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MEAM 535
University of Pennsylvania 1
Geometric Theory of Dynamical SystemsPoincare: Two Key Concepts
State spacePhase portrait
=
x
q
q
n
&,
.2
1
State (mechanical systems)q describes the configuration (position) of the systemx describes the state of the system
Phase PortraitTrajectoryq&
q
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State
Actual (internal) state of the systemMathematical model requires an idealized stateThe idealized state must be observable (in order for results to be practical)
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Modeling: State
EarAttitude
Fang Exposure
=
2
1qq
q
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Modeling: State
EarAttitude
Fang Exposure
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State Space and Time
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Position and Tangent Vectors
C
V V=C/T
V tends to the tangent vector as T tends to zero
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Vector fieldTrajectories State space + trajectories in state space = phase portrait
Vector Field
( )xfx =&
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Integral Curve
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From the vector field to trajectories
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Assumption: Vector fields are smooth!
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Examples
Discontinuous (obviously non smooth) Continuous but non smooth
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Critical Point
Critical points are equilibrium points. The critical points of the vector field, f(x), are found by solving f(x)=0.
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Vector Field near a Critical Point
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Integral Curve
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LimitsLimit points
Limit cycles
Limit sets
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Limit cycles
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Limit sets: Beyond limit points, limit cycles
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Critical point (limit set) and insets
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Limit sets
α - limit setlimit set
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Limit sets (continued)
α - limit set
limit set
ω - limit set Inset of a limit set - initial states that end up at the same equilibrium state (or trajectory). Outset of a limit set - initial states that end up at the same equilibrium state (or trajectory) if time were reversed.
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Attractors are limit sets with “open” insets
Static AttractorThe inset contains a disk around the critical (limit) point. Every initial point in the neighborhood of the critical point will approach it.
Periodic attractorThe inset contains an annulus around the limit cycle. (Almost) Every initial state in the neighborhood of the limit cycle will approach it.
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Separatrix
Limit Cycle Limit Point
Attractors, Basins, and Separatrices
Two attractors, one separatrix, and the basin for the limit pointAny point not in a basin belongs to a separatrix
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Attractors, Basins, and Separatrices
Limit Points (Attractors)
Vagueattractor
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Gradient Systems
φ x
( )( )xxfxφ−∇=
=&
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Example of φ(x)
φ(x) = x4/4 +2x3/3 – 3x2/2 + y2/2
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Equi-potential curves
φ(x) = x4/4 +2x3/3 – 3x2/2 + y2/2
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Contour Plot
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Contour Plot
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Vector Field and Trajectories for a Gradient System
( )( )xxfxφ−∇=
=&
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Phase Portrait for a Gradient System
2 attractors1 separatrix (including a saddle limit point)
x’ = -x3 – 2x2 + 3xy’ = -y
φ(x) = x4/4 +2x3/3 – 3x2/2 + y2/2
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Geometric MechanicsThis was just an introduction!
Very rich fieldAllows “qualitative” studies
Much of the underlying mathematics has been developed in the last century
Poincare (1854-1912)Lie (1842-1899)Lyapunov (1857-1918)
Abraham and Shaw provide a pictorial, easy-to-understand introduction!
