Investigating Non-Linear Resonance in Chua's Circuit

What is non-linear resonance?

Resonance is defined as a physical phenomenon where the response of a dynamical system reaches a maximum amplitude as a result of an applied periodic force. Nevertheless, what are dynamical systems and periodic forces? Brief explanations will be provided in the following lines to lay out a conceptual intuition before diving into the experiments.

**** Note that readers with a background in physics or mathematics can skip the first part of the introduction and dive right into the project objectives and experiments.

1. Dynamical system

Nouns are used to represent everyday objects. For example, if one observes an assembled piece of metal supported by four wheels travelling on the highway, the word car comes to mind. Qualitative description allows us to communicate and describe everyday objects without having to spend too much brainpower. Although being extremely useful, qualitative representation does not take into consideration the full complexity of our surroundings. Quantification becomes necessary when specific details are required. A set of physical values such as dimension, temperature, mass are assigned to bring further clarification. For the most part, the quantified values assign to an object remain constant over time. However, special kinds of objects require a different form of quantification to represent their full complexity. Specifically, the quantified values used as descriptions do not remain constant over time. For instance, the precise dimension of a flower can not take a specific value since it is slowly evolving at any moment in time. It requires a function to describe how the dimension evolves. Namely, the flower is what we referred to as a dynamical system. Many other examples exist in nature, giving rise to fascinating behaviour.

Everyday examples of dynamical systems:

- Swinging Pendulum (Constant periodic change between potential and kinetic energy)

- Population Growth (Constant periodic increase and decrease of the population as a result of predator/prey interactions and reproduction)

- The Human Brain (A constant periodic change of chemical signalling concentration across neuronal cells)

2. Periodic Force

Forces induce change. It is often referred to as a transfer of energy upon which a given object changes its natural state. For example, a cup of tea sitting at a precise position on a table will change if a small force is applied toward a given direction. A force becomes periodic when the applied stimulus is not constant over time. For example, pushing a child sitting on a swinging pendulum every time a maximum height is reached is considered periodic since the external action only occurs at specific moments in time. It is such periodic external actions that induce a change on a system or object that is referred to as periodic forces.

Everyday examples of periodic forces or stimuli:

- Human senses - Light waves interacting with photoreceptors of the eyes and allowing vision. Sound waves stimulating the eardrums and making music enjoyable.

- Seasons - Periodic alteration of environmental conditions such as temperature that affect biological ecosystems.

- Alternating Current - Periodic input signal altering the internal dynamics of an electronic circuit.

Having now gained a brief intuition about the principle of dynamical systems and periodic perturbations, we are forced to wonder about the possible responses a dynamical system can exhibit due to a periodic stimulus. For example, what would happen to the child sitting on a swinging pendulum if the amplitude of the external push keeps increasing to infinity while the applied frequency matches perfectly with the natural swinging frequency of the pendulum? Doubtlessly, nature is capable of shocking responses!

3. Resonance

Resonance occurs when the amplitude of a dynamical system is driven to a maximum value due to an applied periodic force. For example, the amplitude of a swinging pendulum increases dramatically even when a minimal force is applied at the right moment in time. For instance, finding ourselves sitting on a swinging pendulum with a child applying a minimal push would nevertheless increase the swinging amplitude to a maximum if, and only if the child applies the external push periodically when the potential energy of the pendulum is at a maximum. Under such circumstances, the external actions of the child are referred to as inducing resonance on the swinging pendulum.

After all, why should we care about resonance phenomena?

Surprisingly, our life depends on resonance in many aspects:

• Telecommunication - Radio communication occurs by a resonant action of an electromagnetic signal acting on electronic circuits.

• Nuclear Magnetic Resonance - The resonant action of an electromagnetic field on the atomic nucleus allows to gain physical information about various chemicals and biological materials.

• Tacoma Narrow Bridge disaster - Ignoring the power of resonance can lead to unexpected failures. For example, a small wind perturbation led to the collapse of the Tacoma Narrow Bridge. Note that it is still debated whether resonance served as the underlying cause of the disaster. Regardless, the Tacoma Bridge disaster demonstrates the often surprising response nature can exhibit due to small periodic perturbations.

• Chladni Plate Experiment - The induced action of sounds on a vibrating plate having sand creating beautiful geometrical patterns based on the vibrating frequency. The experiment brings to life the geometrical patterns associated with the vibration nature is capable of.

• Wine glass breaking by the action of sound - The experiment demonstrates the ability of sounds waves to induced resonance on a wine glass and consequently leading to shattering of the glass.

