrafted patterns
hard bifurcation
Hartman’s theorem
harvesting
policy
Hausdorff dimension
Hausdorff index
Hausdorff–Besicovich dimension
Heaviside function
Hebb’s learning law
Hebb’s postulate of learning
henry
heteroclinic
bifurcation
orbit
heteroclinic orbit
heteroclinic tangle
heterogeneous
hidden layerHilbert numbers
history
Hodgkin-Huxley equations
Holling-Tanner model
homoclinic
bifurcation
loop
orbit
tangle
homogeneous
homogeneous differential equation
Hopf
bifurcation
singularity
Hopfield network
Hopfield neural network
horseshoe dynamics
host-parasite system
human population
hyperbolic
attracting
critical point
fixed point
repelling
stable limit cycle
unstable limit cycle
hyperbolic iterated function system
hyperpolarize
hyperpolarized
hysteresis
Iid
eal
Ikeda map
image analysis
image compression
incidentindex
inductance
infected population
infectives
inflation unemployment model
information dimension
inhibitory
initial value problem
input vector
insect population
instability
instant physician
integrable
integrate and fire neuron
integrating factor
integrator block
intensity
interacting species
intermittency
route to chaos
invariant
axes
inverse discrete Fourier transform
inverted Koch snowflake
inverted Koch square
isoclines
isolated periodic solution
isothermal chemical reaction
iterated function system (IFS)
iteration
JJa
cobian
Jacobian matrix
Jordan curve
jth point of period i
Julia setKKA
M
theorem
tori
kernel machines
Kerr effect
Kerr type
kinetic energy
Kirchhoff’s
current law
laws
voltage law
Koch
snowflake
Koch curve
Koch square
Llad
ybirds and aphids
laminarize
Landolt clock
Laplace transform
large-amplitude limit cycle
bifurcation
laser
law of mass action
learning process
learning rate
least mean squared (LMS) algorithm
Legendre transformation
Leslie
matrix
model
lexicographical order
Liénardequation
Liénard plane
Liénard system
large parameter
local results
Liénard’s theorem
lie detector
limit cycle
3-D
hyperbolic
neuron
nonexistence
Lindstedt-Poincaré technique
linear differential equation
linear phase shift
linear stability analysis
linear transformation
linearization
linearized system
Lipschitz
continuous
Lipschitz condition
load
local bifurcation
log-log plot
logic gates
logistic equation
logistic growth
logistic map
Lorenz
attractor
equations
Lorenz equations
loss in the fiber
Lotka-Volterra model
low pass filter
low-gain saturation functionlowest common multiple
Lyapunov
quantity
stability
Lyapunov domain of stability
Lyapunov exponent
Lyapunov function, 254, 257, 290, 336, 367
Hopfield network
Lyapunov quantities
Lyapunov stability theorem
lynx and snowshoe hares
MMfile
magnetic field vector
magnetic flux
magnetostrictive ribbon
Mandelbrot
Mandelbrot set
manifold
MATLAB
based exam
MATLAB apps
maximal interval of existence
Maxwell’s equations
Maxwell-Bloch equations
Maxwell-Debye equations
McCulloch-Pitts neuron
mean infectivity period
mean latency period
mechanical oscillator
mechanical system
Melnikov
function
integral
memory devices
memristancememristor
meteorology
method of multiple scales
method of steepest descent
micro-parasite—zooplankton—fish system
minimal chaotic neuromodule
minimal Gröbner basis
mixed fundamental memories
mixing
modulo
monomial
ordering
mortgage assessment
motif
multidegree
multifractal
formalism
Hénon map
Sierpiński triangle
spectra
multistability
multistable
murder
mutual exclusion
mylein sheath
Nnat
ional income
negative limit set
negative semiorbit
negatively invariant
net reproduction rate
network architecture
neural network
neurodynamics
neuromodule
neuronmodule
neuronal model
neurons
neurotransmitters
Newton’s law of cooling
Newton’s law of motion
Newton’s method
noise
NOLM
with feedback
nonautonomous system
nonconvex closed curve
nondegenerate
critical point
nondeterministic chaos
nondeterministic system
nonexistence of limit cycles
nonhyperbolic
critical point
fixed point
nonlinear
center
optics
nonlinear optics
nonlinear phase shift
nonlinear refractive index coefficient
nonlinearity
nonperiodic behavior
nonsimple canonical system
normal form
normalized eigenvector
not robust
numeric::solve
Numerical solutions
Ooc
casional proportional feedback (OPF)OGY method
ohm
Ohm’s law
optical
bistability
computer
fiber
fiber double ring
memories
resonator
sensor
optimal sustainable
orbit
ordinary differential equation
oscillation of a violin string
output vector
ozone production
Ppa
rasitic infection
partial differential equations
partition function
Pascal’s triangle
circuit
Peixoto’s theorem in the plane
pendulum
perceptron
period
bubblings
limit cycle
