Graphical Analysis of 1D Nonlinear ODEs

Introduction to the geometric / graphical approach for analyzing nonlinear ordinary differential equations, including fixed points and their stability. Next, an example: population growth model • Population Growth- The...
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Ross Dynamics Lab: chaotician.com​
► Next: Population growth model (logistic model)
• Population Growth- The...
► See also 2D and 3D dynamical systems
2D • 2D Nonlinear Systems I...
3D • 3D Systems, Lorenz Equ...
► Related videos
Example of over-damped bead in a rotating hoop • Bead in a Rotating Hoo...
Flows on the circle • Flows on the Circle | ...
Flows in 2D • 2D Nonlinear Systems I...
Linearization near fixed points in 2D • Nonlinear Systems: Fix...
► From 'Nonlinear Dynamics and Chaos' (online course).
Online course playlist is.gd/Nonlinea...
► New topics posted regularly.
Subscribe is.gd/RossLabS...
► Course lecture notes (PDF)
is.gd/Nonlinea...
Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 2: Flows on the Line
1D vector field autonomous time-independent nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Points Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field One-Dimensional 1-dimensional Functions
#NonlinearDynamics #DynamicalSystems #DifferentialEquations #dynamics #dimensions #PhaseSpace #Poincare #Strogatz #graphicalmethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #OneDimensional #Functions

[@lowerbound4803]

I want to say thank you from my heart 😻 Very well-explained🙏🙏

[@ProfessorRoss]

Always welcome

[@tiddlywinks497]

Such a fantastic series, thank you so much!

[@marcopivetta7796]

as before, thank you for explaining so thoroughly. As you said, even if the math itself is rather confusing (to me, as i'm not too familiar with the symbology or the more complex grammar), the graphics are pretty intuitive and make sense... So far at least!

[@AnuragSharma-aka-sheiroo]

Another engaging video. I am going slower than I expected to but hopefully the end of this course is the stable x* for my journey.

[@MariaTeresa-cb4jt]

Dear Prof. Ross, thank you for posting this series of lectures. May I ask why is it the case that when x dot > 0, it moves to the right, and when x dot < 0, it moves to the left?

[@ProfessorRoss]

Maria, this is because we usually depict the x-axis such that x increases to the right. So x dot > 0 means x is increasing; thus, motion to the right. And vice-versa.

[@raktimpal641]

Great Lecture. Thanks

[@ProfessorRoss]

I'm glad it was helpful.

[@iitjee1482]

sir are you following the book Strogatz

[@ProfessorRoss]

Indeed. All the details regarding the section and other helpful info are in the video description

[@arijitmalakar3820]

x is stable at -π What does it mean for the population?

[@ProfessorRoss]

The example equation here at 1:23, dx/dt = sin(x), is just a mathematical example. It is not from a population dynamics model, so that's why x can be any real number, positive or negative. In this example, x = -π is just a state of this abstract system that is stable in the sense that any small deviation of the state x from -π from shrink, taking the state back to -π. Hopefully that explanation helps.

[@rupabasu4261]

Sir, just to clarify, this treatment is for equilibrium of motion, not equilibrium of force right?

[@ProfessorRoss]

Rupa, yes, just equilibrium of motion. The treatment done here is general for any system that can be described mathematically by ordinary differential equations, and is it necessarily tied to physics or physical forces.

[@sayanjitb]

Dear sir, at the time stamp around 29:39, you showed how the signs of f'(x*) determines the character of the stability of the point x*. But f'(x*) is evidently the second-order derivative of x(t); hence when its sign is positive, it should indicate local minima, an intuitively stable equilibrium point (opposite for -ve sign of f'(x)). But your qualitative result is yielding the opposite outcome, i.e., unstable point. Don't they contradict each other? Can you please shed some light on it? Thank you

[@sayanjitb]

And one more, at 30:51, can I call this point x* a "semi equilibrium" point?

[@ProfessorRoss]

I think your confusion results from conflating a derivative with respect to TIME t ( d/dt, which is represented here by an over dot) and a derivative with respect to the STATE x ( d/dx, which is represented here by a prime). So while dx/dt = f(x), f'(x) means df/dx, so it does not denote a second derivative of x(t) with respect to time. It is the instantaneous derivative of the state change (dx/dt) with respect to the state.

[@ProfessorRoss]

@@sayanjitb As long as you define a semi equilibrium point this way, yes. It is not a terminology I use in these lectures.

[@sayanjitb]

@@ProfessorRoss yes sir, it's my bad. Now I understand. Thank you sir.