• The importance of Resonance in Music - A very resourceful lecture given by Dr. Walter Lewin from MIT demonstrates the action of resonance in music.

How can we observe resonance experimentally?

Among many methods available to investigate resonance, electronic circuits arise as an efficient experimental model since the dynamical properties of the circuit can easily be modified by changing the electronic properties of the components. In 1983, Leon O. Chua proposed one of the most simple circuit configurations exhibiting chaotic behaviour. In other words, the Chua's Circuit exhibits complex non-linear behaviour and arises as a perfect experimental system to gain further insights into non-linear resonance.

The present project explores non-linear resonance in the Chua's Circuit subjected to a wide range of input signals. Specifically, what are the possible external signals capable of inducing a maximum amplitude response in the Chua's Circuit system? The path to the answer is divided into three sections:

1) First, the theoretical principles underlying various forms of resonances will be reviewed theoretically. Mathematica software will be used as a computational environment allowing to reach theoretical intuition.

2) After clarifying the theoretical parameters of interest for the interpretation of resonance phenomenon, the Chua's Circuit will be explored through circuit simulation using LTspice.

3) Lastly, experimental measurements will be gathered by designing and manufacturing a PCB board following Chua's design. The design process will be undertaken using Eagle software. The production of the circuit will rely on JLC PCB as a manufacturing house. Once assembled, a signal generator will be used to subject the circuit to a wide range of input signals, where the output response will be measured using an oscilloscope.

Overall, the present project guides young students or curious individuals to learn more about resonance phenomena using the Chua's circuit as an experimental system. Additionally, the process of designing the Chua's circuit can become resourceful for individuals wanting to explore chaotic behaviour in electronic circuits. Pieces of literature and tutorials will be provided at each experimental step if one requires further information about the procedures followed and the underlying theories.

Tools and Components Utilized for the Experiment:

• Eagle PCB Design Software

• SPICE for Circuit Simulation

• Oscilloscope - The SDS 1052DL+ from Siglent is utilized for the experiment, but other methods of measurement can also be used.

• Signal Generator - The DG4102 from Rigol is used for the experiment, but any cheaper alternative will work.

• Mathematica - Any other computational environment such as Matlab can be used.

• PCB Manufacturer - PCB boards are ordered through JLC PCB, but any other company can be used. Feel free to shop around.

• Analog Oscilloscope can be useful in order to gain further insight into the chaotic behaviour of the circuit.

Electronics Components:

• OpAmps TL082 (2 OpAmps / IC) or TL084 (4 OpAmps / IC)
• Resistors 22Kohms - 220Ohms - 2.2Kohms - 3.3Kohms - 1Kohms - 100Ohms
• Capacitors 0.1uF - 0.01uF
• Inductor 18mH (Bourns) or 18mH (Fastron)
• Potentiometers 2Kohms or 2.5Kohms
• 9V Battery Connector
• Sliding Switch
• Power Jack Connector
• BNC Connectors

Objectives:

1) What are the known forms of non-linear resonances?

2) What are the parameters used to describe and interpret resonance in the Chua's Circuit?

Establishing the Computational Environment

1) Gain access to Mathematica --> Mathematica 15 days trial

2) After entering your information, an email will be sent providing you with an activation key. If you have not already downloaded Mathematica software on your computer, a download link will also be provided in the email. Alternatively, you can utilize the cloud environment to compile the notebook.

3) Download the Mathematica notebook from the Github Project Repository (ResonanceTheory.nb). Note that the notebook serves as an experimental playground for theoretical ideas and therefore is in constant progress. As a result, the codes underlying each example might not always be optimal.

4) Proceed through the notebook exploring the quantitative interpretation of dynamical systems paired with various non-linear resonant responses. Feel free to experiment with various parameters to gain further insights.

*** Further information about the Wolfram language can be found in the Wolfram Documentation Center.

Discussion & Conclusion from the Analysis

Having now gained some theoretical intuitions in Mathematica, let us reach some preliminary conclusions.

1) Known Types of Resonances

- Stochastic Resonance

NOISE-INDUCED PHENOMENA

QUANTIFICATION METHODS: FREQUENCY RESPONSE (Q), MEAN RESIDENCE TIME, SWITCHING RATE, SIGNAL-TO-NOISE RATIO (SNR)

Noise is intuitively assumed to distract and interfere when precision is required. However, stochastic resonance proves the contrary. Provided a bistable system subjected to an external periodic force of frequency W and a minimal amplitude A, the system will fail to alternate between one stable equilibrium and a lack of external energy. Yet, when noise is added to the external periodic force, the signal-to-noise ratio measured at the input frequency W becomes maximized. As a result, the system jumps periodically from one potential equilibrium to another. This is such noise action that is referred to as stochastic resonance.

