Dynamical Systems 3 Nonlinear systems

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Dynamical Systems 3Nonlinear systems
Ing. Jaroslav Jíra, CSc.
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Stability of Nonlinear SystemsFirst Lyapunov
method linearization
To examine stability of nonlinear systems we have
to use more complicated tools than for linear
ones.
The first step, which frequently works, is the
method of linearization, sometimes called first
Lyapunov method. The principle is expanding of
the right-hand side function in the equation of
motion around the fixed point into the Taylor
series with neglecting of higher order members.
Taylor series
Neglecting the higher order terms we obtain
Being at an equilibrium point, we know, that
f(x) dx/dt 0 , so
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Since the derivative of a constant is zero, then
and consequently
The original system
Linearized system
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Now we have to distinguish between the Jacobian
matrix of the original system and the matrix for
the linearized system at the specific fixed
point, lets say Df and Df(x). While the
Jacobian matrix of the original nonlinear system
Df contains variables and constants, the Jacobian
matrix of the linearized system at the fixed
point Df(x) contains just constants.
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Example of the system linearization
The system is defined by the set of equations
To find fixed points of the system, we have to
solve equations
We find two fixed points
Then we can do necessary calculations for the
Jacobian matrix
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Before examining stability of particular fixed
points we can make a preview of the dynamical
flow by the Mathematica program
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More detailed previews of areas around fixed
points
The area around the xA looks like a saddle
point, i.e. the fixed point should be unstable
The area around the xB looks like a spiral sink,
i.e. the fixed point should be stable
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Jacobian matrix of the original system is
For the first fixed point we have
Eigenvalues of this matrix are approx. ?11.9016
and ?2 -1.4874 Conclusion near xA the system
behaves like a linear system with one positive
and one negative eigenvalue, i.e. the system is
unstable here.
For the second fixed point we have
Eigenvalues of this matrix are approx. ?12
-1.2071 /- 1.171i Conclusion near xB the
system behaves like a linear system with complex
eigenvalues with negative real part, i.e. the
system is stable here.
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What can we do if linearization method cannot
decide? Example 2 unusually damped harmonic
oscillator
We have a harmonic oscillator, where the mass m
moves in very viscous medium, where the damping
force is proportional to the cube of the
velocity.
Set of differential equations describing the
system
After setting vector variable y we can write
Fixed point result from equations
There is only one fixed point
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The Jacobian matrix
Jacobian matrix for linearized system at the
fixed point
Eigenvalues for this system are ?12 /- i, so
they have zero real part and the method of
linearization cannot decide about the stability.
The graph shows phase diagram for µ0.25, x02
and v00. Even from the graph it is not clear if
the trajectory will converge to (0,0) point or if
it will remain at certain non-zero distance from
it.
We will have to use another tool Lyapunov
functions
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Second Lyapunov method Lyapunov functions
If we look on the damped harmonic oscillator from
the point of view of energy, there is clear, that
the system is still losing energy and sooner or
later it must stop at the equilibrium point
(0,0).
The principle of the second Lyapunov method is to
find a function V(x) that represents energy or
generalized energy and satisfies the following
conditions.
1. the function V(x) is continuously
differentiable around the fixed point
2. positive definite V
3. negative definite dV/dt
Additional condition to 3 at any state
where the 3rd condition is
considered satisfied if the system immediately
moves to a state, where
If we succeed in finding such function, then the
fixed point x is stable.
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Application of the second Lyapunov method on our
Example 2
The total energy of a harmonic oscillator
We simplify by taking m1 and k1
This function is continuously differentiable
around zero and is positive for all states except
for the fixed point (0,0), i.e. the first and
second condition are satisfied. This means that
we have a Lyapunov candidate function.
The formula for the derivative of E
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The final result for the time derivation is

• We can notice that dE/dt is always negative
  except for the v0, where dE/dt0. The velocity
  is zero in three situations
• At the fixed point, which is in conformity with
  the third condition
• At the instant, where the spring is maximally
  compressed
• At the instant, where the spring is maximally
  extended
• Situations 2 and 3 satisfy the additional
  condition, because the system immediately moves
  from here to the state where dE/dtlt0. Also the
  third condition is satisfied.

