Phase Line Bifurcation Examples Bifurcation Diagrams Linearization Theorem HartmanGrobman Thm
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Differential Equations and Linear Algebra Lecture 10A. Differential Equations, 4th Edition (by Blanchard, Devaney, and Hall): https://amzn.to/35Wxabr. Phase Line Bifurcation Examples, Bifurcation Diagrams, Linearization Theorem (Hartman-Grobman Theorem). • I work through example bifurcation problems and draw bifurcation diagrams for one-parameter families of autonomous first order nonlinear ordinary differential equations (including one with a cubic function of y). Along the way, I discuss the Linearization Theorem (one-dimensional Hartman-Grobman Theorem) and give the intuition why it is true. Finally, I use the Manipulate function in Mathematica to make animations and bifurcation diagrams. • #bifurcations #billkinneymath #differentialequations • (a.k.a. Differential Equations with Linear Algebra, Lecture 10A, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 10A. #differentialequations). • Google drive link for Differential Equations and Linear Algebra course lecture documents: https://drive.google.com/drive/folder... • Using Mathematica for ODEs (Ordinary Differential Equations) Playlist: • Using Mathematica for Ordinary Differ... • Visual Linear Algebra Online, Section 1.5 (at infinityisreallybig.com), Matrices and Linear Transformations in Low Dimensions: https://infinityisreallybig.com/2019/... • Visual Linear Algebra Online, Section 1.6 (at infinityisreallybig.com), Linear Algebra in Three Dimensions: https://infinityisreallybig.com/2019/... • Visual Linear Algebra Online, Section 1.7 (at infinityisreallybig.com), High-Dimensional Linear Algebra: https://infinityisreallybig.com/2019/... • Infinite Powers, How Calculus Reveals the Secrets of the Universe (by Steven Strogatz): https://amzn.to/2XXRCF6 • Another Differential Equations lecture I made: Differential Equations: As Much As You Can Possibly Learn About in 50 Minutes, especially Population Models: • Differential Equations Crash Course: ... • Check out my blog: https://infinityisreallybig.com/ • Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments. You can also get to know your professors personally. https://www.bethel.edu/ • (0:00) Lecture 9A was about phase lines • (1:02) Bifurcation Example 1: dy/dt = f(y) = y^2 + mu • (2:42) Draw the graphs of f(y) for various values of mu • (4:12) Draw the phase lines for various values of mu. Identify the equilibrium points as sinks, sources, or nodes. • (8:10) Linearization (intuitively) • (8:40) mu = 0 is the bifurcation value • (9:07) Bifurcation diagram with a curve of equilibria (curve of equilibrium points) • (12:21) Linearization Theorem (Hartman-Grobman Theorem in one dimension) • (14:36) Visual intuitive justification of the Linearization Theorem • (16:08) Four things that can happen when f ’(y0) = 0 • (17:35) Wild Example 1: dy/dt = sin(1/y) when y is not 0 and dy/dt = 0 when y = 0 • (19:20) The function f(y) is not continuous at y = 0 (the limit as y approaches zero does not exist) • (19:56) There are infinitely many equilibrium points on any neighborhood of y = 0 • (20:40) Wild Example 2: dy/dt = y*sin(1/y) when y is not 0 and dy/dt = 0 when y = 0 • (21:42) f(y) is continuous at y = 0 but not differentiable there • (22:08) Wild Example 3: dy/dt = y^2*sin(1/y) when y is not 0 and dy/dt = 0 when y = 0 • (22:51) f(y) is differentiable but not continuously differentiable • (23:38) Bifurcation Example 2: dy/dt = f(y) = -y^2 + 4y + mu • (24:23) Use the quadratic formula to find the bifurcation value at mu = -4 • (27:48) Bifurcation diagram • (30:24) Use Linearization Theorem to confirm sinks and sources • (32:34) Bifurcation Example 3: dy/dt = f(y) = y^3 - 2y^2 + y + mu • (33:43) Draw the curve of equilibria by setting f(y) = 0 and solving for mu as a function of y (and drawing a sideways graph) • (37:05) Find the bifurcation values to be mu = 0 and mu = -4/27 • (39:31) Draw phase lines to finish the bifurcation diagram • (41:46) The logistic model with harvesting is a practical application of bifurcation theory • (42:23) Mathematica for Bifurcation Example 1 • (43:36) Animation of graph of f(y) as mu increases • (44:09) Animation of slope field of dy/dt = f(y) as mu increases • (44:40) Animation of phase line of dy/dt = f(y) as mu increases • (45:13) Mathematica for Bifurcation Examples 2 and 3 • (46:55) System of nonlinear equations to find bifurcation values and bifurcation points • AMAZON ASSOCIATE • As an Amazon Associate I earn from qualifying purchases.
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