of limit cycle
undoublings
period-doubling
period-doubling bifurcations to chaos
period-n cycle
period-one behavior
period-three behaviorperiod-two
period-two behavior
periodic
orbit
windows
periodic behavior
periodicity
permittivity of free space
perturbation methods
phase portrait
phase shift
physiology
piecewise linear function
pinched hysteresis
pitchfork bifurcation
pixels
planar manifold
plastics
Poincaré
section
Poincaré compactification
Poincaré map
Poincaré-Bendixson theorem, 229, 321, 364
Poisson brackets
polar coordinates
pole placement technique
pollution
polymer
population
population model
population of rabbits
positive limit set
positive semiorbit
positively invariant
potato man
potential difference
potential energypotential function
power
of a waterwheel
spectra
power law
power-splitting ratio
Prandtl number
preallocate
predation
rate
predator-prey
models
system
probe vector
propagation
psychological profiling
Pyragas’s method
Qqth
moment
qualitative behavior
qualitatively equivalent
quasi-periodicity
quasiperiodic
route to chaos
quasiperiodic forcing
RRö
ssler attractor
Rössler system
radioactive decay
random behavior
rate constant
rate-determining steps
rationally independent
Rayleigh numberRayleigh system
rate equation
real distinct eigenvalues
recurrence relation
recurrent neural network
red and grey squirrels
red blood cells
reduced
reduced Gröbner basis
reflected
refractive index
refractive nonlinearity
refuge
regulator poles
relative permeabilities
relative permittivities
repeated real eigenvalues
repolarization
resistance
resonance terms
resonant
response system
restoring coefficient
restoring force
restrictions in programming
return map
constant
reversed fundamental memories
ring
ringing
RLC circuit
robust
rubbers
SSpolynomial
saddle pointsaddle-node bifurcation
safe bifurcation
scaling
scope
script file
sea lions and penguins
seasonal effects
seasonality
second iterative method
second order linear difference equation
second part of Hilbert’s sixteenth problem
differential equation
secular term
sedimentary rocks
self-similar
self-similar fractal
self-similarity
semistable
limit cycle
semistable critical point
sensitivity to initial conditions, 34, 39, 294
separable differential equation
separation of variables
separatrix
cycle
series solutions
SFR
resonator
sharks and fish
Sierpiński triangle
sigmoid function
signal processing
simple canonical system
simple nonlinear pendulum
simply connected domain
simulation
singletsingular node
Smale horseshoe map
Smale-Birkhoff theorem
perturbation
small perturbation
small-amplitude limit cycle
soft bifurcation
solar system
solution curves
soma
spatial vector
spectrum of Lyapunov exponents
speed of light
spike train
spin-glass states
spirals
spurious steady state
SR flip-flop
stability
diagram
stable
critical point
fixed point
focus
limit cycle
manifold
node
stable critical point
staircases
stationary point
steady state
stiff system
stiffness
stochastic methods
stock market analysis
stoichiometric equations
Stokes’s theoremstrange attractor
stretching and folding
strictly dominant
structurally
stable
unstable
subcritical Hopf bifurcation
subharmonic oscillations
summing junction
supercritical Hopf bifurcation
supervised learning
susceptible population
susceptibles
sustainable
switches
synaptic cleft
synaptic gap
synaptic vesicles
synaptic weights
synchronization
synchronization of chaos
synchronous updating
Ttar
get vector
targeting
τ( q )
Taylor series expansion
tent map
three-dimensional system
threshold
threshold value
time series
chaos detection
plot
Toda Hamiltonian
tolerancetopological dimension
topologically equivalent
torus
total degree
totally connected
totally disconnected
training
trajectory
transcritical bifurcation
transfer function
transient
transmitted
transversal
transversely
travelling salesman problem
triangular input pulse
triangular pulse
trivial fixed point
turbulence
two-neuron module
Uun
constrained optimization problem
uncoupled
uniform asymptotic expansion
uniform harvesting
unipolar activation function
uniqueness theorem
universality
Unix
unstable
critical point
fixed point
focus
limit cycle
manifold
nodeunstable critical point
unsupervised learning
vacuum
value of homes in Boston
van der Pol equation
van der Pol system
vector field
plot
Vvec
torize
velocity of light
Verhulst’s equation
viscosity
viscous fingering
volt
voltage drop
Wwav
e equations
wave vector
wavelength
light
W C
Windows
wing rock
W S
W U
XXray spectroscopy
XOR gate
YYou Tube
youngest class harvesting

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