- Vibrational Resonance

HIGH-FREQUENCY FORCE PHENOMENA

QUANTIFICATION --> FREQUENCY-RESPONSE (Q), MEAN WAITING TIME (Tmw)

As opposed to stochastic resonance, vibrational resonance does not require a bistable system to induce a maximum response. In other words, vibrational resonance can be induced in dynamical systems having possible equilibrium states higher than two. The noise term found in stochastic resonance is replaced by a high-frequency force that allows the system to transition from stability and adventure widely in the phase space. Such transition induced by a high-frequency force is often associated with a maximum response by excitable systems such as biological tissues.

- Ghost Resonance

UNEXPECTED FREQUENCY PHENOMENON

QUANTIFICATION --> FREQUENCY-RESPONSE (Q), MEAN WAITING TIME (Tmw).

What if the principles underlying stochastic and vibrational resonance are combined? Surprisingly, combining two periodic forces paired with a noise threshold induces a maximum response at a frequency missing from the input driving force. Namely, the output signal is amplified at a frequency that is absent from the input. Thus, a ghost frequency response can be triggered by either a noise intensity (ghost-stochastic resonance) or a high-frequency force (ghost-vibrational resonance).

- Coherence Resonance

LONELY NOISE PHENOMENON

QUANTIFICATION --> POWER SPECTRUM, TIME CORRELATION FUNCTION, TIME BETWEEN PULSES (Tp), EXCURSION TIME (Te), ACTIVATION TIME (Ta)

Coherence resonance share similarity with stochastic resonance. The addition of a small noise intensity reaching a specific threshold induces a maximum amplitude response by the system. Contrarily to stochastic resonance, coherence resonance does not require a periodic force. The noise only is capable of inducing resonance on dynamical systems.

- Autoresonance

PHASE-LOCKED PHENOMENON

QUANTIFICATION: FREQUENCY SWEEP-RATE / CHIRP RATE

Autoresonance arises from an external driving force having a frequency that is changing over time. Specifically, the driven system continuously adjusts its oscillation amplitude so that the period of oscillation stays "phased locked" with the period of the driving force. As a result, the system is forced to reach extremely high amplitude. Surprisingly, the phase-locking mechanisms underlying autoresonance occurs even when the parameters of the systems are varied. In fact, parametric autoresonance can also act on one or more system parameters only. The behaviour of a system under autoresonance becomes therefore dependent on the external driving force. One may wonder about the possibility of observing phase-locking phenomena between systems located at large distances.

- Parametric Resonance

PERIODIC CHANGE OF SYSTEM'S PARAMETERS

QUANTIFICATION: POWER SPECTRUM, FREQUENCY RESPONSE (Q)

The characteristic of parametric resonance is fundamentally different from other resonances where an external driving force is necessary. A maximum amplitude response arises when one or more system parameters change periodically over time for parametric resonance. For instance, a system with a natural frequency exhibiting periodic change will have a maximum amplitude response when the rate of change reaches a specific threshold. Parametric resonance can also occur from the periodicity of other parameters such as stiffness, damping, density, etc.

*** Note that the provided examples are only a small portion of the possible ways resonance can occur. Additionally, the examples do not adopt a quantum physics interpretation of oscillator systems, making possible many surprising phenomena. For the curious wanting to learn more, a book providing in-depth information regarding the theory and a wide range of experimental observations can be found HERE.

2) Parameters of Interest

The parameters of interest to interpret resonance in Chua's Circuit system can be divided into two categories. A set of parameters describing the circuit's behaviour (system) and parameters describing the input signal (force).

Systems Parameters:

The parameters describing the state of the circuit over time can be further categorized based on three criteria found to be necessary for chaos to occur in electronic circuits. * :

1) One or more non-linear elements --> Chua's Diode composed of:

- Operational Amplifiers

The values of parameters M0 and M1 found in the provided equations arise from different design configurations and become critical to describe the non-linear property of the circuit.

2) One or more locally active resistors (Negative resistance within Chua's Diode sub-circuit):

- Resistors R1, R2, R3, R4, R5, R6

3) Three or more energy storage elements:

Inductor L1

Capacitor C1

Capacitor C2

Taken together, the dynamical state of the circuit becomes describable with only three variables:

- Voltage Across Capacitor C1 = (V1) or (X)

- Voltage Across Capacitor C2 = (V1) or (Y)

- Current Across Inductor L1 = (I) or (Z)

The time evolution of the circuit is described by three ordinary differential equations found in the provided visuals. In addition, the non-linear property of the resistor is mathematically represented by the function f(V1), which can arise with various levels of complexity based on the chosen circuit configuration.

