Related Experiment Video
Updated: Jun 14, 2025
08:25
Continuous Measurement of Biological Noise in Escherichia Coli Using Time-lapse Microscopy
Published on: April 27, 2021
3.6K
Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems
Risong Li1, Yongjiang Li1, Tianxiu Lu2
1School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China.
Entropy (Basel, Switzerland)
|August 29, 2024
Summary
This study introduces G-scrambled pairs for time-varying discrete dynamical systems (T-VDDS), extending known concepts. It defines and analyzes generically G-chaotic T-VDDS, providing conditions for strong chaos.
Area of Science:
- Dynamical Systems
- Topology
- Metric Spaces
Background:
- Time-varying discrete dynamical systems (T-VDDS) are defined on complete metric spaces.
- The concept of scrambled pairs is a key aspect of chaos theory.
Purpose of the Study:
- To introduce and define G-scrambled pairs for T-VDDS.
- To study properties of G-scrambled pairs and define notions of generic G-chaotic T-VDDS.
- To establish conditions for generic strong G-chaotic T-VDDS.
Main Methods:
- Utilizing Furstenberg families (G) to define G-scrambled pairs.
- Analyzing the properties of the set of G-scrambled pairs within a T-VDDS.
- Defining and investigating generically G-chaotic and generically strongly G-chaotic T-VDDS.
Main Results:
- G-scrambled pairs are provided for T-VDDS, generalizing existing concepts.
- Properties of G-scrambled pairs are investigated.
- Definitions for generically G-chaotic and generically strongly G-chaotic T-VDDS are introduced.
Conclusions:
- The study provides a framework for understanding chaos in T-VDDS using G-scrambled pairs.
- Sufficient conditions for T-VDDS to be generically strongly G-chaotic are presented.
Related Concept Videos
01:24BIBO stability of continuous and discrete -time systems
372
System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system....
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system....
372
01:23Linear time-invariant Systems
229
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
229
01:10Entropy Change in Reversible Processes
2.5K
In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
2.5K
01:18Second Order systems II
96
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.
96
01:21First Order Systems
87
First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
87
01:31Classification of Systems-II
137
Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
137