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¨ ossler model k 1 A 1 + X k − 1 2 X ⇀ − ↽ − − − k 2 X + Y k − 2 2 Y ⇀ ↽ − − − − k 3 A 5 + Y k − 3 A 2 ⇀ ↽ − − − − k 4 X + Z k − 4 A 3 ⇀ ↽ − − − − k 5 A 4 + Z k − 5 2 Z ⇀ − ↽ − − − The A i species are “pool” species, assumed fixed. Willamowski and R¨ ossler, Z. Naturforsch. A 35 , 317 (1980) Marc R. Roussel Period-doubling and chaos September 24, 2019 2 / 4

Example: The Willamowski-R¨ ossler model Dimensionless model equations x = x ( a 1 − k − 1 x − z − y ) + k − 2 y 2 + a 3 ˙ y = y ( x − k − 2 y − a 5 ) + a 2 ˙ z = z ( a 4 − x − k − 5 z ) + a 3 ˙ Let’s study these equations with Xppaut ! Willamowski and R¨ ossler, 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 over 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