Announcements Topics: Work On: sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in your textbook! Work On: Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

Definite (Proper) Integrals Assumptions: f is continuous on a finite interval [a,b]. = real number proper integral finite region

Improper Integrals Why are the following integrals “improper”?

Improper Integrals Type I: Infinite Limits of Integration Definition: Assume that the definite integral exists (i.e., is equal to a real number) for every Then we define the improper integral of f(x) on by provided that the limit on the right side exists.

Improper Integrals Type I: Infinite Limits of Integration Illustration: proper integral finite region

Improper Integrals Type I: Infinite Limits of Integration Examples: Evaluate the following improper integrals. (a) (b)

Improper Integrals Type I: Infinite Limits of Integration When the limit exists, we say that the integral converges. When the limit does not exist, we say that the integral diverges. Rule: is convergent if and divergent if

Illustration infinite area finite area diverges converges

Improper Integrals Type I: Infinite Limits of Integration More Examples: Evaluate the following improper integrals. (a) (b)

Equilibria Definition: A point is called an equilibrium of the discrete-time dynamical system if Geometrically, the equilibria correspond to points where the updating function intersects the diagonal. RECALL

Stability of Equilibria An equilibrium is stable if solutions that start near the equilibrium move closer to the equilibrium. An equilibrium is unstable if solutions that start near the equilibrium move away from the equilibrium.

Checking Stability of Equilibria To determine stability, we can use: Cobwebbing “Graphical Criteria” (if the the updating function is increasing at the equilibrium) “Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

Checking Stability of Equilibria To determine stability, we can use: Cobwebbing “Graphical Criteria” (if the the updating function is increasing at the equilibrium) “Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria Using Cobwebbing

Checking Stability of Equilibria To determine stability, we can use: Cobwebbing “Graphical Criteria” (if the the updating function is increasing at the equilibrium) “Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

Stability Theorem for DTDSs An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e., Example:

Stability Theorem for DTDSs An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e., Example:

Stability Theorem for DTDSs An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e., Example:

Stability Theorem for DTDSs An equilibrium is unstable if the absolute value of the derivative of the updating function is > 1 at the equilibrium, i.e., Example:

Stability Theorem for DTDSs An equilibrium is unstable if the absolute value of the derivative of the updating function is > 1 at the equilibrium, i.e., Example:

Stability Theorem for DTDSs If the slope of the updating function is exactly 1 or -1 at the equilibrium, i.e., then the equilibrium could be stable, unstable, or half-stable. Example:

Stability Theorem for DTDSs If the slope of the updating function is exactly 1 or -1 at the equilibrium, i.e., then the equilibrium could be stable, unstable, or half-stable. Example:

Stability Theorem for DTDSs Example: DTDS for a limited population

Stability Theorem for DTDSs Example: DTDS for a limited population Zoom In

Stability Theorem for DTDSs Example: logistic dynamical system

Stability Theorem for DTDSs Example: logistic dynamical system

Checking Stability of Equilibria To determine stability, we can use: Cobwebbing “Graphical Criteria” (if the the updating function is increasing at the equilibrium) “Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function An equilibrium is stable if the graph of the (increasing) updating function crosses the diagonal from above to below. Example:

Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function An equilibrium is unstable if the graph of the (increasing) updating function crosses the diagonal from below to above. Example:

Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function Example: DTDS for a limited population

Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function Example: DTDS for a limited population Zoom In