Question

Reflect on the concepts of linear and non-linear systems. What concepts (only the names) did you need to accommodate the concept of linear and non-linear systems in your mind? What are the simplest linear system and non-linear system you can imagine? In your day to day, is there any occurring fact that can be interpreted as linear systems and non-linear systems? What strategy are you using to get the graph of linear systems and non-linear systems?

Answer 1

Concepts:Linear and Nonlinear-:

-> The concept of linear relationship suggests that two quantities are proportional to each other: doubling one causes the other to double as well.Linear relationships are often the first approximation used to describe any relationship, even though there is no unique way to define what a linear relationship is in terms of the underlying nature of the quantities. For example, a linear relationship between the height and weight of a person is different than a linear relationship between the volume and weight of a person. The second relationship makes more sense, but both are linear relationships, and they are, of course, incompatible with each other. Medications, especially for children, are often prescribed in proportion to weight. This is an example of a linear relationship.

-> Nonlinear relationships, in general, are any relationship which is not linear. What is important in considering nonlinear relationships is that a wider range of possible dependencies is allowed.Many of the possible nonlinear relationships are still monotonic. This means that they always increase or decrease but not both. Monotonic changes may be smooth or they may be abrupt. For example, a drug may be ineffective up until a certain threshold and then become effective. However, nonlinear relationships can also be non-monotonic. For example, a drug may become progressively more helpful over a certain range, but then may become harmful. Thus the degree of help increases and decreases and this is a non-monotonic, as well as a nonlinear, relationship.

To make the distinction between linearity and nonlinearity a bit more precise, recall that a mathematical equation can be thought of as a function — something that maps inputs to outputs. The equation y = x, for instance, is equivalent to a function that takes as its input a value for x and produces as its output a value for y. The same is true of y = x².The equation y = x is linear because adding together inputs yields the sum of their respective outputs: 1 = 1, 2 = 2, and 1 + 2 = 1 + 2. But that’s not true of y = x²: if x is 1, y is 1; if x is 2, y is 4; but if x is 3, y is not 5.

This example illustrates the origin of the term “linear”: the graph of y = x is a straight line, while the graph of y = x² is a curve. But the basic definition of linearity holds for much more complicated equations, such as the differential equations used in engineering to describe dynamic systems.While linear functions are easy enough to define, the term “nonlinear” takes in everything else. “There’s this famous quote — I’m not sure who said it first — that the theory of nonlinear systems is like a theory of non-elephants,” Parrilo says. “It’s impossible to build a theory of nonlinear systems, because arbitrary things can satisfy that definition.” Because linear equations are so much easier to solve than nonlinear ones, much research across a range of disciplines is devoted to finding linear approximations of nonlinear phenomena.

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