*** In dept theoretical discussion paired with experiments demonstrating the evolution from linear RLC circuit to non-linear Chua's Circuit capable of chaotic behaviour can be found HERE & HERE. Another useful resource providing theoretical insights regarding the implementation of the Chua's Circuit can be found in the following BOOK.

Input Signal Parameters

Having now identified the system parameters describing the dynamical behaviour of the Chua's circuit and the known type of resonances, let us clarify the parameters to be considered when selecting the property of the input signals. As you have probably deduced by now, the range of input parameters become categorized based on the various types of resonances.

Stochastic Resonance--> Periodic force paired with noise.

- Amplitude (A1)

- Frequency (f1)

- Noise Variance / Intensity (D)

** Additional parameters can be necessary to represent complex noise functions.

Vibrational Resonance --> Biharmonic force is composed of a low-frequency and high-frequency force.

- Amplitude (A1)

- Frequency (f1) where (f1 < f2)

- Amplitude (A2)

- Frequency (f2) where (f2 > f1)

Ghost Resonance --> The sum of periodic forces paired with a noise term.

- Amplitude (An) - where n depends on the number of forces.

- Frequency (fn) - where n depends on the number of forces.

- Noise Variance / Intensity (D)

** The external signals inducing Ghost Resonance can arise with various levels of complexity. Additional parameters can be necessary.

Coherence Resonance --> Noise term only.

- Noise variance / Intensity (D)

** Noise can be induced with various levels of complexity and therefore might require additional parameters.

Autoresonance --> The frequency of the driving force changes over time.

- Amplitude (A1)

- Time-Dependent Frequency (f(t))

** The evolution of the force-frequency can arise in various forms and will likely require additional parameters.

Parametric Resonance --> Absence of external perturbations.

The parameters of interest describe the time evolution of the system's parameters rather than external perturbations.

- Time-Evolution of natural frequency (w(t))

- Time-Evolution of sampling (d(t))

*** The wide range of possible complexities dynamical systems can exhibit might therefore require additional parameters.

Objectives:

1) What input signal parameters can induce vibrational resonance in Chua's Circuit system?

2) What signal analysis techniques are required to interpret the response of the circuit?

*** The present section does not serve as an LTSPICE tutorial and will focus primarily on describing the procedures followed during the project. However, links to tutorials and resources are provided below if one wishes to gain further insights into the features offered by LTSPICE.

LTSPICE Ressources:

LTSpice Download Link

Importing Models

Stepping Parameters during Simulation

User-Defined Input Function

Importing / Exporting Results

Output Analysis with Waveform Viewer

LTSpice Forum

Producing the Simulation NETLIST / Circuit Diagram

Luckily, LTSpice comes with a friendly user interface that allows producing a schematic with conventional electronic components without having to represent the circuit through computer code:

1) Gather the necessary components within the circuit diagram work-space:

- Two 22Kohms resistors.

- Two 220ohms resistors.

- One 3.3Kohms resistor.

- One 2.2Kohms resistor.

- One 2.0K resistor. (Potentiometer)

- One 10nF capacitor.

- One 100nF capacitor.

- One 18mH inductor.

- Two voltage inputs.

- One Ground.

- OpAmps: Since the TL082 is not available within the default libraries available in LTSPICE, the LT1057 or the LT1113 can be used to simulate the circuit. Alternatively, the SPICE model of the TL082 can be downloaded and imported within the design environment. Further details about the procedures to follow in order import models can be found HERE & HERE. Also, an already made design of the Chua's Circuit using the TL082 can be downloaded from HERE (Very insightful resources).

2) Connect each component by following the provided Chua's design. Also, ensure to assign the proper values to each component.

3) Since the parameters of interest representing the state of the circuit are the voltage across capacitor C1, capacitor C2 and the current across the inductor L, voltage probes are added accordingly.

4) Let us now explore the effect of varying the value of resistor R7 located between capacitor C1 and capacitor C2. Once a specific resistance threshold value is reached, the circuit exhibit chaotic behaviour. To observe the double scroll chaotic signature, the voltage across capacitor C1 (V(vc1) probe) must be plotted on the horizontal axis and the voltage across capacitor C2 (V(vc2)) on the vertical axis.

