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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”

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