MA354 Long Term Behavior T H 2:45 pm– 4:00 pm Dr. Audi Byrne

Implicit Equations verses Explicit (Analytic) Equations Explicit equations are nice to have... What is a 100 ? SeriesImplicit EqnExplicit Eqn {2,4,6,8,10,...}a n+1 =a n +2, a 0 =2 a n =2n+2

Implicit Equations verses Explicit (Analytic) Equations Explicit equations are nice to have... What is a 100 ? SeriesImplicit EqnExplicit Eqn {2,4,6,8,10,...}a n+1 =a n +2, a 0 =2 a n =2n+2

Implicit Equations verses Explicit (Analytic) Equations Explicit equations are nice to have... What is a 100 ? SeriesImplicit EqnExplicit Eqn {2,4,6,8,10,...}a n+1 =a n +2, a 0 =2 a n =2n+2

Implicit Equations verses Explicit (Analytic) Equations Explicit equations are nice to have... What is a 100 ? SeriesImplicit EqnExplicit Eqn {2,4,6,8,10,...}a n+1 =a n +2, a 0 =2 a n =2n+2

Questions in Dynamic Systems Given a dynamical system defined with a difference equation (an implicit equation), when can you find an explicit solution? What is the long-term behavior of the dynamical solution? –If the explicit equation is known, not so hard. –If the explicit equation is not known?

Long-term Behavior of a Dynamical System A dynamical system is a changing system. Does the system just keep changing forever? Monotonically? Periodically? Erratically? Can a system stop changing?At equilibrium. Can a system start changing again once stopped? Does the dynamical system “settle-down” in the long term?So-called steady-states. How do these answers depend on the initial conditions (and other parameters) of the dynamical system? Global and local sensitivity analyses.

Long-term Behavior of a Dynamical System A dynamical system is a changing system. Does the system just keep changing forever? Monotonically? Periodically? Erratically? Can a system stop changing?At equilibrium. Can a system start changing again once stopped? Does the dynamical system “settle-down” in the long term?So-called steady-states. How do these answers depend on the initial conditions (and other parameters) of the dynamical system? Global and local sensitivity analyses.

Long-term Behavior of a Dynamical System A dynamical system is a changing system. Does the system just keep changing forever? Monotonically? Periodically? Erratically? Can a system stop changing?At equilibrium. Can a system start changing again once stopped? Does the dynamical system “settle-down” in the long term?So-called steady-states. How do these answers depend on the initial conditions (and other parameters) of the dynamical system? Global and local sensitivity analyses.

Long-term Behavior of a Dynamical System A dynamical system is a changing system. Does the system just keep changing forever? Monotonically? Periodically? Erratically? Can a system stop changing?At equilibrium. Can a system start changing again once stopped? Does the dynamical system “settle-down” in the long term?So-called steady-states. How do these answers depend on the initial conditions (and other parameters) of the dynamical system? Global and local sensitivity analyses.

Long-term Behavior of a Dynamical System A dynamical system is a changing system. Does the system just keep changing forever? Monotonically? Periodically? Erratically? Can a system stop changing?At equilibrium. Can a system start changing again once stopped? Does the dynamical system “settle-down” in the long term?So-called steady-states. How do these answers depend on the initial conditions (and other parameters) of the dynamical system? Global and local sensitivity analyses.

Example:  P=b Consider the following generalized dynamical system: P 0 = P 0, where P 0  [-∞,+∞]Initial Condition  P = b, where b  [-∞,+∞]Rule 1. Can we write an explicit (analytic) function for P n ? 2. What is the long-term behavior of the dynamical system? (I.e., what is the limit of P n as n  ∞?)

Example:  P=b Consider the following generalized dynamical system: P 0 = P 0, where P 0  [-∞,+∞]Initial Condition  P = b, where b  [-∞,+∞]Rule 1. Can we write an explicit (analytic) function for P n ? 2. What is the long-term behavior of the dynamical system? (I.e., what is the limit of P n as n  ∞?)

Example:  P=b Consider the following generalized dynamical system: P 0 = P 0, where P 0  [-∞,+∞]Initial Condition  P = b, where b  [-∞,+∞]Rule Can we write an explicit (analytic) function for P n ? Implicit equation: P n+1 = P n + b, (and P 0 = P 0 ) Explicit equation: ???

Can we write an explicit function for P n ? Using implicit equation: P n+1 = P n + b, (and P 0 = P 0 ) P 0 = P 0 P 1 = P 0 + b P 2 = P 1 + b = (P 0 + b) + b = P 0 + 2b P 3 = P 2 + b = (P 0 + 2b) + b = P 0 + 3b P n = P n-1 + b = ??? …

Can we write an explicit function for P n ? Using implicit equation: P n+1 = P n + b, (and P 0 = P 0 ) P 0 = P 0 P 1 = P 0 + b P 2 = P 1 + b = (P 0 + b) + b = P 0 + 2b P 3 = P 2 + b = (P 0 + 2b) + b = P 0 + 3b P n = P n-1 + b = ??? …

1.Can we write an explicit function for P n ? Using implicit equation: P n+1 = P n + b, (and P 0 = P 0 ) P 0 = P 0 P 1 = P 0 + b P 2 = P 1 + b = (P 0 + b) + b = P 0 + 2b P 3 = P 2 + b = (P 0 + 2b) + b = P 0 + 3b P n = P n-1 + b = P 0 + n*b Explicit Equation …

2. What is the long term behavior? P n = P 0 + n*b There are two parameters: P 0 and b. P 0 : shifts the function vertically b : most important parameter Explicit Equation Long term behavior b>0 b=0 b<0

Fixed Point A number x is called a fixed point (or equilibrium point) of a dynamical system a n+1 =f(a n ) if f(x)=x. –If a k =x then a k+1 =x, a k+2 =x, … –If a k =x then a k+s =x for all s ≥ 0. –To find fixed points, we may solve the equality f(x)=x.

Example: Finding Fixed Points To find fixed points, we may solve the equality f(x)=x. Find the fixed points of the dynamical system P n+1 = P n + b. Solution: P n+1 = f(P n ) = P n + b f( P n )= P n P n = P n + b… only if b=0.

2. What is the long term behavior? There exists a fixed point only if b=0. Long term behavior b>0 b=0 b<0