What are 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.

Is the van der Pol oscillator chaotic?

Van der Pol’s Equation was first given in 1926. 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.

What is the van der Pol problem?

Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of Nature that at certain drive frequencies an irregular noise was heard, which was later found to be the result of deterministic chaos.

Is Van der Pol equation linear?

The van der Pol oscillator is a non-conservative oscillator with non-linear damping.

What is the stable limit cycle?

Stable limit cycles are examples of attractors. They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.

What is limit cycle oscillator?

Limit cycle is an oscillation peculiar to nonlinear systems. The oscillatory behavior, unexplainable in terms of linear theory, is characterized by a constant amplitude and frequency determined by the nonlinear properties of the system.

How do you determine the stability of a limit cycle?

The usual approach is to consider small disturbances of the hmit cycle and to find out if these die away by looking at their first order effects in terms of the so-called characteristic exponents. If all but one of the characteristic exponents are negative, the limit cycle is stable.

What is limit cycle in DSP?

A limit cycle, sometimes referred to as a multiplier roundoff limit cycle, is a low-level oscillation that can exist in an otherwise stable filter as a result of the nonlinearity associated with rounding (or truncating) internal filter calculations.

What is limit cycle differential equations?

A limit cycle is a closed trajectory such that at least one other trajectory spirals into it (or spirals out of it). For example, the closed curve in the phase portrait for the Van der Pol equation is a limit cycle. The limit cycle in the Van der Pol oscillator is asymptotically stable.

What is van der Pol oscillator?

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation :

What is the van der Pol equation?

Actually, the van der Pol equation is a particular case of more general Liénard equation: named after the French physicist Alfred-Marie Liénard (1869–1958). The Van der Pol equation has no exact, analytic solution, but it has a limit cycle.

What is a relaxation oscillator?

An example of a Relaxation oscillator. In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation :

Is the van der Pol system a Liénard system?

Actually, the van der Pol system ( 1) satisfies the Liénard’s theorem ensuring that there is a stable limit cycle in the phase space .The van der Pol system is therefore a Liénard system . which can be regarded as a special case of the FitzHugh-Nagumo model (also known as Bonhoeffer-van der Pol model ). where the transformation was used.