^{1}

^{*}

^{1}

^{2}

^{3}

Nonlinearity is required to produce autonomous oscillations without external time dependent source, and an example is the pendulum clock. The escapement mechanism of the clock imparts an impulse for each swing direction, which keeps the pendulum oscillating at the resonance frequency. Among nature’s observed autonomous oscillators, examples are the quasi-biennial oscillation of the atmosphere and the 22- year solar oscillation [1]. Numerical models simulate the oscillations, and we discuss the nonlinearities that are involved. In biology, insects have flight muscles, which function autonomously with wing frequencies that far exceed the animals' neural capacity. The human heart also functions autonomously, and physiological arguments support the picture that the heart is a nonlinear oscillator.

Among nature’s oscillators, the lunar tide is a wellunderstood familiar example. The sea level rises and falls with a period of about 12 hours, which is produced by the gravitational interaction of the moon circling the Earth. Another example is the diurnal atmospheric tide, which is produced by variable solar heating. Generated by external time dependent forcing, the tides are not autonomous oscillators. Oscillations that are forced by a time dependent source can be understood in terms of a linear system. Take n linear equations with n unknowns, A(n, n)·X(n) = S(n), A the square matrix of terms that account for dissipation, X the unknown linear column vector, and S the source vector. With X(n) = A^{−1}(n, n)·S(n), A^{−1} the inverse matrix, it follows that an oscillation X ≠ 0 is produced with a source S ≠ 0 (Finite differencing produces the matrix elements for A to obtain a solution for a system of linear differential equations). In space physics, there is a large scientific body of literature featuring linear mathematical models that can generate and reproduce a variety of observed oscillations applying known excitation sources.

The above linear analysis demonstrates, with X(n) = A^{−1}(n, n)·S(n), that for S = 0, X = 0. This means that without an external time dependent source, a linear system cannot produce an oscillation. Autonomous oscillators must be nonlinear—and a familiar example of such an oscillator is the pendulum clock. Once initiated, the clock continues oscillating at the resonance frequency that is determined by the length of the pendulum. The external source for the clock is the steady force of the weight pulling on the drive chain that activates the escapement wheel. With sufficient energy from the suspended weights to overcome friction, the escapement mechanism provides the impulse/nonlinearity that keeps the pendulum oscillating, as illustrated in

In mathematical terms, the variables (x, y) interact in a nonlinear system, like a_{1}·xy + b_{1}·xy = s_{1}, a_{2}·xy + b_{2}·xy = s_{2}, or for example a_{1}·x^{2} + b_{1}·y^{3} = s_{1}, a_{2}·x^{2} + b_{2}·y^{3} = s_{2}, where a_{12} and b_{12} account for dissipation such as friction, and s_{1}, s_{2} are external sources that would be zero for the mechanical clock. A simple example of nonlinear interaction is illustrated with the temperature feedback of snow. In winter at high latitudes, as it gets colder, the precipitation turns into snow. White snowflakes, covering the ground, reflect incoming solar radiation to make it still colder. The interaction between diminishing radiation and resulting formation of ice crystals, two variables of climate change, say x, y, produce the nonlinear term, xy, that accelerates cooling. Another example of climate change is the melting of glaciers due to CO_{2}-induced global warming. As the glaciers melt, the dark surface gets exposed and absorbs more solar radiation to acelerate the warming. Emptying a bottle with fluid would

be a hands-on experiment demonstrating a nonlinear oscillation. The fluid pouring out, with sufficient speed, is compressed in the bottleneck. The resulting increase in pressure caused by the flow, a nonlinear feedback, slows down the flow to produce a glugging oscillation.

[

Stretch-activation of muscle contraction is the mechanism that produces the high frequency oscillation of autonomous insect flight, briefly discussed in Section 3. The same mechanism is also invoked to explain the functioning of the cardiac muscle. In Section 4, we present a tutorial review of the cardio-vascular system, heart anatomy, and muscle cell physiology, leading up to Starling’s Law of the Heart, which supports our notion that the human heart is also a nonlinear oscillator. In Section 5, we offer a broad perspective of the tenuous links between the modeled fluid dynamical oscillators and the human heart physiology.