Conclusion our Lyapunov candidate function
satisfies all conditions for the Lyapunov
function, so examined fixed point (0,0) is
stable.
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How to estimate Lyapunov candidate functions?
If we examine a physical system, we should
calculate the energy. If the state vector is x
and the fixed point is 0, we could try
If the fixed point is not at the origin of
coordinate system, we have to modify
If we are not successful, then we can try
If it still does not work, we can use a general
quadratic form
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Classification of Stability of Nonlinear Systems
1. Lyapunov stability fixed point x is a stable
equilibrium if for every neighborhood U of x
there is a neighborhood such that
every solution x(t) starting in V remains in U
for all times
Lyapunov stability of an equilibrium means that
solutions starting "close enough" to the
equilibrium remain "close enough" forever. Such
fixed point is considered Lyapunov stable or
neutrally stable.
In this case the third condition for Lyapunov
function is satisfied, when dV/dt lt0. The time
derivative must be negative semidefinite.
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2. Asymptotic stability fixed point x is
asymptotically stable if it is Lyapunov stable
and additionally V can be chosen so that
for all
Asymptotic stability means that solutions that
start close enough not only remain close enough
but also eventually converge to the
equilibrium Such fixed point is considered
asymptotically stable.
In this case the third condition for Lyapunov
function is satisfied, when dV/dt lt 0. The time
derivative must be negative definite.
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3. Exponential stability fixed point x is
exponentially stable if there is a neighborhood V
of x and a constant agt0 such that
Exponential stability means that solutions not
only converge, but in fact converge faster than
or at least as fast as the exponential function
Exp(-at).
Exponentially stable equilibria are also
asymptotically stable, and hence Lyapunov stable.
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Bifurcations
Bifurcation is a qualitative change of phase
portrait in the area of attraction which can be
achieved by change of driving parameter when
passing through the critical value. We
distinguish two principal types of
bifurcations Global bifurcation its effects
are not limited by the neighbourhood of a point
or cycle in the phase space. It can not be
detected by analysis of fixed point stability.
Local bifurcation its effects are limited by
the neighbourhood of a point or cycle in the
phase space. Fixed points may appear or disappear
due to the parameter change, they change their
stability, or even break apart into periodic
points. Such bifurcation can be analysed entirely
through changes in the local stability properties
of equilibria, periodic orbits or other invariant
sets.
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The Logistic Equation
The logistic equation, also known as Verhulst
equation, is a formula for approximating the
evolution of an animal population over time.
Contrary to the bacteria model, living conditions
for animals significantly vary during the year.
Some species are fertile just for particular
season of year, not every existing animal
reproduce etc. For this reason, the system might
be better described by a discrete difference
equation than a continuous differential equation.
where xn is an actual population in the current
year, xn1 is population in the next year and r
is combined rate for reproduction and for
starvation. Zero value for the x means dead
population and x1 means population on its limit.
Now we will examine what happens with stability
and fixed points, if we try to change r. The only
sure thing without any computations is, that for
x(0)0 we have a stable fixed point meaning dead
population for any r.
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Development of the population for various values
of r.
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Bifurcation diagram for the logistic equation
The graph is an output of the Mathematica
program. The initial value was x(0)0.1 and 300
iterations were calculated for every r. For the
value r3 we can observe the first bifurcation
(doubling of the functional dependence). Another
bifurcations follow for r3.449, r3.544 etc.
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• The bifurcation diagram can be divided into 4
  parts
• Extinction (rlt1) if the growth rate is less than
  1 the system "dies
• Fixed point area (1ltrlt3) the series tends to a
  single value for any initial x0
• Oscillation area (3ltrlt3.57) The series jumps
  between two or more discrete states.
• Chaos area (3.57ltrlt4) the system can evaluate to
  any position at all with no apparent order
• For higher values of r (rgt4) all solutions zoom
  to infinity and the modeling aspects of this
  function become useless.

