# I Does chaos exist in circuits with linear elements?

Tags:
1. Aug 10, 2017

### student-engineer

I have heard that chaos exists in all dynamical systems.Does this mean that chaos exists in circuits with linear elements too?Which software is best for analyzing chaos in electric circuits?

2. Aug 10, 2017

### Svein

Thermal noise (which you will get from a physical resistor) is sort of chaotic.

3. Aug 10, 2017

### student-engineer

But I think chaos is deterministic (i.e not noise,not random), non-periodic flow(i.e changing with respect to time).I think chaos is not noise.Chaos exists because of sensitive dependence on initial conditions.

4. Aug 10, 2017

### Svein

That was an unusual definition of chaos...

I do not disagree, but I am not quite sure what you mean. When you say "Chaos exists because of sensitive dependence on initial conditions", I usually call that an unstable or divergent solution. It is treated extensively in the theory of differential equation.

Now look at my avatar. It is part of an image of a Julia set. Wikipedia has this to say about a Julia set: "the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'." Check out https://books.google.no/books?id=uG...EIVTAF#v=onepage&q=julia set physical&f=false for physical structures described by Julia sets.

5. Aug 10, 2017

### sophiecentaur

There is a massive difference between Random (as in thermal noise) and Chaotic, which is deterministic but gives very big swings in output results (same result every time for the same input values in the mathematical model). Noise otoh is Gaussian (or similar), about a mean value.
I read somewhere (no citation but no surprise either) that the weather can sometimes be chaotic and sometimes not. 'They' can identify which.

6. Aug 10, 2017

### student-engineer

Yes,I was learning about chaos,lorenz equations.Yes,but may be that perturbation is initial condition and when the solutions do not converge,it is called chaos.Chaos is a form of non-linear dynamics of a dynamical system.It is discussed in non-linear control theory and methods for the control of chaos have also been made so far.Control systems in electrical engineering is all about the control of output voltage,current,frequency.Most of the or may be all systems in reality are non-linear,correct me if I am incorrect.The dynamical systems are linearized for the design of controller.

7. Aug 10, 2017

### student-engineer

I learned the same.

8. Aug 10, 2017

### student-engineer

E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. https://doi.org/JAHSAK20, 130–141 (1963). Lorenz called chaos deterministic a-periodic flow.
What is perturbation?I am not able to understand it exactly.Is it a collective terms used to refer to mathematical modeling methods?A set of differential equations may describe a mathematical model.Then what could be the difference of chaos from divergent solution.In control theory,the aim is to make a system stable i.e a solution which converges to a point,this is the rationale behind controller design.Controllers are designed to retain the steady state behavior of a system.But,it will be nice if you share more of your concepts or correct any of mine.

9. Aug 10, 2017

### Svein

Perturbation is any deviation from the calculated reference point. Imagine a thin stick set upright in the vertical position. It may stand there as long as nothing disturbs it, like a slight vibration in the floor, a slight air gust... All these introduce perturbations in the system (consisting on the stick resting on the floor).

10. Aug 10, 2017

### student-engineer

Can perturbation thus simply be an initial condition and that initial condition could be of any of the parameters of the system and thus an initial condition giving divergent response implies a chaotic system too?

11. Aug 10, 2017

### Filip Larsen

There is no deterministic chaos in any system which has only linear elements, i.e. systems that follows the superposition principle for all its solutions. It is thus a necessary, but not sufficient, condition for chaos (at least one positive Lyapunov exponent) that some parts of the system has non-linear behaviour that allows for the "folding" and "stretching" dynamics similar to Horseshoe maps.

Apparently, it is technical possible to get chaos without non-linear dynamics (in this case with a linear filter) provided you have access to a random pulse train, but the linear circuit itself is still not able to exhibit chaos on its own.

In electronic systems, Chua's circuit is a early well known example of a practical system that can exhibit chaos. Another example is Van der Pol oscillators. In these, one or more components exhibit non-linear behaviour.