A mathematical explanation of hysteresis?

[lucerne]

I wish to know why and how does hysteresis occur from a purely mathematical perspective, using analytical functions and multivariable calculus. Thank you!

 

[Vanadium 50]

Since some phenomena and materials experience this and some don't, a "purely mathematical perspective" is impossible. Some physics will need to enter the problem.

 

[HallsofIvy]

From a purely mathematical point of view, hysteresis results when there is a cubic (or higher degree odd) dependence. If, in the phase plane (y versus dy/dt) y has a cubic dependence on dy/dt and we have the central part (in red in the graph below) unstable, then as x approaches 2 from below, it jumps to the other blue branch, going from 2 up. But as we come back down, x will stay on that section until it gets down to x= -2 before it jumps to the branch. (The x-axis is vertical in this graph.)

 

[Studiot]

Hysteresis refers to an number of unrelated physical phenomena (and to some related ones) in mechanical systems, digital electronics, elasticity, electromagnetism to name but a few.

So if you would like to tell us which area you referring to please?

 

[pmacias]

Thank you for your question. Hysteresis is a phenomenon in which the output of a system is not directly proportional to the input. It can be observed in various physical systems, such as magnetic materials, electrical circuits, and mechanical systems.

From a mathematical perspective, hysteresis can be explained using analytical functions and multivariable calculus. To understand hysteresis, we need to first understand the concepts of input and output variables. In a system, the input variable is the independent variable that is controlled or varied, while the output variable is the dependent variable that is affected by the input.

In a hysteresis system, the output variable does not change linearly with respect to the input variable. This means that the rate of change of the output variable is not constant for different values of the input variable. Mathematically, this can be represented as a nonlinear function of the input variable.

One way to understand hysteresis mathematically is through the use of differential equations. These equations describe the relationship between the input and output variables in a system. In hysteresis systems, the differential equations are often nonlinear, meaning that they cannot be solved using basic algebraic techniques. Instead, multivariable calculus is used to analyze and solve these equations.

Another mathematical tool used to explain hysteresis is the concept of a hysteresis loop. This is a graphical representation of the relationship between the input and output variables in a hysteresis system. It is a closed curve that can be plotted using the input and output data points. The shape of the hysteresis loop can provide insights into the behavior of the system and can be analyzed using multivariable calculus techniques.

In summary, hysteresis can be explained mathematically using analytical functions and multivariable calculus. By understanding the nonlinear relationship between the input and output variables in a system, we can use mathematical tools such as differential equations and hysteresis loops to analyze and predict the behavior of hysteresis systems. I hope this helps to answer your question.