Impossibility of Oscillations in 1D Flows
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- เผยแพร่เมื่อ 11 ม.ค. 2025
- What kinds of end states exist for vector fields on the real line? A graphical approach is used to show that only fixed points are possible. Oscillations are impossible. For vector fields on the circle, oscillations are possible • Flows on the Circle | ...
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► From 'Nonlinear Dynamics and Chaos' (online course).
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Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 2: Flows on the Line
nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field One-Dimensional 1-dimensional Functions oscillations line 1D circle curve
#NonlinearDynamics #DynamicalSystems #DifferentialEquations #dynamics #dimensions #PhaseSpace #Poincare #Strogatz #graphicalmethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #OneDimensional #Functions
the fact this playlist directly covers the book used in my course is life saving. Although the book I argue is actually one of the more digestible ones.
Question: @ 5:53 when we cut the x-axis and form a circle, conventionally the right direction is positive and in a circle the counterclockwise rotation is positive. I understand this detail isn't too imperative to the message being explained at the moment but I am simply curious about how the direction of the vector field on the circle was concluded.
thank you
-a student returning to college who took diff eq in 2017
I'm glad the playlist is useful, and thanks for watching. As you rightly point out, the main point here wasn't to get the vector direction correct, but just to illustrate the difference between the topology of a circle, versus the real line. I follow the standard convention of the angle increasing counter-clockwise in my flows on the circle video: th-cam.com/video/Q_0oB1DHyQU/w-d-xo.html
@@ProfessorRoss thank you professor!
Nice explanation