Differential Equations Lec 31 (Class 34), Hyperbolicity, Stability, Hamiltonian & Lyapunov Functions

preview_player
Показать описание

Differential Equations Course, Lecture 31. (0:00) Encouragement to pay attention and take notes for subtle things in this lecture and mention one more reading assignment. (0:57) Review hyperbolic equilibrium points and Hartman-Grobman theorem (linearization). (4:32) Comments about hyperbolic equilibrium points being structurally stable. (5:46) Lyapunov stable and unstable equilibrium points. Draw pictures to help understand the definitions. (18:24) Physical importance of stable versus unstable equilibrium points. Relate it to the pendulum and competing species models. (21:40) Hamiltonian systems and the ideal (undamped) pendulum. (23:25) Basic consequences of the nonlinearity, small-angle linear approximation, and solution curves in the phase plane. (28:57) Find the Hamiltonian function for the ideal pendulum (correction from the end of Class #33 (Lecture #30)) and graph the level curves with ContourPlot along with solution curves on top of the level curves. (36:34) Construct a Hamiltonian system by starting with the Hamiltonian function and then study the resulting system. (42:39) Saddles and centers are the only kinds of nondegenerate equilibrium points that can occur in Hamiltonian systems. (43:19) Adding friction to the pendulum model results in a dissipative system. The old Hamiltonian function for the system without friction now becomes a Lyapunov function for the system with friction. (43:54) Comment about the possibility of doing an experiment as part of the project as a way of getting more points. (44:27) The corresponding system. (44:57) See solutions in the phase plane and how they move from higher to lower values of the Lyapunov function. (48:18) Confirm symbolically with the Chain Rule that the Lyapunov function decreases along solution curves.

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
  1. Bill Kinney
  2. hyperbolic equilibrium
  3. hartman grobman theorem
  4. Lyapunov Stability
  5. Lyapunov Function
  6. Hamiltonian function
Рекомендации по теме
Differential Equations Lec 0:52:09

Differential Equations Lec 31 (Class 34), Hyperbolicity, Stability, Hamiltonian & Lyapunov Funct...

Differential Equations || 0:33:58

Differential Equations || Lec 55 || Ex: 4.7: Q31 - 35 || Cauchy's Euler Differential Equations

Differential Equations || 0:05:30

Differential Equations || Lec 30 || Ex: 4.1, Q 31 || Verification of a General solution

Engineering Mathematics | 1:15:27

Engineering Mathematics | Differential Equation - 5 | Lec 31 | GATE 2021 Crash Course

Differential Equations || 0:08:20

Differential Equations || Lec 31 || Ex: 5.1: Q :- 3 || Free Undamped Motion, Spring Mass System

Lec 31: Differential 0:48:01

Lec 31: Differential Equations of Equilibrium in Cylindrical shell using membrane hypothesis

Differential Equations || 0:13:57

Differential Equations || Lec 15 || Exercise No 2.3: Q 31

Lec-31 (Differential equation) 1:30:23

Lec-31 (Differential equation) Basic Mathematics (JC223), JNCASR, IISC

Lec 31 Semi-group 0:32:40

Lec 31 Semi-group arising from the heat equation

Engineering Mathematics | 0:59:39

Engineering Mathematics | Lec 31 | Differential Equation: Solution of Differential Equation - 12

DIFFERENTIAL EQUATIONS (LEC-31)(SEM-2)(B.SC/B.TECH/B.E)#sachinchandra#(9693967379) 0:06:16

DIFFERENTIAL EQUATIONS (LEC-31)(SEM-2)(B.SC/B.TECH/B.E)#sachinchandra#(9693967379)

Differential Equations || 0:13:15

Differential Equations || Lec 31 || Ex: 4.2 || Formula for Reduction of Order, 2nd Solution

MTH-401 Differential Equations 0:26:56

MTH-401 Differential Equations Lec#31,,, after grand quiz 8th lecture

Mod-06 Lec-31 Stability 0:54:05

Mod-06 Lec-31 Stability Equilibrium Points Continued I

Differential Equation(lec-31) Important 0:39:35

Differential Equation(lec-31) Important Questions Of 12th

Linear Homogeneous Difference 0:23:51

Linear Homogeneous Difference Equations with Constant Coefficients (Lec#31)

Lec 31 0:29:15

Lec 31

Diff Eqs Lec 0:51:43

Diff Eqs Lec #30, Hyperbolic Equilibria, Hamiltonian Systems, Pendulum

Differential Equations || 0:20:16

Differential Equations || Lec 67 || Ex: 5.1: Q 29, 30 || Spring Mass System with Driven Force

Differential Equations Lec 0:05:43

Differential Equations Lec 9

partial differential equation 0:53:52

partial differential equation lec no 31

Lec 2 : 2:03:46

Lec 2 : Introduction to Partial Differential equations

Competitive Mathematics | 0:15:53

Competitive Mathematics | 31 JULY 2022 | Differential equation | Lec - 61

Lec 31 Unidirectional 0:31:10

Lec 31 Unidirectional transport. Common form of transport equations

Комментарии
Автор

Nice Close up of the Lyapunov Stability explanation.

videofountain
Автор

At 42:50, you say that in non degenerate Hamiltonian case we always get equilibrium points as centres or saddles. I think the close up of the slide at 37:27 proves that ---- because the Trace of the Jacobian at equilibrium point is zero, hence the point (T, D) is always on the vertical D axis in the TD plane.

swastiktiwari