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Outline Outline 4 Definitions 4 Definitions 4 Phase Plane 4 Phase Plane 4 Attractor and Stability 4 Attractor and Stability 4 Bifurcation 4 Bifurcation ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Engineering Equation of

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Outline Outline 4 4Definitions Definitions 4 4Phase Plane Phase Plane 4 4Attractor and Stability Attractor and Stability 4 4Bifurcation Bifurcation

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ME 639-Turbulence

Simple Pendulum Simple Pendulum

mg

2

Engineering Systems Engineering Engineering Systems Systems Dynamical Dynamical Systems Systems State State State

θ θ & ,

Phase Plane Phase Plane Phase Plane

θ θ versus

  • f

Plot &

l

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ME 639-Turbulence

Simple Pendulum Simple Pendulum Simple Pendulum Equation of Motion Equation of Equation of Motion Motion System of Differential System of Differential Equations Equations

) f(x x t , = &

⎩ ⎨ ⎧ θ ω − ω ζω − = ω ω = θ sin 2

2

  • &

&

l g

2

  • =

ω

⎩ ⎨ ⎧ θ ω − ω ζω − = ω ω = θ

2

  • 2

& &

Linear Pendulum Linear Pendulum Linear Pendulum

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Time Does Not Time Does Not Appear Explicitly Appear Explicitly Autonomous Systems Autonomous Autonomous Systems Systems Nonautonomous Systems Nonautonomous Nonautonomous Systems Systems Time Appears Time Appears Explicitly Explicitly Orbit/ Trajectory Orbit/ Orbit/ Trajectory Trajectory Curves in Phase Space Curves in Phase Space Solutions of Equation Solutions of Equation

  • f Motion
  • f Motion
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Vectors Tangent Vectors Tangent to Trajectories in to Trajectories in Phase Space Phase Space

Flow Flow Flow Null Cline Null Cline Null Cline

Lines in Phase Space Lines in Phase Space for for

xi = &

⎩ ⎨ ⎧ ζ θ ω − = ω = ω = θ 2 / sin

  • &

& Pendulum Pendulum

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ME 639-Turbulence

Intersection

  • f Clines

Intersection Intersection

  • f Clines
  • f Clines

Equilibrium Points Equilibrium Points

Poincare Section Poincare Map Poincare Section Poincare Section Poincare Map Poincare Map

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ME 639-Turbulence

Non-Wandering Set

A set of points that orbits starting form this set come arbitrary close and arbitrary often to any point in the set

Non Non-

  • Wandering Set

Wandering Set

A set of points that orbits starting form A set of points that orbits starting form this set come arbitrary close and this set come arbitrary close and arbitrary often to any point in the set arbitrary often to any point in the set

  • Fixed (stationary) points
  • Limit cycle (periodic)
  • Quasi periodic
  • Chaotic (bounded, non-periodic)
  • Fixed (stationary) points

Fixed (stationary) points

  • Limit cycle (periodic)

Limit cycle (periodic)

  • Quasi periodic

Quasi periodic

  • Chaotic (bounded, non

Chaotic (bounded, non-

  • periodic)

periodic)

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Nonlinear Dynamical Systems

  • Qualitative Behavior
  • Non-wandering sets
  • Stability of non-wandering sets
  • Changes in the number of non-wandering sets

Nonlinear Dynamical Systems Nonlinear Dynamical Systems

  • Qualitative Behavior

Qualitative Behavior

  • Non

Non-

  • wandering sets

wandering sets

  • Stability of non

Stability of non-

  • wandering sets

wandering sets

  • Changes in the number of non

Changes in the number of non-

  • wandering sets

wandering sets

Bifurcation

Appearance and Disappearance of Non-wandering sets

Bifurcation Bifurcation

Appearance and Disappearance of Appearance and Disappearance of Non Non-

  • wandering sets

wandering sets

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Lyapunov (Marginal) Stability

A non-wandering set (NWS) is Lyapunov stable if every orbit starting in its neighborhood stays in its neighborhood.

Lyapunov Lyapunov (Marginal) Stability (Marginal) Stability

A non A non-

  • wandering set (NWS) is

wandering set (NWS) is Lyapunov Lyapunov stable if every orbit starting in its stable if every orbit starting in its neighborhood stays in its neighborhood. neighborhood stays in its neighborhood.

Asymptotic Stability

A NWS is asymptotically stable if in addition to Lyapunov stability every orbit in its neighborhood approaches the NWS.

