Related Experiment Video
Updated: May 27, 2025
10:17
Patterning of Microorganisms and Microparticles through Sequential Capillarity-assisted Assembly
Published on: November 4, 2021
3.1K
Structure of random Turing-like patterns in discrete-time systems is determined by the initial conditions
Huimin Zhang1, Jian Gao1, Changgui Gu2
1Anqing Normal University, School of Mathematics and Physics, Anqing 246011, People's Republic of China.
Physical Review. E
|February 20, 2025
Summary
The structure of Turing-like patterns in discrete-time systems is determined by initial state ratios, not control parameters. Changing this ratio shifts patterns from spots to labyrinth and inverse spots.
Area of Science:
- Complex Systems
- Pattern Formation
- Non-linear Dynamics
Background:
- Spatiotemporal ordered structures, or patterns, emerge from complexity in diverse systems.
- Turing patterns, explained by activator-inhibitor mechanisms, are a key model for pattern formation.
- Turing-like patterns share structural similarities but arise from different mechanisms.
Purpose of the Study:
- To investigate the determining factors for Turing-like pattern structures in discrete-time systems.
- To identify a novel parameter controlling pattern transitions.
- To elucidate the underlying mechanism of pattern formation in these systems.
Main Methods:
- Analysis of discrete-time systems exhibiting pattern formation.
- Systematic variation of initial condition state ratios.
- Introduction and application of a quantitative structure parameter.
Main Results:
- Turing-like pattern structure in discrete-time systems is solely determined by the ratio of initial states.
- Pattern transitions (spots to labyrinth to inverse spots) are induced by altering the initial state ratio.
- A direct proportionality was established between the structure parameter and the initial state ratio.
Conclusions:
- The structure of Turing-like patterns in discrete-time systems is controllable via initial conditions.
- Pattern transitions are linked to the traversability of multiperiodic states.
- This research offers new insights into discrete-time pattern formation mechanisms.
Related Concept Videos
01:21First Order Systems
82
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...
82
01:31Classification of Systems-II
133
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,
133
01:16Basic Discrete Time Signals
189
The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is...
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is...
189
01:23Linear time-invariant Systems
208
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...
208
01:20Discrete-Time Fourier Series
211
The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
For a discrete-time periodic signal x[n]...
211
01:24Convolution: Math, Graphics, and Discrete Signals
223
In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
223