Dynamical Systems 2 Topological classification

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Dynamical Systems 2Topological classification
Ing. Jaroslav Jíra, CSc.
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More Basic Terms
Basin of attraction is the region in state space
of all initial conditions that tend to a
particular solution such as a limit cycle, fixed
point, or other attractor.
Trajectory is a solution of equation of motion,
it is a curve in phase space parametrized by the
time variable.
Flow of a dynamical system is the expression of
its trajectory or beam of its trajectories in the
phase space, i.e. the movement of the variable(s)
in time
Nullclines are the lines where the time
derivative of one component of the state variable
is zero.
Separatrix is a boundary separating two modes of
behavior of the dynamical system. In 2D cases it
is a curve separating two neighboring basins of
attraction.
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A Simple Pendulum

• Differential equation

After transformation into two first order
equations
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An output of the Mathematica program Phase
portratit of the simple pendulum
Used equations
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A simple pendulum with various initial conditions
Stable fixed point f00
f045
f090
f0135
Unstable fixed point f0180
f0170
f0190
f0220
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A Damped Pendulum

• Differential equation

After transformation into two first order
equations
Equation in Mathematica NDSolvex't yt,
y't -.2 yt - .26 Sinxt, and phase
portraits
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A Damped Pendulumcommented phase portrait
Nullcline determination
At the crossing points of the null clines we can
find fixed points.
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A Damped Pendulumsimulation
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Classification of Dynamical SystemsOne-dimensiona
l linear or linearized systems
Time Derivative at x Fixed point is
Continous f(x)lt0 Stable
Continous f(x)gt0 Unstable
Continous f(x)0 Cannot decide
Discrete f(x)lt1 Stable
Discrete f(x)gt1 Unstable
Discrete f(x)1 Cannot decide
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Verification from the bacteria example
Bacteria equation
Derivative
1st fixed point - unstable
2nd fixed point - stable
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Classification of Dynamical SystemsTwo-dimensiona
l linear or linearized systems
Set of equations for 2D system
Jacobian matrix for 2D system
Calculation of eigenvalues
Formulation using trace and determinant
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Types of two-dimensional linear systems1.
Attracting Node (Sink)
Equations
Jacobian matrix
Eigenvalues ?1 -1 ?2 -4
Eigenvectors
Solution from Mathematica
Conclusion there is a stable fixed point, the
attracting node (sink)
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A quick preview by the Vectorplot function in the
Mathematica
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Meaning of the Eigenvectorexample of modified
attracting node
Equations
Jacobian matrix
Eigenvalues ?1 -3.62 ?2 -1.38
Eigenvectors
Eigenvector directions are emphasized by black
arrows
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2. Repelling Node
Equations
Jacobian matrix
Eigenvectors
Eigenvalues ?1 1 ?2 4
Solution from Mathematica
Conclusion there is an unstable fixed point, the
repelling node
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3. Saddle Point
Equations
Jacobian matrix
Eigenvectors
Eigenvalues ?1 -1 ?2 4
Solution from Mathematica
Conclusion there is an unstable fixed point, the
saddle point
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4. Spiral Source (Repelling Spiral)
Equations
Jacobian matrix
Eigenvectors
Eigenvalues ?1 12i ?2 1-2i
Solution from Mathematica
Conclusion there is an unstable fixed point, the
spiral source sometimes called unstable
focal point
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5. Spiral Sink
Equations
Jacobian matrix
Eigenvectors
Eigenvalues ?1 -12i ?2 -1-2i
Solution from Mathematica
Conclusion there is a stable fixed point, the
spiral sink is sometimes called stable
focal point
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6. Node Center
Equations
Jacobian matrix
Eigenvectors
Eigenvalues ?1 1.732i ?2 -1.732i
Solution from Mathematica
Conclusion there is marginally stable (neutral)
fixed point, the node center
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Brief classification of two-dimensional dynamical
systems according to eigenvalues
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Special cases of identical eigenvalues
A stable star (a stable proper node)
Equations and matrix
Eigenvalues eigenvectors
Solution
An unstable star (an unstable proper node)
Equations and matrix
Eigenvalues eigenvectors
Solution
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Special cases of identical eigenvalues
A stable improper node with 1 eigenvector
Equations and matrix
Eigenvalues eigenvectors
Solution
An unstable improper node with 1 eigenvector
Equations and matrix
Eigenvalues eigenvectors
Solution
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Classification of dynamical systems usingtrace
and determinant of the Jacobian matrix
1.Attracting node p-5 q4 ?9 2.
Repelling node p5 q4 ?9 3. Saddle
point p3 q-4 ?25 4. Spiral source
p2 q5 ?-16 5. Spiral sink p-2 q5
?-16 6. Node center p0 q5 ?-20 7.
Stable/unstable star p-/ 2 q1 ?0 8.
Stable/unstable improper node p-/ 2
q1 ?0
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Several configurations of damped oscillator
Equation of motion
Rewritten into a set of 1st order equations
Overdamped oscillator, d2 s-1, ?1 s-1
Attracting node
Underdamped oscillator, d1 s-1, ?2 s-1
Spiral sink
Critically damped osc. , d1 s-1, ?1 s-1
Stable improper node
Simple harmonic osc. , d0 s-1, ?1 s-1
Node center
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Example 1 a saddle point calculation in
Mathematica
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Classification of Dynamical SystemsLinear or
linearized systems with more dimensions
Time Eigenvalues Fixed point is
Continous all Re(?)lt0 Stable
Continous some Re(?)gt0 Unstable
Continous all Re(?)lt0 some Re(?)0 Cannot decide
Discrete all ?lt1 Stable
Discrete some ?gt1 Unstable
Discrete all ?lt1 some ?1 Cannot decide
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Basic Types of 3D systems
Node all eigenvalues are real and have the same
sign
Attracting Node all eigenvalues are negative
?1lt ?2lt ?3lt 0
Repelling Node all eigenvalues are positive
?1gt ?2gt ?3gt 0
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Basic Types of 3D systems
Saddle point all eigenvalues are real and at
least one of them is positive and at least one is
negative Saddles are always unstable
?1lt ?2lt 0 lt ?3
?1 gt ?2 gt 0 gt ?3
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Basic Types of 3D systems
Focus-Node there is one real eigenvalue and a
pair of complex-conjugate eigenvalues, and all
eigenvalues have real parts of the same sign.
Stable Focus-Node real parts of all eigenvalues
are negative Re(?1)ltRe(?2)ltRe(?3)lt0
Unstable Focus-Node real parts of all
eigenvalues are positive Re(?1)gtRe(?2)gtRe(?3)gt0
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Basic Types of 3D systems
Saddle-Focus Point there is one real eigenvalue
with the sign opposite to the sign of the real
part of a pair of complex-conjugate eigenvalues
This type of fixed point is always unstable.
Re(?1)gt Re(?2) gt 0 gt ?3
Re(?1) lt Re(?2) lt 0 lt ?3