# Nonlinear Systems

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## Nonlinear Systems

oscillatory systems whose properties depend on processes taking place in them. The oscillations of such systems are described by nonlinear equations, and the systems themselves are called nonlinear.

Examples of nonlinear systems are mechanical systems in which (1) the elastic moduli of the bodies depend on their deformations, (2) the coefficient of friction between the surfaces of the bodies depends on the relative velocity of the bodies (the sliding velocity), or (3) the masses of the bodies depend on their velocities. Nonlinear electric systems are those containing ferroelectrics whose dielectric constant depends on the intensity of the electric field.

In mechanical systems the abovementioned dependencies lead to (1) nonlinear relationships between stresses and deformations (a violation of Hooke’s law), (2) to a nonlinear relationship between frictional forces and sliding velocity, or (3) to a nonlinear relationship between the force acting on a body and the acceleration imparted to it (if the magnitude of the body’s velocity changes in the process). Analogously, in electrical systems the relationships between electric charges and the intensity of the field created by the charges, between the voltage across the leads of a conductor and the current flowing through the conductor (in violation of Ohm’s law), or between a current and the intensity of the magnetic field (magnetic induction) created by the current in a magnetic material are found to be nonlinear. Each of these nonlinear relationships causes the differential equations that describe the behavior of nonlinear systems to be nonlinear; hence their name.

Strictly speaking, all physical systems are nonlinear. The behavior of nonlinear systems differs fundamentally from that of linear systems. One of the most characteristic properties of nonlinear systems is the violation in them of the superposition principle: the effect of each action on the system in the presence of another action is different from its effect in the absence of the other action. Many important properties in the behavior of nonlinear systems manifest themselves if oscillations are excited in the system; this determines the main area of practical application of such systems. One of the most important applications is the

generation of undamped oscillations through conversion of energy from a constant source, using the nonlinear properties of resistance (friction). Because of the distortion of the shape of an external harmonic influence in nonlinear systems and the inapplicablility of the superposition principle to such systems, various types of oscillation conversion may be achieved in them, such as rectification, frequency multiplication, and modulation of oscillations.

### REFERENCES

Gorelik, G. S.*Kolebaniia i volny*, 2nd ed. Moscow, 1959. Chapter IV.

Andronov, A. A., A. A. Vitt, and S.E. Khaikin.

*Teoriia kolebanii*, 2nd ed. Moscow, 1959. Chapter 2, § 1–4, 6–7; chap. 3, § 1–3, 6–7.

S. E. KHAIKIN