Lecture Notes 19

Lecture Notes 19 - Lecture19 Lecture 19 Application of...

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Lecture19 1 November 14, 2007 Outline: 1) Linear Dynamical systems du /dt=Au ; u (0)=u 0 Definitions and Interpretation General Solution and the matrix exponential e A Interpretation of General Solution as a change of Basis 2) Examples: The Romeo and Juliet Problems The Reactive Model The Contrarian Model 3) General Classification of fixed points for 2x2 systems Lecture 19: Application of Diagonalization: Linear Dynamical systems and the dynamics of love affairs Definition: an autonomous, linear dynamical system can be written as du /dt = Au u (0)=u 0 where u is a state vector, Au is a vector that describes how u changes with time and u 0 is the initial state at time t=0 Physical example: u =[ x y ]' is the position of a particle Au is the velocity of the particle u 0 =[ x0 y0]' is the initial position Linear Dynamical systems: Geometric interpretation: u (t) is a trajectory (parameterized curve) where Au is the vector tangent to the curve at any point. Linear Dynamical systems:
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Lecture Notes 19 - Lecture19 Lecture 19 Application of...

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