Announcements

Topics: - 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].

proper integral finite region

= real number

Improper Integrals

Why are the following integrals “improper”?

Improper Integrals Type I: Infinite Limits of Integration

Definition:Assume that the definite integralexists (i.e., is equal to a realnumber) 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 areafinite area

convergesdiverges

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 systemif

Geometrically, the equilibria correspond to points where the updating function intersects the diagonal.

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:1. Cobwebbing2. “Graphical Criteria” (if the the updating

function is increasing at the equilibrium) 3. “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:1. Cobwebbing2. “Graphical Criteria” (if the the updating

function is increasing at the equilibrium) 3. “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:1. Cobwebbing2. “Graphical Criteria” (if the the updating

function is increasing at the equilibrium) 3. “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

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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:1. Cobwebbing2. “Graphical Criteria” (if the the updating

function is increasing at the equilibrium) 3. “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

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