dynamical systems 1 introduction ing. jaroslav jíra, csc
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Dynamical Systems 1
Introduction
Ing. Jaroslav Jíra, CSc.
Definition
Dynamical system is a system that changes over time according to a set of fixed rules that determine how one state of the system moves to another state. Dynamical system is a state space (phase space) together with a set of functions describing change of the system in time.
A dynamical system has two partsa) a State space, which determines possible values of the state
vector. State vector consists of a set of variables whose values can be within certain interval. The interval of all possible values form the entire state space.
b) a Function, which tells us, given the current state, what the state of the system will be in the next instant of time
A state vector can be described by
)](),......,(),([)( 21 txtxtxtx n
),....,,(),....,,....,,(),,....,,( 21212211 nnnn xxxfxxxfxxxf
),....,,(
.
.
),....,,(
),....,,(
21
21222
21111
nnnn
n
n
xxxfxdt
dx
xxxfxdt
dx
xxxfxdt
dx
A function can be described by a single function or by a set of functions
Entire system can be then described by a set of differential equations – equations of motion
Classification of Dynamical Systems
Dynamical system can beeither or
Linear Nonlinear
Autonomous Nonautonomous
Conservative Nonconservative
Discrete Continous
One-dimensional Multidimensional
Linear system – a function describing the system behavior must satisfy two basic properties
• additivity
• homogeneity )()()( yfxfyxf
)()( xfxf
222
22222
5)(525)5()5(
)()(2)()(
xxfxxxf
yxyfxfyxyxyxyxf
For example f(x) = 3x; f(y) = 3y;• additivity f(x+y) = 3(x+y) = 3x + 3y = f(x) + f(y)• homogeneity 5 * f(x) = 5* 3x = 15x = f(5x)
Nonlinear system is described by a nonlinear function. It does not satisfy previous basic properties
For example f(x) = x2; f(y) = y2;
Example: we have two systemsSystem A: System B:
Clearly , therefore the system is not time-invariant or is nonautonomous.
Autonomous system is a system of ordinary differential equations, which do not depend on the independent variable. If the independent variable is time, we call it time-invariant system.
Condition: If the input signal x(t) produces an output y(t) then any time shifted input, x(t + δ), results in a time-shifted output y(t + δ)
)(10)( txty
System A:Start with a delay of the input
Now we delay the output by δ
Clearly therefore the system is time-invariant or autonomous.
System B:
Conservative system - the total mechanical energy remains constant, there are no dissipations present, e.g. simple harmonic oscillator
Start with a delay of the input
Now delay the output by δ
Nonconservative (dissipative) system – the total mechanical energy changes due to dissipations like friction or damping, e.g. damped harmonic oscillator
The system can be solved by iteration calculation. Typical example is annual progress of a bank account. If the initial deposit is 100000 crowns and annual interest is 3%, then we can describe the system by
)()(
))(()1(
)0(
0
0
xfkx
kxfkx
xx
k
100000*03.1)(
)(03.1)1(
100000)0(
kkx
kxkx
x
Discrete system – is described be a difference equation or set of equations. In case of a single equation we are also talking about one-dimensional map. We denote time by k, and the system is typically specified by the equations
Continous system – is described by a differential equation or a set of equations.
For example, vertical throw is described by initial conditions h(0), v(0) and equations
)(´
)0( 0
xfx
xx
gtv
tvth
)´(
)()´(
where h is height and v is velocity of a body.
Definition from the Mathematica: When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system.
Multidimensional system is described by a vector of functions like
where x is a vector with n components, A is n x n matrix and B is a
constant vector
One-dimensional system is described by a single function like
where a,b are constants.
btaxtx
bkaxkx
)()´(
)()1(
BA
BA
)()´(
)()1(
txtx
kxkx
Repetition From the Matrix Algebra
Matrix A Matrix B Vector C
333231
232221
131211
aaa
aaa
aaa
A
333231
232221
131211
bbb
bbb
bbb
B
3
2
1
c
c
c
C
333323321331323322321231313321321131
332323221321322322221221312321221121
331323121311321322121211311321121111
bababababababababa
bababababababababa
bababababababababa
A.B
333232131
323222121
313212111
cacaca
cacaca
cacaca
CA.
Identity matrix (unit matrix),
we use symbol E or I
100
010
001
E
312213332112322311322113312312332211)det( aaaaaaaaaaaaaaaaaa A
3231
2221
1211
333231
232221
131211
aa
aa
aa
aaa
aaa
aaa
A
2212
2111
dd
ddD
12212211)det( dddd D
Determinant
Inversion matrix – 2x2
dc
baA
ac
bd
cbadac
bd
dc
ba 1
)det(
11
1
AA
Inversion matrix – 3x3
The basic matrix operations in the Mathematica
uu A
For the eigenvalue calculation we use the formula
0)det( EA
Eigenvalues and Eigenvectors
The eigenvectors of a square matrix are the non-zero vectors which, after being multiplied by the matrix, remain proportional to the original vector (i.e. change only in magnitude, not in direction). For each eigenvector, the corresponding eigenvalue λ is the factor by which the eigenvector changes when multiplied by the matrix.
