MA354 Long Term Behavior T H 2:30pm– 3:45 pm Dr. Audi Byrne

Recall: Dynamical System A dynamical system is a system that changes over time. It is described by an evolution rule (that describes the change that occur in terms of the relevant parameters, such as time). The evolution rule can be given implicitly (by describing the dynamical system at time t by using information from time t-1) or explicitly (a single expression for the system at any time.)

Example: Implicit Verses Explicit Descriptions of Dynamical Systems A plant grows 3 leaves per day. The dynamical system L(d) is the number of leaves per day d. Implicit description: L(d + 1) = L(d) + 3 Explicit description: L(d) = 3*d

Implicit Equations verses Explicit (Analytic) Equations Explicit equations are nice to have... What is a 100 ? SeriesImplicit EqnExplicit Eqn {2,5,8,11,14,...}a n+1 =a n +3, a 0 =0 a n =3n+2

Long Term Behavior Recall that computing the dynamical system for different time points is called solving the system or integrating the system. We are often interested in the long-term behavior of a dynamical system: the ‘trend’ of the system over time. The long-term behavior of the system is generally called the solution of the system.

Questions About 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.

Finding Long-term Behavior/Solutions If we have an explicit expression for the dynamical system, we can find the long-term behavior by considering the limit of the system as t  . L(d) = 3*d

Finding Long-term Behavior/Solutions If we have an explicit expression for the dynamical system, we can find the long-term behavior by considering the limit of the system as t  . L(t) = 3*t

Finding Long-term Behavior/Solutions If we have an explicit expression for the dynamical system, we can find the long-term behavior by considering the limit of the system as t  . L(t) = 3*t

Finding Long-term Behavior/Solutions If we have an implicit expression for the dynamical system, it’s generally more difficult to determine the long term behavior. L(d + 1) = L(d) + 3 We know the next state, but what about the state after that, and the state after that? Integrating the system for 100s of steps may or may not give us an idea of what the system is doing in the long run.

Implicitly defined dynamical system: D(t+1) = f(D(t))

Implicitly defined dynamical system: D(t+1) = f(D(t))

Implicitly defined dynamical system: D(t+1) = f(D(t)) When graphing, how do we know When we’ve found the “solution”?

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?

Finding Long-term Behavior/Solutions If we have an implicit expression for the dynamical system, you can try to solve it by (a) integrating it (e.g., graphing the system for many time steps), (b) finding the equivalent explicit description, or (c) finding solution using an analytic trick. We’ll compare methods (a), (b) and (c).

(a) Solving dynamical systems by integrating the dynamical system.

B. Example: a n+1 =r*a n A. Example: a n+1 =a n +b C. Example: a n+1 =r*a n +b Excel

(b) Solving dynamical systems by finding the explicit description.

B. Example: a n+1 =r*a n A. Example: a n+1 =a n +b C. Example: a n+1 =r*a n +b On Board

(c) Solving dynamical systems by finding fixed points.

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.

HW for Class Day After We Cover Fixed Points Section 1.3: #1 (b,e,f) – explicit eqns #2 (b,c,g,h) –fixed points #5 – explicit eqns