The simulation parameters are set as follow:

- Type: Transient

- Stop Time: 0.10

- Time to start saving data: 0

- Maximum Timestep: 1E-5

The chaotic signatures obtained from the simulation confirm that the netlist produced adheres to Chua's Circuit properties. The resistance value from R7 inducing optimal Chaotic behaviour appeared to be centred around 1.5KOhms. However, the system's dynamic changes drastically at resistance values below 1.485 KOhms and above 1.965 KOhms. Although the objective of the present simulation is not to investigate chaos, feel free to explore how the resistance value of R7 affects the chaotic behaviour of the circuit.

Observing Resonance in LTSPICE

The following procedures will primarily focus on exploring vibrational resonance to minimize the length of the project. However, the other types of resonances can also be investigated by adjusting the input signal parameters. (A1, f1, A2, f2)

Vibrational Resonance (HIGH-FREQUENCY FORCE)

The action of vibrational resonance is an enhanced amplitude of the output signal found at a frequency matching the external frequency (f1) of the low-frequency driving force (F1) by the addition of a high-frequency (f2) periodic force (F2).

Simulation Parameters:

- Type: Transient

- Stop Time: 100ms

- Start Time: 50ms

- Time Step Max: 1*10^-5

System's Parameters:

The system's parameters selected for the simulation can be derived from the provided simulation schematic.

Input Parameters:

FORCE 1 (F1)

- Amplitude (A1) = (0.3 V, 0.8 V, 1.3 V, 1.8V)

- Frequency (f1) must be <(f2) = (50 Hz, 100 Hz, 150 Hz, 200 Hz)

FORCE 2 (F2)

- Amplitude (A2) = (0.5 V, 1.5 V, 2.5 V, 3.5 V)

- Frequency (f2) must be >(f1) = (500 Hz, 1500 Hz, 2500 Hz, 3500 Hz)

Expected Outcomes / Hypothesis:

- Under Fast Fourier Transform analysis (FFT), the amplitude of the output signal at frequency matching (f1) will be maximized when the parameters (A2) and (f2) reach specific values.

- Under amplitude response analysis, the amplitude of the output signal will reach a maximum only when parameters (A1), (A2), (f1), (f2) reach specific threshold values.

- Under time series analysis, the voltage state across capacitor 1 (V1) will periodically switch between (V1 < 0) and (V1 > 0) with a period equal to half the period of frequency (f1) once parameters (A2) and (f2) reach specific values.

Observations:

Fast Fourier Transform (FFT) Analysis:

1) As parameter (f2) increases from 500 Hz to 3500 Hz incremented by 1000 Hz, the amplitude of the output signal (V1) at (f1 = 100Hz) reaches a maximum when (f2) is equal to 1500 Hz.

2) As parameter (A2) increases from 0.5 V to 3.5 V incremented by 1.0 V, the amplitude of the output signal (V1) at (f1 = 100 Hz) reaches a maximum when (A2) is equal to 0.5 V.

3) As parameter (f1) increases from 50 Hz to 200 Hz incremented by 50 Hz, the amplitude of the output signal (V1) at (f1) reaches a maximum when (f1) reaches 200 Hz.

4) As parameter (A1) increases from 0.3 V to 1.8 V incremented by 0.5V, the amplitude of the output signal (V1) at (f1 = 200 Hz) reaches a maximum when (A1) is equal to 1.8 V.

Amplitude Response (Q) Analysis:

- Gather FFT data where the amplitude is reported in dBV.

- Convert the amplitude from dBV to V. (V = 10exp(dBV / 20))

- Calculate the Amplitude Response (Q) where Q = (Output Amplitude (V) / Input Amplitude (A1))

1) The results obtained from the first analysis correlate with the data obtained during the simulation. Specifically, the Response Amplitude Q reaches a maximum threshold when the high-frequency force value (f2) is around 1500 Hz.

2) Similarly, the result obtained during the second amplitude response analysis shows the Amplitude Response Q increasing as the amplitude of the high-frequency force (A2) decreases.

*** The Excel sheet used to gather the simulation results from LTSpice and produce the Amplitude Response graph can be downloaded HERE. The same analysis procedures can be followed to gain further insights into the optimal values for the other parameters.

Time Series Analysis:

As the amplitude of the high-frequency force (A2) is increased from 0.00V to 1.50V, the system undergoes a periodic switching from V1 < 0 and V1 > 0. Calculating the mean residence time for V1 = + and V1 = - demonstrate that once (A2) reaches a critical value, MRT+ becomes equal to MRT-. Furthermore, at such a critical value of (A2), the periodic switching between V1>0 and V1<0 becomes half the period of the low-frequency force (Periodic Switching = 1/2*(1/f1)).

