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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