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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?
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.Svein said:Thermal noise (which you will get from a physical resistor) is sort of chaotic.
That was an unusual definition of chaos...student-engineer said: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.
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.Svein said:Thermal noise (which you will get from a physical resistor) is sort of chaotic.
Svein said: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.
I learned the same.sophiecentaur said: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.
Svein said: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=uGDmBwAAQBAJ&pg=PA300&lpg=PA300&dq=julia+set+physical&source=bl&ots=uX4C6Ud0Rc&sig=ksl23FpMbaNKhtbvgAb6mjW_x2c&hl=no&sa=X&ved=0ahUKEwjw4r6yiM3VAhXChrQKHYqZAxAQ6AEIVTAF#v=onepage&q=julia set physical&f=false for physical structures described by Julia sets.
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).student-engineer said:What is perturbation?
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?Svein said: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).
Chaos in circuits with linear elements refers to the phenomenon where a small change in initial conditions can lead to drastically different outcomes in circuit behavior. This is often seen in complex circuits with multiple feedback loops and nonlinear components.
In regular circuit behavior, small changes in initial conditions lead to small changes in the output. However, in chaotic circuits, even tiny changes in initial conditions can lead to drastically different outputs, making it difficult to predict the behavior of the circuit.
The main cause of chaos in circuits with linear elements is the presence of multiple feedback loops and nonlinear components. These elements can amplify small changes and create complex patterns of behavior in the circuit.
While it is difficult to completely eliminate chaos in circuits with linear elements, it can be controlled to some extent by carefully designing the circuit and reducing the number of feedback loops and nonlinear components. Advanced control techniques such as chaos control and synchronization can also be used.
Yes, chaos in circuits with linear elements has been found to have practical applications in fields such as secure communication, data encryption, and signal processing. It can also be used to generate complex waveforms for scientific research and testing purposes.