*** Further information into the time-series signature of vibrational resonance can be gained HERE.

Objectives:

1) Designing the Chua's Circuit PCB using Eagle.

2) Providing the manufacturing files to JLC PCB for production and order the required components from Mouser.

3) Assembling the circuit.

Eagle General Tutorial Links:

For further insights into the features of Eagle, feel free to explore the tutorial links below.

• Getting Started Part 1
• Getting Started Part 2
• Circuit Simulation

Circuit Schematic in Eagle

1) Create a new folder under the projects section of Eagle's main page.

2) Create a schematic design file by right-clicking the project folder and selecting NEW --> SCHEMATIC.

3) The first task when creating a new schematic is to import a frame inside the design workspace. Select the ADD-PART button and search for the frame library. (FRAME_A_L with dimension 8 1/2 * 11 INCH, Landscape) was selected for the project.

4) The required electronic components must now be selected from available libraries and imported into the design workspace. Select the ADD-PART button and gather the components to reproduce Chua's Circuit design. A list of all the components used during the project can be accessed HERE.

*** Note that different components from other libraries could have also been used. Since we are not doing any simulation inside eagle, the critical criteria when selecting your components is to ensure the footprint and package type match the components ordered from electronic suppliers. Failing to do soo will lead to difficulties when soldering the chips onto the PCB board since the soldering pin layout will not match those components.

5) Assign names and values to all components.

6) Proceed to create the circuit schematic by connecting each component following Chua's Circuit design.

PCB Routing and Manufacturing

1) Once the components have been connected and the circuit diagram fully completed, click the GENERATE/SWITCH TO BOARD button to generate a PCB layout workspace.

2) To control board layout and dimension, Fusion 360 is used for the design process.

3) Once the PCB layout is completed in Fusion 360, it is imported into Eagle's PCB layout workspace by clicking the FUSION 360 vertical button on the right. Further insights into the procedure required to import the PCB from Fusion 360 can be found HERE.

4) Having all the components and the desired PCB layout within the Eagle environment, we must place each component onto the PCB board. To ease the routing process, ensure the location of each component follows the logic of the schematic.

5) Each air wire connecting the components must be routed onto the surface of the PCB. You can either manually route each connection or utilize the auto-router option to have EAGLE automatically create all your connections. Given the simplicity of the schematic, manual routing was used to ensure all pins were routed correctly. In addition, a polygon was created to provide collective ground to all components. Further information on routing methods as well as creating the polygon can be found HERE. Note that before proceeding to the next section, one can delete or add text on the PCB board for descriptive purposes.

6) Once all routing connections have been completed, we review all design parameters to ensure the PCB layout produces the intended output. Also, the manufacturing parameters must be adjusted to comply with the requirements of the manufacturing house. All details about the procedures of finalizing the design parameters and producing the required Gerber file to send for manufacturing can be found HERE.

7) Once the Gerber manufacturing ZIP file has been generated, a manufacturing house must be selected to order the board. More details about the manufacturing options to take into consideration can be found HERE. For the current project, the Gerber files were uploaded to the JLC PCB ordering page.

Assembling the Circuit

- Given the size of the components, a microscope is necessary to solder the components appropriately onto the PCB. Alternatively, you can select the SMT assembly option when ordering your PCB from JLC PCB and have them assemble all the components for you. Further details can be found HERE.

- The + and - pins of the operational amplifier IC are not easily noticeable. Therefore, ensure to solder the IC with the reference line-oriented horizontally on the opposite side of the potentiometer.

- When soldering the potentiometer, utilizing the left and middle pin will increase the resistance when the knob is rotated clockwise. Contrarily, using the right and middle pin will increase the resistance when the knob is counter-clockwise.

*** Since the present research project is in constant progress, different circuit configurations will be made available in the future on the Github Project Repository.

Objectives:

1) What resistance value is required across the potentiometer to observe chaotic behaviour in the circuit?

2) How does the amplitude value (A2) of the high-frequency force (F2) affect the time-series evolution of the circuit under vibrational resonance?

Observing Chaos:

1) Connect channel #1 of the oscilloscope to output 1 of the circuit, reporting the voltage across capacitor C1.

2) Connect channel #2 of the oscilloscope to output 2 of the circuit, reporting the voltage across capacitor C2.

3) Connect the two 9V batteries to their respective battery jacks.

4) Adjust the oscilloscope settings to X-Y mode to plot the voltage across capacitor C1 on the X-axis and the voltage across capacitor C2.

5) Adjust the scale of each axis until the measurement resolution is appropriate.

