## Answer in 60

### Input interpretation

### Phase portrait with limit cycle

**Key moments in video**

From 00:53 — Introduction and analysis of the van der Pol oscillator

From 01:23 — Van der Pol oscillator limit cycle behavior

From 02:18 — Multiple – scale expansion for the Van der Pol oscillator

From 05:59 — Initial Conditions

From 07:06 — Imposing conditions on the slow time evolution

From 10:47 — Example: First order non – linear ODE

From 14:21 — Frequency shifts on different scales

**Key moments in video**

From 01:09 — Introduction to the Van der Pol oscillator

From 02:24 — Vacuum tube analogy

From 04:51 — Theory of negative damping

From 06:23 — Unique, stable limit cycle

From 09:56 — Motivation and explanation of transformation

From 26:50 — Violin bow and stick slip oscillations

From 29:21 — Graph of waveform

From 34:06 — Estimate of period

From 52:53 — Nonlinear damping

From 01:05:48 — Averaging theory

**Key moments in video**

From 04:25 — Second-Order Differential Equation into a Set of First Order Equations

From 06:14 — Long Term Stability Analysis

From 06:19 — Long Term Stability Analysis Two Equations

## Related querie

### What is van der Pol equation used for?

The Van der Pol equation is now concerned as **a basic model for oscillatory processes in physics, electronics, biology, neurology, sociology and economics** [17]. Van der Pol himself built a number of electronic circuit models of the human heart to study the range of stability of heart dynamics.

### How does a relaxation oscillator work?

A relaxation oscillator is a repeating circuit (like the flasher circuit illustrated above) which **achieves its repetitive behavior from the charging of a capacitor to some event threshold**. The event discharges the capacitor, and its recharge time determines the repetition time of the events.

### What does a phase portrait show?

A phase portrait graph of a dynamical system depicts **the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a state space**.

### Why is the van der Pol equation important?

Since then it has been used by scientists **to model a variety of physical and biological phenomena**. For instance, in biology, the van der Pol equation has been used as the basis of a model of coupled neurons in the gastric mill circuit of the stomatogastric ganglion [18, 19] (see also Appendix).

### Is the van der Pol equation stiff?

Example: The van der Pol Equation, **µ = 1000 (Stiff)**

This example presents a stiff problem. For a stiff problem, solutions can change on a time scale that is very short compared to the interval of integration, but the solution of interest changes on a much longer time scale.

### How does an Op-Amp relaxation oscillator work?

For the relaxation oscillator, **the opamp is configured to run “open loop” in which no feedback is used to limit the gain**. Because the amplifier is essentially unstable, it will move as quickly as possible to either its largest possible positive or negative output. These limits are set by the power supply voltage.

### What is used as a relaxation oscillator?

Relaxation oscillator consists of **a feedback loop with a switching device such as a transistor, Op-Amp, relay,etc.. that repetitively charges and discharges the capacitor through a resistor**. In UJT Relaxation Oscillator, UJT is used as the switching device.

### What do phase planes represent?

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is **a visual display of certain characteristics of certain kinds of differential equations**; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc.

### What is phase trajectory and phase portrait?

We think of this as describing the motion of a point in the plane (which in this context is called the phase plane), with the independent variable. as time. **The path travelled by the point in a solution is called a trajectory of the system.** **A picture of the trajectories is called a phase portrait of the system**.

### Is Van der Pol chaotic?

Van der Pol's Equation was first given in 1926. It gives limit cycles. The present paper reports the chaotic behavior of modified Van der Pol's Equation with forcing function. **In three of six cases, CHAOS is found, while three other cases give limit cycles**.