Nonlinear Systems
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State Space description
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Equilibrium Points
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Linearization
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A equilibrium point will be "locally asymptotically stable" if the linearized system is asymptotically stable. If all the eigenvalues of the linearized system are not on the then the equilibrium point is said to be hyperbolic.
Local Stability for hyperbolic equilibrium points
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Examples
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Pendulum
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Both the eigenvalues are negative as are both positive
is positive as are both positive
The equilibrium point is unstable.
Multiple equilibrium points and Linearization
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The region of attraction of the stable equilibrium point is the entire
space excluding
Nevertheless, the equilibrium point
only locally asymptotically stable.
Region of attraction
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Tunnel Diode Circuit
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Equilibrium Points and Linearization
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Equilibrium points
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Hysteretic Behavior
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Atomic Force Microscopes
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Modeling
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Cantilever: One Mode Model
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Tip-sample Interaction
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Intermolecular and Surface Forces, Third Edition: Revised Third Edition by Jacob N. Israelachvili (Jun 27, 2011)
Lennard-Jones Potential
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Force Curves
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Equilibrium Points
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Linearization
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movieForceCurve
Hysteretic behavior
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Experiments
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Phase portraits
Duffings Oscillator
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Van-der pol worked on vacuum diodes and their modeling
The physics is described by
Differentiating the last equation with time we have
Thus the physics is described by
Let
Then the equation with derivatives with respect to time yield
Van-der Pol oscillator results when
Van-der Pol Oscillator
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State Space representation
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Structurally stable periodic orbits
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Aspects of Nonlinear Systems
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Consider
Non-unique solutions
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Finite Escape Time
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Systems that are not linear can have periodic orbits that are structurally stable.
Example
Van-der pol oscillators
Periodic Orbits
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