Linear versus nonlinear phenomena
When the change in the output of a system is directly proportional to the change in input, the system exhibits a linear phenomenon. All other phenomena are nonlinear. Linear phenomena are much simpler and coherent than nonlinear phenomena. Maxwell’s theory of Electromagnetism [1] is a linear phenomenon. Millions of electromagnetic signals travel in space among cell phones through satellites, but they don’t interfere with each other. The data and videos sent via the signals are neither lost nor created during the transmission. On the other hand, when we listen to two music tracks at the same time, they interfere with each other resulting in some noise. Since elements of nonlinear systems interact with each other, the results might add or reduce in comparison to their components. Another example of nonlinear phenomena is our natural ecosystem. One case study in [2] shows how small changes in interactions between predators and prey cascaded among trophic levels causing drastic climate changes in the forests.
| PROPERTIES | LINEAR SYSTEMS | NONLINEAR SYSTEMS |
|---|---|---|
| Representation | Represented by a line or an equation with single degree variables. | Represented by a curve or equations with more than one degree of variables/ trigonometric functions. |
| Superposition Principle | Addition: F (x1 + x2) = F (x1) + F (x2), Scaling: F (αx) = α F (x) |
Nonlinear systems cannot be broken into subproblems. [4] |
| Dependencies | Systems which have fewer elements or dependencies. | Nonlinear systems are composed of a large number of elements. |
| Motion | Motion is regular in space and time. | These systems can show arbitrary and random motion. [3] |
| Sensitivity and determinism | The response is directly proportional to the input parameters. The past and future values can be calculated from linear calculations. | A small change in input may cause no |
| effect or huge differences in the system. These systems are not deterministic. [3] | ||
| Examples | Family expenditures increase linearly with the number of family members, the current is linearly dependent on voltage across two points, etc. | All the natural phenomena like the motion of a pendulum, climate prediction or spread of a pandemic. |
Relationship
Nonlinear systems are complex because their elements interact with each other arbitrarily. There- fore, the superposition principle does not apply to such systems. But, since linear systems are easier to solve and understand, most of the research is into approximations [5] and transformations [6] of nonlinear equations into linear equations. We can trace the study of the relationship between linear and nonlinear equations back to 1932 when Carleman linearization expressed finite-dimensional ordinary differential nonlinear equations in infinite-dimensional linear equations [7].