In space physics, like in other areas of environmental science, mathematical models have been widely used to provide a physical understanding of the observations. Mentioned in the introduction, a large number of phenomena can be understood in terms of linear systems, like the diurnal atmospheric tide that is produced by variable solar heating. But there is no such external time dependent source known, which could produce the observed quasi-biennial oscillation in the atmosphere or the 22-year oscillation of the solar magnetic field. Numerical models generate the oscillations through nonlinear interactions that produce periodicities determined by the internal dynamical properties of the fluids.

In the zonal circulation of the terrestrial stratosphere at low latitudes, the quasi-biennial oscillation (QBO) dominates, and [

In a seminal paper published in 1968, [

From that paper, we present in

The nonlinearity is displayed in

be written in the approximate form [dU(ω,r)/dr]^{3}, of odd power. With complex notation, a source in the form [Exp(iωt)]^{3} produces the term with Exp[i(ω + ω + ω)t] for the higher order frequency, 3ω. Such a nonlinear source also generates Exp[i(ω + ω − ω)t] for the fundamental frequency ω, which maintains the oscillation.

Zonal-mean oscillations with periods around 2 months have been seen in ground based and satellite wind measurements in the Earth atmosphere [12-14]. The Numerical Spectral Model generates such oscillations

[

The similarity between the BMO and QBO is also evident in ^{3}. Such a non-linearity can generate an oscillation without external time-dependent forcing. Like the QBO, the meridional BMO can be understood as a nonlinear oscillator.

Like the QBO, the BMO is dissipated by viscosity. But unlike the zonal winds of the QBO, the meridional winds of the BMO produce pressure variations that counteract, and dampen, the winds to produce a shorter dissipative time constant. This additional thermodynamic feedback explains why the period of the BMO is much shorter than that of the QBO. In both cases, the oscillation periods are determined by the dissipation rates.

The solar magnetic field is observed varying with a period of about 22 years. During periods of enhanced

magnetic fields, irrespective of polarity, sunspots form around the equator that are associated with enhanced solar radiation. The resulting 11-year cycle of solar activity produces variations that strongly depend on the wavelength of the emitted radiation. At lower altitudes in the atmosphere near the ground, the solar cycle effect is relatively small. But with increasing altitude, the effect increases due to the shorter wavelength ultraviolet radiation that is absorbed, and above 150 km, the medium of artificial satellites, the radiative input can vary by as much as a factor of 2 during the solar cycle. The solar cycle and its periodicity are variable, and empirical models that employ the observed precursor polar magnetic field during the minimum have been very successful in predicting the magnitude and duration of the solar maximum [

Pars pro toto, we refer here to the magneto hydro-dynamic (MHD) model of [

[_{f}(r_{c},θ,t), where B_{f} represents the toroidal magnetic field, (r,θ) are spherical polar coordinates (r_{c}, for the tachocline region), and B_{o} is taken to be constant,

(1)

The nonlinear source terms of Eq.1 have the property that they are of odd (e.g., 3^{rd}) power, a similar nonlinearity also appears in the classical dynamo model of [

Insect flight is an outstanding example of autonomous oscillation, and the muscle physiology involved has served as a model for understanding the human heart.

In an advanced class of insects, wing frequencies as large as 1000 Hz are produced, which are determined by the inertia of the wings and far exceed the frequency capacity of the animals’ neural system [

muscles work in tandem through stretch-activation. When the dorsoventral muscle contracts to lift the wings up (

A tutorial review is presented of the human heart function, which provides the framework for the notion that the autonomous heart can be understood as a nonlinear oscillator.

In the cardiovascular system [_{2}-poor blood, returning from the lungs in veins, is pumped in arteries by the left side of the heart to the tissue cells of the human body. The returning blood in veins, oxygen-poor and CO_{2}-rich, is pumped in arteries by the right heart into the lungs that pick up oxygen and shed CO_{2}.