.ChaoticCircuits.UsersGuide.Concept .ChaoticCircuits.UsersGuide.ConceptChaotic circuits have received considerable interest in the literature because they have been applied in numerous areas such as secure communications, robotics, image processing or random bit generator. One of the most important research directions is constructing robust chaotic oscillators with simple structures. There are two kinds of simple chaotic oscillators: non-autonomous and autonomous oscillators.
Note that an oscillator is not a perpetuum mobile but needs energy supply either an explicitely modeled source voltage or implicitely. The operational amplifiers might have a supply voltage not explicitely modeled.
The default parameters lead to periodic results. Possible parameter changes to obtain chaotic behavior are noted in the documentation of the respective subpackage.
Be curios, try different parameter settings to explore the path from periodic behaviour to chaos - in many cases you see bifurcation, i.e. different attractors.
Note that at the beginning a transient process takes place, dependent on the initial conditions. It might be necessary to continue the simulation, as shown for the analytic equations of the ChaoticOscillator proposed by [Tamasevicius2005] :
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| Fig. 1 a=0.75: y vs. x initial transient | Fig. 2 a=0.75: y vs. x after one continuation |
The results meant for plotting are labeled. Normally plot one result variable versus the other, in this example y vs. x and / or z vs. x. Of course the initial transient depends on the initial conditions.
For default parameters, the results converge to a fixpoint. In certain parameter regions only one fixpoint is present. Changing parameters, it might happen that one of two fixpoitns is reached, dependent on the inintial conditions. This is called bifurcation and might happen mutliple times. In the end this leads to chaotic behaviour: Smallest differences in initial conditions lead to dramatically different results. Note that of course the results are strongly dependent on the implementation details of the solver, too.
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| Fig. 3 a=0.35: y vs. x initial transient | Fig. 4 a=0.35: y vs. x after one continuation |
In these examples simple autonomous systems are presented:
| Component | IdealCircuit | ImprovedCircuit |
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| Diode model | A simple diode according to Shockley equation is used. | A more sophisticated diode model with linear continuation of the characteristic and optional temperature dependency. |
| Operational amplifiers | ideal 3 pin model: input currents are zero, differential input voltage is zero, amplification is infinite, output voltage is not limited | more sophisticated opAmp model: input currents are zero, differential input voltage is zero, amplification is finite, output voltage is limited by suppply voltage |
Note that for the more sophisticated OpAmp-model IdealizedOpAmp3Pin from this library is used
instead of IdealizedOpAmpLimited from MSL
to get rid of the implementation with smooth (which allows tools to avoid events when saturating!) and noEvent (which suppresses events!).
All circuits can be built in reality with simple electronic components. If an inductance is used, its resistance is modeled, too. The ohmic losses of an inductance can't be neglected.
The equations are summarized in this document.
Note: In the subdirectory Resources.Reference reference results for all examples are stored.