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Basic types of local bifurcations
1. Saddle node (fold) bifurcation
2. Period doubling (flip) bifurcation
3. Pitchfork bifurcation
4. Transcritical bifurcation
5. Hopf bifurcation
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1. Saddle node (fold) bifurcation
In this bifurcation two fixed points collide and
annihilate each other.
Differential equation of the system
For rlt0 there are two fixed points a stable at
and unstable at
For rgt0 there are no fixed points
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Example for two-dimensional system, where r -2
Differential equations
Fixed points
The Jacobian matrix
Linearized Jacobian matrix for xA
Eigenvalues for this Jacobian matrix
Conclusion the fixed point xA is a saddle point
since there are real eigenvalues of various
signs.
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Linearized Jacobian matrix for xB
Eigenvalues for this Jacobian matrix
Conclusion the fixed point xB is an attracting
node since there are real eigenvalues, both of
negative signs.
The name of this bifurcation is derived from the
pair of these two types of fixed points saddle
and node.
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An example of the saddle-node bifurcation for
Fixed points are given by
unstable stable
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2. Period doubling (flip) bifurcation
This type of bifurcation can be observed at the
logistic equation. If we accept also negative
values, we observe period halving in the left
part of the graph and period doubling in the
right part.
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3a. Pitchfork bifurcation Supercritical case
Differential equation of the system
In this bifurcation one fixed point splits into
three various ones.
For rlt0 there is just one stable fixed point at
x0
For rgt0 there is one unstable fixed point at x0
and two stable fixed points at
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3b. Pitchfork bifurcation Subcritical case
In this bifurcation three various fixed points
annihilate into one fixed point.
Differential equation of the system
For rgt0 there is just one unstable fixed point at
x0
For rlt0 there is one stable fixed point at x0
and two unstable fixed points at
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4. Transcritical bifurcation
In this bifurcation there is one stable and one
unstable fixed point and they exchange their
stability when they collide.
Differential equation of the system
For rlt0 there is one stable fixed point at x0
and one unstable fixed point for xr
For rgt0 there is one unstable fixed point at x0
and one stable fixed point for xr
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5. Hopf bifurcation
This bifurcation is a two-dimensional one. In
this bifurcation a single fixed point changes
into a limit cycle or vice versa.
Differential equations of the system
There is only one fixed point
Jacobian matrix
linearized Jacobian matrix
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Eigenvalues determination
Characteristic equation yields two complex
conjugated roots
According to our previous experience we can say,
that for rlt0 there is a stable fixed point and
for rgt0 there is an unstable fixed point.
To be able to decide about r0 we have to use a
Lyapunov function, choosing candidate function
We can clearly see, that for r0 the dV/dt is
outside the fixed point always negative, hence we
have a Lyapunov function. This also tells us,
that for r0 the fixed point is Lyapunov stable
and also asymptotically stable.
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Phase diagram for r-0.1, x101 and x200
Phase diagram for r0, x101 and x200
Phase diagram for r0.1, x101 and x200
Phase diagram for r0.1, x100.2 and x200
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3D interpretation of the Hopf Bifurcation. For
negative values of r the system converges to the
fixed point (0,0), for r0 the system still
converges to the fixed point, but very slowly.
For positive values of r the attractor is not an
unstable fixed point (0,0) but a limit cycle
regardless the starting point, i.e. it does not
matter whether we start inside or outside the
cycle. Diameter of the limit cycle raises with
raising parameter r.
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A program in Mathematica, which calculates the
Hopf bifurcation
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General rules concerning bifurcations.
Continuous systems a local bifurcation appears,
when an eigenvalue has zero real part. If the
eigenvalue is zero, then there is a saddle-node,
pitchfork or transcritical bifurcation. If
eigenvalues have zero real part, but they are
complex conjugated, then there is a Hopf
bifurcation.
Discrete systems a local bifurcation appears,
when the modulus of eigenvalue is equal to one.
If the eigenvalue is 1, then there is a
saddle-node, pitchfork or transcritical
bifurcation. If the eigenvalue is -1, then there
is a period doubling bifurcation. If there are
two complex conjugated eigenvalues with modulus
equal to one, then there is a Hopf bifurcation.