6) Slowly turn the knob of the potentiometer to find the appropriate resistance value for chaotic behaviour to occur. This process can be extremely tedious if you have any unstable connections or improper potentiometer resistance.

7) Although not the most precise method, a multimeter can measure the resistance value induced by the potentiometer.

Chaos Measurement #1:

Resistance Value (R) = 1.194 KOhm

Horizontal Axis (V1) = 100 mV/div

Vertical Axis (V2) = 500 mV/div

Chaos Measurement #2:

Resistance Value (R) = 0.750 KOhm

Horizontal Axis (V1) = 50 mV/div

Vertical Axis (V2) = 500 mV/div

Chaos Measurement #3:

Resistance Value (R) = 1.240 KOhm

Horizontal Axis (V1) = 100 mV/div

Vertical Axis (V2) = 500 mV/div

Chaos Measurement #4:

Resistance Value (R) = 1.220 KOhm

Horizontal Axis (V1) = 200 mV/div

Vertical Axis (V2) = 1.00 V/div

Observation & Discussion

The state-space representation of the circuit produced by the oscilloscope in X-Y mode allows analyzing the dynamical behaviour of the circuit by plotting the voltage across capacitor C1 on the X-axis and the voltage across capacitor C2 on the Y-axis. You can imagine a small point within the X-Y plot of the oscilloscope as the state of the circuit at an arbitrary time T. As time evolves, the collection of state measurements (points) gives rise to a geometrical representation where the state evolution of the system produce geometrical patterns. For example, from measurement #1 with a resistance value around 1.194 KOhm, we observe the state of the system orbiting around an equilibrium point through a total of four different orbits. The number of orbits increases as the resistance value across the potentiometers approaches the threshold required to induce chaos in the circuit, which is found to be around 1.220 KOhm. As observed in measurement #4, the system's state under chaotic behaviour can jump from positive orbits to negative orbits. Theoretically, a specific set of parameters values where the dynamical evolution of the state from one attractor to another becomes completely "random." Surprisingly, such chaotic property is useful for many applications such as cryptography. Further theoretical insights into the evolution of chaos in the Chua's Circuit can be found HERE & HERE.

The size difference between the two scroll attractors is due to input power asymmetry. Specifically, asymmetry arises when there is a small voltage difference between the batteries positive and negative input sources. For instance, the negative power provided by one battery might be 0.7 V less in magnitude than the positive input. Therefore, a power supply can be utilized to compensate for the power difference by applying 9V and -9.7V through the input of the operational amplifiers.

Additionally, the type of diode utilized in the circuit is critical for chaos to occur. A wide range of diodes led to the inability to observe any chaotic behaviour in the circuit. The 18mH 5% inductor from Fastron was found to allow the best results.

Vibrational Resonance Experiment:

1) Connect channel #1 of the signal generator into the input jack #1 of the circuit.

2) Connect channel #2 of the signal generator into the input jack #2 of the circuit.

3) Connect channel #1 of the oscilloscope into output #1 of the circuit, reporting the voltage across capacitor C1.

4) Connect channel #2 of the oscilloscope into output #2 of the circuit, reporting the voltage across capacitor C2.

5) Ensure to firmly mount the circuit to avoid any loose connections during the measurement process.

6) Connect the two 9V batteries to their respective battery jack.

7) Set the oscilloscope to time series more and briefly adjust the scale of each axis as well as the sampling rate until you can observe a clear sine wave for both channel #1 and channel #2.

8) For time series analysis, we want to measure one channel output at a time to maximize clarity. Therefore, turn off the channel #1 display on the oscilloscope.

9) Adjust the resistance value of the potentiometer to 1.350 KOhm.

*** To minimize the length of the project, only the effect of the amplitude (A2) of the high-frequency force (F2) will be explored. Also, only a time series analysis of the output signal will be presented. Adjusting the oscilloscope mode to FFT could allow exploring the frequency component of the output signal. Feel free to experiment with different input parameters and measurement methods.