Asymptotic Stability Asymptotic Stability

A NWS is asymptotically stable if in A NWS is asymptotically stable if in addition to addition to Lyapunov Lyapunov stability every orbit in stability every orbit in its neighborhood approaches the NWS. its neighborhood approaches the NWS.

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Attractors

Asymptotically stable non-wandering sets are called attractors.

Attractors Attractors

Asymptotically stable non Asymptotically stable non-

  • wandering

wandering sets are called attractors. sets are called attractors.

Basin of Attraction

The set of all initial states that approach the attractor.

Basin of Attraction Basin of Attraction

The set of all initial states that The set of all initial states that approach the attractor. approach the attractor.

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Given an Orbit Given an Orbit Given an Orbit ) f(x x t ,

  • =

&

) t (

  • x

xo(t) is asymptotically stable if ∆x(t) decays x xo

  • (t

(t) is asymptotically ) is asymptotically stable if stable if ∆ ∆x(t) decays x(t) decays x x x f x

  • ∆

⋅ = ∆ d ) ( d dt ) ( d

s t

s

e x x ∆ = ∆

λ

I x f λ − d d del

s

λ

Solution Solution

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Stable Orbit if all are negative Stable Orbit if all are negative Stable Orbit if all are negative

d d tr d d del < > x f x f

s

λ

Routh-Hurwitz Theorem For two-D all the roots are negative if Routh Routh-

  • Hurwitz Theorem

Hurwitz Theorem For two For two-

  • D all the roots

D all the roots are negative if are negative if

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Two real eigenvalues

  • f the same sign

Two real Two real eigenvalues eigenvalues

  • f the same sign
  • f the same sign

Nodes Nodes

Pair of complex conjugate eigenvalues Pair of complex Pair of complex conjugate conjugate eigenvalues eigenvalues

Spirals Spirals

Two real eigenvalues with opposite sign Two real Two real eigenvalues eigenvalues with opposite sign with opposite sign

Saddle Saddle

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A change in the number of attractors of a A change in the number of attractors of a nonlinear dynamical systems with the change nonlinear dynamical systems with the change

  • f a system parameter is called
  • f a system parameter is called bifurcation

bifurcation. .

Bifurcation Bifurcation is associated with the is associated with the change of stability of an change of stability of an attractor attractor. . In a bifurcation point, at least one In a bifurcation point, at least one eigenvalue eigenvalue of the

  • f the Jacobian

Jacobian will attain a will attain a zero real part. zero real part.

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In a stationary bifurcation, a single In a stationary bifurcation, a single real real eigenvalue eigenvalue crosses the boundary crosses the boundary

  • f stability.
  • f stability.
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Hopf Hopf bifurcation occurs when a bifurcation occurs when a conjugated complex pair crosses the conjugated complex pair crosses the boundary of stability. boundary of stability.

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u ) u ( u

2

− µ = &

For For µ µ<0, u=0 is a stable equilibrium solution. <0, u=0 is a stable equilibrium solution. For For µ µ>0, u=0 is an unstable equilibrium >0, u=0 is an unstable equilibrium solution, and u= solution, and u= ±µ ±µ1/2

1/2 are stable solutions.

are stable solutions. At At µ µ=0, a =0, a suppercritical suppercritical Pitchfork Pitchfork bifurcation occurs. bifurcation occurs.

Consider a dynamical system Consider a Consider a dynamical system dynamical system

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For For µ µ>0, u=0 is an unstable equilibrium solution. >0, u=0 is an unstable equilibrium solution.

For For µ µ<0, u=0 is a stable equilibrium solution, <0, u=0 is a stable equilibrium solution, and u= and u= ± ±( (-

  • µ

µ) )1/2

1/2 are unstable solutions.

are unstable solutions. At At µ µ=0, a =0, a subcritical subcritical Pitchfork Pitchfork bifurcation occurs. bifurcation occurs.

u ) u ( u

2

+ µ = &

Consider a dynamical system Consider a Consider a dynamical system dynamical system

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For For µ µ<0, u=0 is a stable equilibrium solution, <0, u=0 is a stable equilibrium solution, and u= and u=µ µ is an unstable solution. is an unstable solution.

u ) u ( u − µ = &

Consider a dynamical system Consider a Consider a dynamical system dynamical system At At µ µ=0, the two solution exchange stability =0, the two solution exchange stability and a and a transcritical transcritical bifurcation occurs. bifurcation occurs. For For µ µ>0, u=0 is an unstable solution, and u= >0, u=0 is an unstable solution, and u=µ µ is is a stable equilibrium a stable equilibrium solution. solution.