# Approximating dynamics in continuous-time with discrete-time

I'm not sure if this is the best place to ask this question, so please direct me to the appropriate community if this question seems out of place here.

I am working on a project in which I'm trying to generate chaotic behavior in a system consisting of three RLC circuits interacting with each other nonlinearly with feedback. The system is intended to be a highly simplified model of a real-world phenomenon. Each circuit is driven with two inputs:

1. a sinusoidal voltage source
2. measurements made at nodes on one circuit which are amplified and fed into the others

Chaotic behavior in circuits has been demonstrated by Chua et al. in the so-called Chua circuit.

I have two ways of generating the sinusoidal inputs to the circuits. First, I can use standalone function generators, which, according to Wikipedia, generate signals by charging and discharging a capacitor. Second, I can use a data acquisition system (or DAQ), which interfaces with my computer and generates signals by sampling a virtual waveform and outputting the samples. The value that is outputted is held for 1/F_s seconds, where F_s is the sampling rate.

For convenience, I would prefer to use the latter because the input can be easily controlled (switched on or off and the amplitude or frequency adjusted) based on measurements of voltage that I take across different elements in my system. However, because the signal generated by the DAQ is not truly continuous-time (unlike the real world in which time is continuous) in the sense that the outputted value is only updated every 1/F_s seconds, I'm wondering if the interactions and feedback will be able to accurately recreate the same type of chaotic behavior that truly continuous time inputs (such as those generated by a function generator) would produce.

Importantly, my sinusoidal input signals are band-limited to 100 Hz and the sampling rate of the DAQ can be up to 80 kHz. At low sampling rates (less than 1 kHz), I can clearly see jumps in voltage between successive samples when I measure the DAQ-generated signal on an oscilloscope. At higher sampling rates (tens of kHz), these jumps essentially disappear (even though I know they are there and just cannot be seen).

• The answer is Yes but try to show more specs. transfer function , schematic, errors, acceptance criteria. Feb 7, 2018 at 18:55
• Your DSO sampling interval must be much smaller than dt for a slew rate of dV/dt for accurate readings. so don't assume they are there. It must exceed the Nyquist Criteria to get all the harmonics of step function by a wide margin Feb 7, 2018 at 19:01
• Consider to add a low pass filter on your output while using the maximal DAC sampling rate. Feb 8, 2018 at 10:20
• MarkoBuršič and @TonyStewart.EEsince'75, thank you for your comments. I decreased my sampling interval and once it was fine enough, I was able to see the steps just as I was for the signal generated with the lower slew rate. I then did exactly what MarkoBuršič suggested by adding an RC filter to the output of the DAQ card and using the voltage across the capacitor as my output to smooth my signal. As long as the cutoff frequency is greater than my bandwidth but smaller than the slew rate of the DAC, I get a smooth signal. Feb 8, 2018 at 12:20
• Yes that is essential. ... To suppress excess bandwidth noise. As @laptop wrote. Curious .What are you simulating ? Feb 8, 2018 at 12:58

## 1 Answer

I'm wondering if the interactions and feedback will be able to accurately recreate the same type of chaotic behavior that truly continuous time inputs (such as those generated by a function generator) would produce.

Yes, it will be able to recreate the chaotic behavior, but only if the response of the non-linear system is limited to 100Hz (which it is probably not).

Since the input to your digital system is 100Hz that is limiter for the response of the digital system.

If you can prove that the response of the nonlinear systems response is bounded to below 100Hz then you don't need to be worried about what the digital system is doing to the analog system.

One method that might help determine this is to insert one or more low pass filters in the nonlinear circuit and observing the result. One problem I foresee is the circuit may suffer from bandlimiting because nonlinear circuits change behavior from tiny fluctuations (butterfly effect).

Another method that might help determine if the nonlinear circuit would suffer from limiting it to 100Hz would be to simulate this circuit in SPICE and change the the simulation time to a fixed time step with a rate of 100Hz

If the system works the DAQ will need a low pass filter on the output (probably at 100Hz) to recreate the signal and reduce the chopping from the zero order hold.

Source: RealHD audio

There are also different kinds of holds that you can use with a digital system to more accurately recreate a signal and the bi-linear transform to determine the response:

Source: Slideplayer

• Thanks for the information about zero-order hold. I had learned about that in one of my classes back in college, but I had forgotten about it. What I'm looking for is a signal that is bandlimited to 100 Hz, so that any ringing effects that occur when samples are delivered by the DAC are eliminated. Feb 8, 2018 at 14:48
• To clarify, you suggest simulating the circuit on SPICE to determine the bandwidth of the outputs of the system, which may be larger than the bandwidth of the inputs due to chaotic fluctuations? Is this what you mean by, “the circuit may suffer from bandlimiting because nonlinear circuits change behavior from tiny fluctuations?” Could you please explain why I should be concerned with the bandwidth of my output if it's only the sinusoidal input that I want to be continuous? Feb 8, 2018 at 14:49
• I added RC filters at the output of my DAC with a cutoff frequency of 400 Hz. With the roll-off, I think this cutoff should be close enough to 100 Hz to remove the ringing effects. Feb 8, 2018 at 14:51