Discrete and Continuous Dynamical Systems: Applications ...
Discrete and Continuous Dynamical Systems: Applications and Examples
Yonah Borns-Weil and Junho Won Mentored by Dr. Aaron Welters
Fourth Annual PRIMES Conference May 18, 2014
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 1 / 32
Overview of dynamical systems
What is a dynamical system? Two flavors:
Discrete (Iterative Maps) Continuous (Differential Equations)
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 2 / 32
Iterative maps
Definition (Iterative map)
A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R R. Basic Ideas:
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 3 / 32
Iterative maps
Definition (Iterative map)
A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R R. Basic Ideas:
Fixed points
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 3 / 32
Iterative maps
Definition (Iterative map)
A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R R. Basic Ideas:
Fixed points Periodic points (can be reduced to fixed points)
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 3 / 32
Iterative maps
Definition (Iterative map)
A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R R. Basic Ideas:
Fixed points Periodic points (can be reduced to fixed points) Stability of fixed points
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 3 / 32
Iterative maps
Definition (Iterative map)
A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R R. Basic Ideas:
Fixed points Periodic points (can be reduced to fixed points) Stability of fixed points By approximating f with a linear function, we get that a fixed point x is stable whenever |f (x)| < 1.
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 3 / 32
Getting a picture: "cobwebbing"
J. Won, Y. Borns-Weil (MIT)
Discrete and Continuous Dynamical Systems
May 18, 2014 4 / 32
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