What is Discrete Compounding?

Composite discrete chaotic system is a special type of chaotic system. It consists of two (or more) discrete chaotic systems through certain rules. Compared with discrete chaotic systems, the iterative process of composite discrete chaotic systems is not only sensitive to initial conditions. It also has the flexibility to select iterative functions according to double sequences, so the iterative process also has some randomness.

Chaos is a deterministic, random-like process that appears in nonlinear dynamical systems. This process is neither periodic nor convergent, and has a sensitive dependence on the initial value.

According to the nature of dynamical systems, chaos can be divided into four types:

1) time chaos;

2) space chaos;

3) space-time chaos;

4) Functional chaos;

Composite discrete chaotic system is a special type of chaotic system. It consists of two (or more) discrete chaotic systems through certain rules. Compared with discrete chaotic systems, the iterative process of composite discrete chaotic systems is not only sensitive to initial conditions. It also has the flexibility to select iterative functions according to double sequences, so the iterative process also has some randomness.

Assume

Are two discrete chaotic dynamic systems, for arbitrary sequences

,then

The composite discrete chaotic dynamic system of the two iterative systems under the sequence R is denoted as

Where R is called a compound sequence. For q = 0 or q = 1,

Called its subsystem.

The compound iterative systemic dynamic behavior is related to the compound sequence R. If i is sufficiently large,

As a constant, the composite iterative system degenerates into a single chaotic system. Generally, a composite iterative system maintains the chaotic characteristics of all sub-iterative systems, and its behavior is much more complicated than a single sub-iterative system ^[1]

Chaos applications can be divided into chaos synthesis and chaos analysis. The former uses artificially generated chaos to obtain possible functions from chaotic dynamic systems, such as associative memory of artificial neural networks, etc .; the latter analyzes chaotic signals obtained from complex artificial and natural systems and finds hidden deterministic rules, such as Non-linear deterministic prediction of time series data, etc.

The specific potential applications of chaos can be summarized as follows:

(1) Optimization: Use the randomness, ergodicity and regularity of chaotic motion to find the best advantages, which can be used in many aspects such as system identification and optimal parameter design.

(2) Neural network: Fusion of chaos and neural network, so that the neural network gradually degenerates from the original chaotic state to the general neural network. The dynamic characteristics of the chaotic state in the middle process are used to make the neural network escape from the local minimum, thereby ensuring that Global optimization, which can be used for associative memory, robot path planning, etc.

(3) Image data compression: Replace complex image data with a set of simple dynamic equations that can generate chaotic attractors, so that only the parameters of this set of dynamic equations need to be memorized and stored, and the data volume is larger than the original image data Reduced, thereby achieving image data compression.

(4) High-speed retrieval: The chaos ergodicity can be used for retrieval, that is, while changing the initial value, the data to be retrieved is compared with the value that has just entered the chaotic state, and the state close to the data to be retrieved is retrieved. This method has higher retrieval speed than random retrieval or genetic algorithm.

(5) Prediction of non-linear time series: Any time series can be regarded as an input-output system determined by non-linear mechanism. If the irregular motion phenomenon is a chaotic phenomenon, then the decision theory using chaos phenomenon is adopted. Non-linear techniques enable short-term predictions with high accuracy.

(6) Pattern recognition: Using the sensitivity of chaotic trajectories to initial conditions, it is possible to make the system recognize different patterns with only slight differences.

(7) Fault diagnosis: The fault diagnosis can be performed by comparing the set characteristics of the attractor composed of time series with the sampled time series data ^[3] .