2
1
2
1
2221
1211
u
u
u
u
aa
aaTwo-dimensional example:
This represents a system of two linear equations:
2222121
1212111
uuaua
uuaua
After small rearrangement:0)(
0)(
222121
212111
uaua
uaua
0)(
uEA In the matrix form
This is a system of linear homogeneous equations. Such system has a nontrivial (nonzero) solution only in case when the matrix (A-λE) is singular.
The matrix is singular when its determinant is equal to zero.
0)det( EA
Short explanation
0)(
uEA For the eigenvector calculation we use the formula:
0
0
2
1
2221
1211
u
u
aa
aa
Two-dimensional example:
After finding eigenvalues we can say, that we found a diagonal matrix, which is similar to the original matrix. The diagonal matrix has the same properties like the original one for the purpose of solving dynamical system stability. We write the diagonal matrix in the form:
n
000
0...00
000
000
2
1
What are advantages of diagonal matrix?
1. Multidimensional discrete system.
)()1( kxkxA )0()( xkx k
A
The typical formula is: To obtain k-th element:
Raising matrices to the power is quite difficult, but in case of diagonal matrix we simply have:
kn
k
k
k
000
0...00
000
000
2
1
A
2. Multidimensional continous system.
)(txdt
xd A )exp()( 0 txtx A
The typical differential equation: Solution of the equation:
t
t
t
ne
e
e
t
000
0...00
000
000
)exp(2
1
A
Using matrices as an argument for the exponential function is much more difficult than raising them to the power, but in case of diagonal matrix we can write:
Example for eigenvalue and eigenvector calculation
51
24A
1
1
0
022
0
0
11
22
51
24
1
21
21
2
1
2
1
1
1
u
uu
uu
u
u
u
u
Initial matrix
1
2
02
02
0
0
21
21
51
24
2
21
21
2
1
2
1
2
2
u
uu
uu
u
u
u
u
Any vector that satisfies condition u1=u2 is an eigenvector for the λ=6
Any vector that satisfies condition u1=-2u2 is an eigenvector for the λ=3
Characteristic equation
Eigenvalues
Eigenvector calculation
051
24det)det(
EA
3;6
01892)5)(4(
21
2
Automatic eigenvalue and eigenvector calculation in the Mathematica
Trace of a matrix is a sum of the elements on the main diagonal
nnaaaTr ....)( 2211A
),....,,(
.
.
),....,,(
),....,,(
21
21222
21111
nnnn
n
n
xxxfxdt
dx
xxxfxdt
dx
xxxfxdt
dx
n
nnn
n
n
x
f
x
f
x
f
x
f
x
f
x
fx
f
x
f
x
f
...
.........
...
...
21
2
2
2
1
2
1
2
1
1
1
J
Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. This matrix is frequently being marked as J, Df or A.
6
248
457
013
Tr
Phase Portraits
A phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
The phase space of a two-dimensional system is called a phase plane, which occurs in classical mechanics for a single particle moving in one dimension, and where the two variables are position and velocity.
A phase portrait is a plot of single phase curve or multiple phase curves corresponding to different initial conditions in the same phase plane.
A phase curve is a plot of the solution of equations of motion in a phase plane (generally in a phase space).
A phase portrait of a simple harmonic oscillator
Differential equation
)sin(
)cos(
tA
v
tA
x
)(sin
)(cos
22
22
tA
v
tA
x
02 xx
)sin(
)cos(
tAv
tAx
Now we separate sine and cosine functions, raise both equations to the power of two and finally we add them.
where x is displacement, v is velocity, A is amplitude and ω is angular frequency.
1)(sin)(cos 2222
ttA
v
A
x
122
A
v
A
x
Final equation describes an ellipse
Solution of the equation
The following figure shows a phase portrait of a simple harmonic oscillator with ω=10 s-1 and initial conditions x(0)=1 m; v(0)=0 m/s
1101
22
vx
122
A
v
A
x
Oscillator with critical dampingω= 10s-1; δ= 10s-1
x(0)=1m; v(0)=0 m/s
Overdamped oscillatorω= 10s-1; δ= 20s-1
x(0)=1m; v(0)=0 m/s
Underdamped oscillatorω= 10s-1; δ= 1s-1
x(0)=1m; v(0)=0 m/s
Simple harmonic oscillatorinitial amplitudes are 1,2, …,10 m
Creating a phase portrait of an oscillator in Mathematica
Stability and Fixed PointsA fixed point is a special point of the dynamical system which does not change in time. It is also called an equilibrium, steady-state, or singular point of the system.
