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Second Order systems II

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.

Second-Order Circuits

Second-Order Circuits

Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Design Example: Underdamped Parallel RLC Circuit

Design Example: Underdamped Parallel RLC Circuit

Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
Starting with a fixed...

RLC Circuit as a Damped Oscillator

RLC Circuit as a Damped Oscillator

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...

Types of Damping

Types of Damping

If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...

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Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns

Max L Trostel¹, Moses Z R Misplon¹, Andrés Aragoneses¹

¹Department of Physics and Astronomy, Carleton College, Northfield, MN 55057, USA.

Entropy (Basel, Switzerland)

|December 3, 2020

Summary

We analyzed the complex dynamics of the driven double-well Duffing oscillator using ordinal pattern analysis. This method reveals hidden dynamical regimes within chaos, offering new insights into classical-to-quantum transitions.

Keywords:

Duffing oscillator chaos complex dynamics permutation entropy semiclassical transition

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Area of Science:

Nonlinear Dynamics
Chaos Theory
Quantum Mechanics

Background:

The driven double-well Duffing oscillator is a fundamental model exhibiting complex dynamics, including chaos.
It is relevant for understanding the transition from classical to quantum chaos.
Traditional statistical tools have limitations in characterizing all dynamical regimes.

Purpose of the Study:

To explore the complexity of Duffing oscillator dynamics in classical and semi-classical regimes.
To apply ordinal pattern analysis for a deeper understanding of chaotic behavior.
To identify novel dynamical regimes not detectable by conventional methods.

Main Methods:

Utilized ordinal pattern analysis to probe system dynamics.
Investigated the driven double-well Duffing oscillator across classical and semi-classical regimes.
Analyzed hierarchies and probabilities of ordinal patterns.

Main Results:

Unveiled distinct dynamical regimes within the chaotic range.
Characterized these regimes by unique ordinal pattern hierarchies and probabilities.
Revealed a correlation between Lyapunov exponent and permutation entropy.

Conclusions:

Ordinal pattern analysis provides a powerful tool for uncovering hidden dynamics in chaotic systems.
Dips in the Lyapunov exponent signify transitions between different dynamical regimes.
This research offers valuable insights for experiments in the semi-classical regime and quantum chaos.