Ван дер Поль осцилляторы: Nonlinear серпіліс және хаостық жүйелер

Oscillating dynamical system with nonlinear damping

In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non conservative, oscillating system with non linear damping. It evolves in time according to the second order differential equation

where x is the position coordinate—which is a function of the time t—and μ is a scalar parameter indicating the nonlinearity and the strength of the damping.

The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation oscillations and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes. When these circuits are driven near the limit cycle, they become entrained, i. e. the driving signal pulls the current along with it. 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. The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh and Nagumo extended the equation in a planar field as a model for action potentials of neurons. The equation has also been utilised in seismology to model the two plates in a geological fault, and in studies of phonation to model the right and left vocal fold oscillators.

As μ moves from less than zero to more than zero, the spiral sink at origin becomes a spiral source, and a limit cycle appears "out of the blue" with radius two. This is because the transition is not generic: when 1=ε = 0, both the differential equation becomes linear, and the origin becomes a circular node. Knowing that in a Hopf bifurcation, the limit cycle should have size we may attempt to convert this to a Hopf bifurcation by using the change of variables which givesThis indeed is a Hopf bifurcation.

The quantum van der Pol oscillator, which is the quantum mechanics version of the classical van der Pol oscillator, has been proposed using a Lindblad equation to study its quantum dynamics and quantum synchronization. Note the above Hamiltonian approach with an auxiliary second order equation produces unbounded phase space trajectories and hence cannot be used to quantize the van der Pol oscillator. In the limit of weak nonlinearity (i. e. μ→0) the van der Pol oscillator reduces to the Stuart–Landau equation. The Stuart–Landau equation in fact describes an entire class of limit cycle oscillators in the weakly nonlinear limit. The form of the classical Stuart–Landau equation is much simpler, and perhaps not surprisingly, can be quantized by a Lindblad equation which is also simpler than the Lindblad equation for the van der Pol oscillator. The quantum Stuart–Landau model has played an important role in the study of quantum synchronisation (where it has often been called a van der Pol oscillator although it cannot be uniquely associated with the van der Pol oscillator). The relationship between the classical Stuart–Landau model (μ→0) and more general limit cycle oscillators (arbitrary μ) has also been demonstrated numerically in the corresponding quantum models. The circuit contains: a triode, a resistor R, a capacitor C, a coupled inductor set with self inductance L and mutual inductance M. In the serial RLC circuit there is a current i, and towards the triode anode ("plate") a current ia, while there is a voltage ug on the triode control grid. The Van der Pol oscillator is forced by an AC voltage source Es.]] Author James Gleick described a vacuum tube Van der Pol oscillator in his book from 1987 Chaos: Making a New Science. According to a New York Times article, Gleick received a modern electronic Van der Pol oscillator from a reader in 1988.