Resonance Measurement [#1 - #8]

Resistance Value (R): R = 1.350 KOhm

Input Signal Parameters:

CH1 (Force #1) Amplitude = 0.3 V

CH1 (Force #1) Frequency = 50 Hz

CH2 (Force #2) Amplitude = [0.0 V - 3.5 V - 4.5 V - 5.5 V - 6.5 V - 7.5 V - 8.5 V ]

CH2 (Force #2) Frequency = 500 Hz

Observation & Discussion:

Once a threshold value of (A2) is reached, the system can jump from positive values to negative values. The time spent by the system in the positive and negative regions is referred to as the residence time (T+, T-). Taking all the residence times over a period of time allows us to calculate the mean residence time (Tmr+, Tmr-). In measurement number #3, the mean residence time between positive (T+) and negative (T-) differ drastically. The system is found to output negative values during a short period of time compared with the time spent in the positive region. As the input amplitude (A2) increases from 4.5 V in measurement #3 up to an amplitude (A2) of 8.5 V in measurement #8, the difference between the positive mean residence time (Tmr+) and the negative mean residence time (Tmr-) is found to decrease. For a specific threshold value of amplitude (A2_max), the mean residence time between positive and negative values becomes equal. Further analysis could demonstrate that the periodic switching from positive region to negative region at amplitude value equal to (A2_max) occurs with a period equal to half the period of the low-frequency input (f1). It is at such an input threshold that vibrational resonance occurs. Given the parameters selected for the experiment, A2_max is found to be around 8.5V. The relatively high voltage threshold compared with the results obtained during the simulation is likely due to the resistance value (R) set at 1.350 KOhm during the experiment. Furthermore, an amplitude response analysis could have been calculated to find the specific value of A2_max for a range of values close to 8.5V.

*** Note that the present research project is still ongoing, and future updates regarding experiments will be added on the Github Project Page and Research Gate.

Summary

Together, the present project lays the groundwork for future experiments investigating non-linear resonance using Chua's Circuit as an experimental system. Here's below a summary of the preliminary conclusions reached during the project:

Theory

• A total of six different types of resonances were reviewed:

- Stochastic Resonance (Periodic force paired with noise)

- Vibrational Resonance (High-frequency biharmonic force)

- Coherence Resonance (Noise Only)

- Ghost Resonance (Biharmonic force paired with noise leading to unexpected output frequency)

- Parametric Resonance (Periodic system's parameters)

- Auto-Resonance (Phase-locking with the external force)

• The time evolution of the Chua's Circuit was modelled with three parameters of interest and three ordinary differential equations:

- Voltage across capacitor 1 (V1)

- Voltage across capacitor 2 (V2)

- Current across the inductor (L)

Simulation

• For the simulation parameters selected, specific input values were found to induce a maximum amplitude response by the circuit under vibrational resonance:

- Force 1 Amplitude (A1): 1.8 V

- Force 2 Amplitude (A2): 0.5 V

- Force 1 Frequency (f1): 200 Hz

- Force 2 Frequency (f2): 1500 Hz

• Three types of signal analysis methods were reported to interpret the resonant response of the circuit:

- Fast Fourier Transform (FFT) Analysis: Allowing to decompose the output signal into its frequency components.

- Amplitude Response Analysis (Q): Allowing to derive the maximum amplitude value for a range of input parameters.

- Time-series Analysis: Allowing to observe the evolution of the system over time.

Hardware Design

• The following electronics components properties were selected:

Capacitor C1: 100 nF

Capacitor C2: 10 nF

Inductor L: 18 mH

Variable Resistor R: 2 KOhm

Resistor R2: 220 Ohms

Resistor R3: 220 Ohms

Resistor R4: 2.2 KOhm

Resistor R5: 22 KOhm

Resistor R6: 22 KOhm

Resistor R7: 3.3 KOhm

Operational Amplifier: TL082

Experiments

• Under the selected experimental parameters, chaotic behaviour by the circuit was found to occur at a specific resistance value of:

- Resistance Value (R) = 1.220 KOhm

• Under vibrational resonance, a periodic switching between positive and negative values was found to be induced by a threshold amplitude value of the high-frequency force:

- Force 2 Amplitude (A2) = 7.5 V (+- 1.0V) (Approximate Solution)

Future Directions

• Exploring other types of resonances in the Chua's Circuit by exploring various input parameters. For example, can other types of non-linear resonance be observed in the Chua's Circuit system?

• Investigating the propagation of resonant input signals between multiple Chua's Circuit coupled following different network topologies. How would the state evolution of each coupled system differ over time? For example, would the maximum response amplitude by the circuit subjected to the input signal indirectly induce the other systems in the network?

• Exploring resonance in biological systems. Given the non-linear properties of biological organisms, can similarity be found between the resonant response of the Chua's circuit and the response of living systems to external stimulus? Can the principles underlying non-linear resonance allow us to gain further insights into the physical mechanisms allowing long-range communication between living organisms?

• Turning non-linear resonance into practicality. How can non-linear resonance be used in future design to solve practical problems?

"If you want to find the secrets of the universe, think in terms of energy, frequency, and vibrations." - Nikola Tesla.