If a system is defined by an equation dx/dt = f(x), then the fixed point x~ can be found by examining of condition f(x~)=0. We need not know analytic solution of x(t).
For discrete time systems we examine condition x~ = f(x~)
An attractor is a set towards which a dynamical system evolves over time. It can be a point, a curve or more complicated structure
A stable fixed point: for all starting values x0 near the x~ the system converges to the x~ as t→∞.
A marginally stable (neutrally stable) fixed point: for all starting values x0 near x~, the system stays near x~ but does not converge to x~ .
An unstable fixed point: for starting values x0 very near x~ the system moves far away from x~
A perturbation is a small change in a physical system, most often in a physical system at equilibrium that is disturbed from the outside.
Phase portraits of basic three types of fixed points
STABLE MARGINALLY STABLE UNSTABLE
Example 1– bacteria in a jar
A jar is filled with a nutritive solution and some bacteria. Let b (for birth) be relative rate at which the bacteria reproduce and p (for perish) be relative rate at which they die. Then the population is growing at the rate r = b−p.
If there are x bacteria in the jar, then the rate at which the number of bacteria is increasing is (b − p)x, that is, dx/dt = rx. Solution of this equation fox x(0)=x0 is
rtextx 0)( This model is not realistic, because bacteria population goes to the ∞ for r>0. Actually, as the number of bacteria rises, they crowd each other, produce more toxic waste products etc. Instead of constant relative perish rate p we will assume relative perish rate dependent on their number px. Now the number of bacteria increases by bx and their number decreases by px2.
New differential equation will be
2pxbxdt
dx
2pxbxdt
dx
0)~(~0~~ 2
xpbx
xpxb
Differential equation,
initial number of bacteria x(0)=x0
To be able to find a fixed point, we have to set the right-hand side of the differential equation to zero.
There are two possible solutions,
i.e. we have two fixed points:
p
bx
x
2
1
~
0~
Analytic solution provided by Mathematica
First fixed point x~1= 0;
There are no bacteria, so none can be born, none can die, but after small contamination of the jar (perturbation), but smaller than b/p, we can see, that the number of bacteria will increase by dx/dt = bx-px2>0 and will never return to the zero state.Conclusion: this fixed point is unstable.
Second point x~2= b/p;
At this population level, bacteria are being born at a rate bx~=b(b/p) = b2/p and are dying at a rate px~2 = p(b/p)2 = b2/p, so birth and death rates are exactly in balance.
If the number of bacteria would be slightly increased, then dx/dt = bx-px2<0 and would return to equilibrium.
If the number of bacteria would be slightly decreased, then dx/dt = bx-px2>0 and would return to equilibrium.
Small perturbations away from x~ = b/p will self-correct back to b/p.
Conclusion: this fixed point is stable and is also an attractor of this system.
0 10 20 30 40t0.0
0.2
0.4
0.6
0.8
1.0x
b /p
0.2 0.4 0.6 0.8 1.0x
0.10
0.05
0.00
0.05
0.10
dx
dt
Graphical solution from Mathematica
Input parameters: b=0.2, p=0.5
Initial conditions: x0= 0.9 for blue curves
x0= 0.01 for red curves
Number of bacteria in time Phase portrait
Example 2 – predator and prey
Now we have a biological system containing two species – predators (wolves) and prey (rabbits).
Population of rabbits in time is r(t) and population of wolves is w(t).
Rabbits, left on their own, will reproduce with velocity dr/dt= ar, where a>0
Wolves, without rabbits, will starve and their population will decline with velocity dw/dt= -bw, where b>0
hrwbwdt
dw
grwardt
dr
Here are differential equations describing the closed system
When brought into the same environment, wolves will catch and eat rabbits. Loss to the rabbit population will be proportional to number of wolves w and number of rabbits r by a constant g (aggressivity of predators). Gain to the wolf population will be also proportional to r and w, this time by a constant h (effectivity of transformation of prey meat into the predator biomass).
Here is time dependence and phase diagram for both populations
a=0.3; b=0.1; g=0.002; h=0.001, initial number of rabbits r0=100, wolves w0=25
Number of populations over time Phase portrait
An attractor of this system drawn on the phase diagram is a limit cycle.
With higher rabbit natality and higher wolf mortality together with higher wolf aggressivity the changes are quicker
a=0.75; b=0.2; g=0.03; h=0.01
Number of populations over time Phase portrait
With extremely low rabbit natality and low wolf mortality togehter with high wolf aggressivity both populations will vanish
a=0.01; b=0.05; g=0.05; h=0.05
Number of populations over time Phase portrait
Calculation of the predator-prey system in the Mathematica
The phase portrait of the predator-prey system in the Mathematica