Chemistry 4010 Lecture 6: Period-doubling bifurcations and chaos - - PowerPoint PPT Presentation

Chemistry 4010 Lecture 6: Period-doubling bifurcations and chaos

Marc R. Roussel September 24, 2019

Marc R. Roussel Period-doubling and chaos September 24, 2019 1 / 4

Example: The Willamowski-R¨

• ssler model

A1 + X

k1

− − ⇀ ↽ − −

k−1 2 X

X + Y

k2

− − ⇀ ↽ − −

k−2 2 Y

A5 + Y

k3

− − ⇀ ↽ − −

k−3 A2

X + Z

k4

− − ⇀ ↽ − −

k−4 A3

A4 + Z

k5

− − ⇀ ↽ − −

k−5 2 Z

The Ai species are “pool” species, assumed fixed.

Willamowski and R¨

• ssler, Z. Naturforsch. A 35, 317 (1980)

Marc R. Roussel Period-doubling and chaos September 24, 2019 2 / 4

Example: The Willamowski-R¨

• ssler model

Dimensionless model equations

˙ x = x(a1 − k−1x − z − y) + k−2y2 + a3 ˙ y = y(x − k−2y − a5) + a2 ˙ z = z(a4 − x − k−5z) + a3 Let’s study these equations with Xppaut!

Willamowski and R¨

• ssler, Z. Naturforsch. A 35, 317 (1980)

Marc R. Roussel Period-doubling and chaos September 24, 2019 3 / 4

Things we learned from this example

Period-doubling bifurcation: bifurcation of a limit cycle in which the period doubles Period-doubling cascade: often, period-doubling bifurcations are repeated

• ver a finite parameter interval

Sensitive dependence on initial conditions: Exponential divergence of trajectories over time Chaotic solution characterized as follows: Non-periodic solution in a bounded region of phase space Displays sensitive dependence on initial conditions Fractal attractor Often reached as the accumulation of a period-doubling cascade

Marc R. Roussel Period-doubling and chaos September 24, 2019 4 / 4