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 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 jf 0(x∗)j < 1: Iterative maps Definition (Iterative map) A (one-dimensional) iterative map is a sequence fxng 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 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 jf 0(x∗)j < 1: Iterative maps Definition (Iterative map) A (one-dimensional) iterative map is a sequence fxng 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 Stability of fixed points By approximating f with a linear function, we get that a fixed point x∗ is stable whenever jf 0(x∗)j < 1: Iterative maps Definition (Iterative map) A (one-dimensional) iterative map is a sequence fxng 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 By approximating f with a linear function, we get that a fixed point x∗ is stable whenever jf 0(x∗)j < 1: Iterative maps Definition (Iterative map) A (one-dimensional) iterative map is a sequence fxng 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 fxng 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 jf 0(x∗)j < 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 If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it's stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r p p r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on. A famous example: the logistic map We consider xn+1 = rxn(1 − xn) on the interval [0; 1]: Properties vary based on r: J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r p p r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on. A famous example: the logistic map We consider xn+1 = rxn(1 − xn) on the interval [0; 1]: Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it's stable. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 p p r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on. A famous example: the logistic map We consider xn+1 = rxn(1 − xn) on the interval [0; 1]: Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it's stable. 1 If 1 < r ≤ 3, 0 is unstable, 1 − r is stable. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 2-cycle becomes 4-cycle, then 8-cycle, and so on. A famous example: the logistic map We consider xn+1 = rxn(1 − xn) on the interval [0; 1]: Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it's stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r p p r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 A famous example: the logistic map We consider xn+1 = rxn(1 − xn) on the interval [0; 1]: Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it's stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r p p r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 The case of a stable 2-cycle J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 6 / 32 The orbit diagram We can plot the points in stable cycles with respect to r: J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 7 / 32 The first Feigenbaum constant n Let rn be where stable 2 cycle begins. The distance between rn's converges roughly geometrically, up to r1. Definition (δ) The first Feigenbaum constant is defined as r − r δ = lim n n−1 ≈ 4:669 ::: n!1 rn+1 − rn J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 8 / 32 The first Feigenbaum constant: not just for one map? Yeah, but why do we care about δ? Consider the sine map xn+1 = r sin πxn: Guess what its orbit diagram looks like? J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 9 / 32 Sine map orbit diagram No, I didn't accidentally repeat the previous image... ...it looks exactly the same! Not only that, if you try to calculate δ here, you'll get the same number! J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 10 / 32 Univerality of δ Theorem 1 (Universality of δ) If f 00 0 1 f 00(x)2 D f (x) = (x) − < 0 sch f 0 2 f 0(x) in the bounded interval and f experiences period-doubling, then letting frng be defined for this new map, r − r lim n n−1 = δ: n!1 rn+1 − rn Essentially, δ is a \universal constant!" J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 11 / 32 Now, the continuous case ... Continuous dynamical systems involve analyzing differential equations. They describe systems that change over time. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 12 / 32 Belousov's discovery in 1950's exhibits a periodical behavior. Oscillating chemical reactions Chemical reactions: governed by differential equations involving concentrations of the reactants and products. Multi-step reactions can exhibit complicated dynamical behaviors. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 13 / 32 Oscillating chemical reactions Chemical reactions: governed by differential equations involving concentrations of the reactants and products. Multi-step reactions can exhibit complicated dynamical behaviors. Belousov's discovery in 1950's exhibits a periodical behavior. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 13 / 32 Continuous dynamical systems and oscillating chemical reactions Figure: Periodic behavior of an oscillating chemical reaction. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 14 / 32 Continuous dynamical systems: one{dimensional case x_ = f (x) The continuous time dynamics_x of a system is governed by its current statex. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 15 / 32 Flow and vector fields Stable and unstable fixed points (x _ = 0) Continuous dynamical systems: one{dimensional case Example:x _ = r + x2, where r is a parameter. Figure: The phase portrait of the systemx _ = r + x 2. J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 16 / 32 Continuous dynamical systems: one{dimensional case Example:x _ = r + x2, where r is a parameter. Figure: The phase portrait of the systemx _ = r + x 2. Flow and vector fields Stable and unstable fixed points (x _ = 0) J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 16 / 32 Continuous dynamical systems: bifurcations Example:x _ = r + x2, where r is a parameter.