Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems)...
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Transcript of Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems)...
![Page 1: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/1.jpg)
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”)
![Page 2: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/2.jpg)
Definite (Proper) Integrals
Assumptions:
f is continuous on a finite interval [a,b].
proper integral finite region
= real number
![Page 3: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/3.jpg)
Improper Integrals
Why are the following integrals “improper”?
![Page 4: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/4.jpg)
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.
![Page 5: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/5.jpg)
Improper Integrals Type I: Infinite Limits of Integration
Illustration:
proper integral
finite region
![Page 6: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/6.jpg)
Improper Integrals Type I: Infinite Limits of Integration
Examples:Evaluate the following improper integrals.
(a)(b)
![Page 7: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/7.jpg)
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
![Page 8: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/8.jpg)
Illustration
infinite areafinite area
convergesdiverges
![Page 9: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/9.jpg)
Improper Integrals Type I: Infinite Limits of Integration
More Examples:Evaluate the following improper integrals.
(a)(b)
![Page 10: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/10.jpg)
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.
![Page 11: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/11.jpg)
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.
![Page 12: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/12.jpg)
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)
![Page 13: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/13.jpg)
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)
![Page 14: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/14.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 15: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/15.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 16: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/16.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 17: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/17.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 18: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/18.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 19: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/19.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 20: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/20.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 21: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/21.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 22: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/22.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 23: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/23.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 24: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/24.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 25: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/25.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 26: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/26.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 27: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/27.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 28: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/28.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 29: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/29.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 30: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/30.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 31: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/31.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 32: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/32.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 33: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/33.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 34: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/34.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 35: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/35.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 36: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/36.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 37: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/37.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 38: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/38.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 39: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/39.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 40: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/40.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 41: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/41.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 42: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/42.jpg)
Checking Stability of Equilibria Using Cobwebbing
![Page 43: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/43.jpg)
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)
![Page 44: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/44.jpg)
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:
![Page 45: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/45.jpg)
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:
![Page 46: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/46.jpg)
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:
![Page 47: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/47.jpg)
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:
![Page 48: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/48.jpg)
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:
![Page 49: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/49.jpg)
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:
![Page 50: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/50.jpg)
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:
![Page 51: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/51.jpg)
Stability Theorem for DTDSs
Example: DTDS for a limited population
![Page 52: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/52.jpg)
Stability Theorem for DTDSs
Example: DTDS for a limited population
Zoom In
![Page 53: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/53.jpg)
Stability Theorem for DTDSs
Example: logistic dynamical system
![Page 54: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/54.jpg)
Stability Theorem for DTDSs
Example: logistic dynamical system
![Page 55: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/55.jpg)
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)
![Page 56: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/56.jpg)
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:
![Page 57: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/57.jpg)
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:
![Page 58: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/58.jpg)
Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
Example: DTDS for a limited population
![Page 59: Announcements Topics: -sections 7.7 (improper integrals), 6.7 + 6.8 (stability of dynamical systems) * Read these sections and study solved examples in.](https://reader038.fdocuments.us/reader038/viewer/2022103023/56649e595503460f94b52895/html5/thumbnails/59.jpg)
Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
Example: DTDS for a limited population
Zoom In