Dynamical systems and van der Waerden’s theorem David M. McClendon Ferris State University Big Rapids, MI Hope College Colloquium March 10, 2015 David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. Things modeled by dynamical systems 1 (Economics) the value of a stock or commodity 2 (Biology) the deer population in western Michigan 3 (Meteorology) the temperature at a fixed spot 4 (Astronomy) the position of a comet 5 (Physics) the motion of a pendulum David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. To formulate such an object mathematically, we need two things: David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. To formulate such an object mathematically, we need two things: 1. The phase space The phase space X of a dynamical system is the set of all possible “positions” or “states” of the system. For example, if the system is keeping track of the price of a stock as time passes, X is the set of all possible stock prices. David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. To formulate such an object mathematically, we need two things: 2. The evolution rule The evolution rule or transformation T of a dynamical system is a function T : X → X that tells you, given your current state x , your state one unit of time from now. For example, if the system is keeping track of a stock price, if the current price is 30, then T (30) would be the price of the stock tomorrow (if time is measured in days). David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. Definition A (discrete) dynamical system is a pair ( X , T ) where X is some set and T is a function from X to itself. David McClendon Dynamics and vdW’s theorem
Dynamical systems Loosely speaking, a “dynamical system” is a mathematical model for anything that changes as time passes. Definition A (discrete) dynamical system is a pair ( X , T ) where X is some set and T is a function from X to itself. Unfortunately, this is too general a situation to say much mathemat- ically, so usually one requires that X and T have some additional “structure”. David McClendon Dynamics and vdW’s theorem
Additional structures Each additional “structure” you might require on X and T gives rise to a different subfield of dynamical systems: Subfields within dynamical systems 1 One-dimensional dynamics: X ⊆ R or S 1 2 Smooth dynamics: X is a manifold; T differentiable 3 Complex dynamics: X = C ; T rational map 4 Ergodic theory: X is a measure space; T is a measure-preserving transformation 5 Algebraic dynamics: X is a quotient of a Lie group; T is a translation 6 Topological dynamics: X is a compact metric space; T is continuous David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. A set X is a metric space if there is a function d which measures the distance between points in a reasonable way: d ( x , y ) = the distance between x and y David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. I won’t tell you exactly what compact means here, but think of a compact space as one that is “closed” (i.e. contains all its bound- ary points) and “bounded” (i.e. you can enclose the set in a cir- cle/sphere of finite radius). David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. x • ❍❍❍❍ d ( x , y ) • y ❍ X David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. A function T : X → X is called continuous if whenever points x and y are sufficiently close to one another, the points T ( x ) and T ( y ) can’t be too far apart. More precisely, this means that for every number ǫ > 0, there is a corresponding number δ > 0 such that whenever d ( x , y ) < δ , it must be that d ( T ( x ) , T ( y )) < ǫ . David McClendon Dynamics and vdW’s theorem
Topological dynamical systems Definition A topological dynamical system (t.d.s.) is a pair ( X , T ) where X is a compact metric space and T is a continuous function from X to itself. T ( x ) x ✲ • ✲ • • • y T T ( y ) X David McClendon Dynamics and vdW’s theorem
Iterates Given a dynamical system ( X , T ) and a point x ∈ X : x = your present state T ( x ) = your state one unit of time from now T ( T ( x )) = T ◦ T ( x ) = your state two units of time from now T ( T ( T ( x ))) = T ◦ T ◦ T ( x ) = T 3 ( x ) etc. Definition We define T n ( x ) = T ◦ T ◦ · · · ◦ T ( x ); therefore T n ( x ) is the state n units of time from now if x is your current state. T n is called the n th iterate of T . David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems Prediction problems Given a dynamical system ( X , T ) and a point x ∈ X , predict T n ( x ) for large values of n . Do the numbers x , T ( x ) , T 2 ( x ) , T 3 ( x ) , ... follow a pattern? Do the numbers T n ( x ) have a limit as n → ∞ ? If x is changed slightly, do the numbers x , T ( x ) , T 2 ( x ) , T 3 ( x ) , ... stay pretty much the same, or do they become drastically different? David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems Prediction problems Frequently it is impossible to predict T n ( x ) for large n , in which case the question becomes one of explaining why such prediction is impossible (chaos theory). Prediction problems have applications in math, physics, biology, computer science, economics, etc. David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems An example where prediction is easy Let X = [0 , ∞ ) and let T ( x ) = x 2 . Then: If x = 1, then T n ( x ) = 1 for all n . If x < 1, then T n ( x ) → 0 as n → ∞ . If x > 1, then T n ( x ) → ∞ as n → ∞ . David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems An example where prediction is hard Let X = [0 , 1] and let T ( x ) = 4 x (1 − x ). Then if x = . 345, the iterates of x are ... David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems An example where prediction is hard { 0 . 345 , 0 . 9039 , 0 . 347459 , 0 . 906925 , 0 . 337648 , 0 . 894567 , 0 . 377268 , 0 . 939747 , 0 . 226489 , 0 . 700766 , 0 . 838772 , 0 . 540934 , 0 . 993298 , 0 . 0266299 , 0 . 103683 , 0 . 371731 , 0 . 934188 , 0 . 245922 , 0 . 741777 , 0 . 766176 , 0 . 716602 , 0 . 812334 , 0 . 60979 , 0 . 951784 , 0 . 183564 , 0 . 59947 , 0 . 960421 , 0 . 152052 , 0 . 515728 , 0 . 999011 , 0 . 00395398 , 0 . 0157534 , 0 . 0620209 , 0 . 232697 , 0 . 714197 , 0 . 816479 , 0 . 599364 , 0 . 960507 , 0 . 151732 , 0 . 514838 , 0 . 999119 , 0 . 00351956 , 0 . 0140287 , 0 . 0553275 , 0 . 209065 , 0 . 661428 , 0 . 895764 , 0 . 373485 , 0 . 935976 , 0 . 2397 , 0 . 728977 , 0 . 790279 , 0 . 662953 , 0 . 893786 , 0 . 379731 , 0 . 942142 , 0 . 218042 , 0 . 682 , 0 . 867505 , 0 . 459761 , 0 . 993523 , 0 . 0257389 , 0 . 100306 , 0 . 360978 , ... } David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems An example where prediction is hard So if X = [0 , 1], T ( x ) = 4 x (1 − x ) and x = . 345, the iterates of x have no discernable pattern. What’s more, is that if you change x from . 345 to something like . 346, the iterates you obtain from the new x look nothing like the iterates you obtain from the old x . David McClendon Dynamics and vdW’s theorem
Major problems in dynamical systems Classification problems Given two dynamical systems, are they the same up to a change of language (i.e. isomorphic) or different? Are they same up to some weaker notion of equivalence? What are their commonalities? What are their differences? David McClendon Dynamics and vdW’s theorem
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