On three dimensional fractal dynamics with fractional inputs and applications

Emile Franc Doungmo Goufo; Abdon Atangana; Emile Franc Doungmo Goufo; Abdon Atangana

doi:10.3934/math.2022114

AIMS Mathematics

2022, Volume 7, Issue 2: 1982-2000. doi: 10.3934/math.2022114

Previous Article Next Article

Research article Special Issues

On three dimensional fractal dynamics with fractional inputs and applications

Emile Franc Doungmo Goufo ^{1
,
,},
Abdon Atangana ²

1.
Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa
2.
Institute for Groundwater Studies, University of the Free State, Bloemfontein 9300, South Africa

Received: 19 August 2021 Accepted: 24 October 2021 Published: 05 November 2021
MSC : 65P20, 65P30, 28A80, 65J15, 26A33

The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.

Keywords:

Citation: Emile Franc Doungmo Goufo, Abdon Atangana. On three dimensional fractal dynamics with fractional inputs and applications[J]. AIMS Mathematics, 2022, 7(2): 1982-2000. doi: 10.3934/math.2022114

Related Papers:

[1]	Jian Gao, Hao Liu, Yang Zhang . Intelligent traffic safety cloud supervision system based on Internet of vehicles technology. Electronic Research Archive, 2023, 31(11): 6564-6584. doi: 10.3934/era.2023332
[2]	Mohd. Rehan Ghazi, N. S. Raghava . Securing cloud-enabled smart cities by detecting intrusion using spark-based stacking ensemble of machine learning algorithms. Electronic Research Archive, 2024, 32(2): 1268-1307. doi: 10.3934/era.2024060
[3]	Mohammed Alshehri . Blockchain-assisted cyber security in medical things using artificial intelligence. Electronic Research Archive, 2023, 31(2): 708-728. doi: 10.3934/era.2023035
[4]	Majed Alowaidi, Sunil Kumar Sharma, Abdullah AlEnizi, Shivam Bhardwaj . Integrating artificial intelligence in cyber security for cyber-physical systems. Electronic Research Archive, 2023, 31(4): 1876-1896. doi: 10.3934/era.2023097
[5]	Gang Cheng, Yijie He . Enhancing passenger comfort and operator efficiency through multi-objective bus timetable optimization. Electronic Research Archive, 2024, 32(1): 565-583. doi: 10.3934/era.2024028
[6]	Liling Huang, Yong Tan, Jinzhu Ye, Xu Guan . Coordinated location-allocation of cruise ship emergency supplies under public health emergencies. Electronic Research Archive, 2023, 31(4): 1804-1821. doi: 10.3934/era.2023093
[7]	Linsen Song, Yichen Du . A real-time pricing dynamic algorithm for a smart grid with multi-pricing and multiple energy generation. Electronic Research Archive, 2025, 33(5): 2989-3006. doi: 10.3934/era.2025131
[8]	Kun Han, Feng Jiang, Haiqi Zhu, Mengxuan Shao, Ruyu Yan . Learning cooperative strategies in StarCraft through role-based monotonic value function factorization. Electronic Research Archive, 2024, 32(2): 779-798. doi: 10.3934/era.2024037
[9]	Ming Wei, Congxin Yang, Bo Sun, Binbin Jing . A multi-objective optimization model for green demand responsive airport shuttle scheduling with a stop location problem. Electronic Research Archive, 2023, 31(10): 6363-6383. doi: 10.3934/era.2023322
[10]	Peiqun Lin, Chenxing He, Lingshu Zhong, Mingyang Pei, Chuhao Zhou, Yang Liu . Bus timetable optimization model in response to the diverse and uncertain requirements of passengers for travel comfort. Electronic Research Archive, 2023, 31(4): 2315-2336. doi: 10.3934/era.2023118

Abstract

1. Introduction

In the existing application domain of intelligent systems for multi-cloud resource scheduling in cyberspace, academia and industry are mainly concerned with efficiently realizing information processing and knowledge reasoning. However, the number of published system vulnerabilities is increasing ^[1], and cyberspace system failures and disruptions caused by exploited vulnerabilities may have catastrophic consequences for those affected. Once information processing and knowledge reasoning systems are exploited by attackers due to known or unknown vulnerabilities, the effectiveness of information processing and knowledge reasoning is greatly reduced. Therefore, we need an information processing and knowledge reasoning methodology that improves the efficiency of information processing and knowledge reasoning while guarding against a variety of known and unknown risks and vulnerabilities. Nazir et al. ^[2] have investigated a number of tools and techniques for mining system vulnerabilities. However, there is no method to mine and discover all unknown system risks and vulnerabilities in information processing and knowledge reasoning systems.

The theory of endogenous security in cyberspace ^[3] may be able to solve the above problems. The root of the cyberspace security problem lies in the insufficient consideration of security requirements in architecture design, which needs to be solved by designing architectures with their own security attributes and security capabilities.

This paper is the first to apply endogenous security theory to a multi-cloud resource scheduling intelligent system. Due to the polymorphic nature of the information processed by the system, its information can be resolved into multiple knowledge representations, such as meta-knowledge, rule-knowledge, instance-knowledge, artificial intelligence (AI) model feature-knowledge, etc., which leads to the natural polymorphism of the information knowledge representations, and different forms of knowledge representations require compatible knowledge reasoning subsystems. Therefore, our first-of-its-kind multi-cloud resource intelligent scheduling system architecture should be endogenously secure due to the mechanism we call Dynamic, Heterogeneous, and Redundant (DHR). We propose an endogenously secure, multi-channel redundant approach and system for intelligent scheduling of multi-cloud resources. The system has three distinctive features (DHR): First, redundancy (R), the four parallel intelligent reasoning channels of the system have redundancy to ensure that while one reasoning channel is processing in a working state, the other is in a backup standby state; second, heterogeneity (H), each of the reasoning systems and inference machines of these four channels also have different knowledge representations, and the heterogeneity increases the difficulty of the attacker; third, dynamism (D), the system has a dynamic immune ability—when the intelligent reasoning channel in the work process encountered external threats using known or unknown system vulnerabilities to attack the failure, one of the standby reasoning channels will be timely transformed into a normal working state, while loading security components to repair the failed channel, to ensure that the system always has the ability to self-heal.

The main contributions of this work are as follows:

First, a system model of endogenous security knowledge reasoning has been designed. It provides a polymorphic knowledge representation of the information to be processed and multiple knowledge reasoning methods, which are separately compatible.

Second, we have developed a dual-channel redundant endogenous security knowledge reasoning system application plan utilizing rule-based and algorithmic reasoning techniques based on AI models to meet the needs of intelligent scheduling for multi-cloud resources.

Finally, we carried out experiments on the DHR characteristics of the multi-cloud resource intelligent scheduling system that we have designed. The findings indicate that the endogenous security DHR characterization implemented in our application scheme is effective and significantly enhances system reliability.

The paper is structured as follows: Section 2 presents an overview of related research and Section 3 introduces a knowledge reasoning system model with endogenous secure DHR features featuring multiple channels and a methodology for assessing system reliability. Then, Section 4 systematically outlines the practical design of application scenarios for an intelligent scheduling system in multi-cloud resource management with endogenous security. Section 5 details the research methodology and analysis while interpreting the results. Finally, Section 6 summarizes the findings of the paper.

2. Related work

2.1. Multi-cloud resource scheduling intelligent system

In the area of multi-cloud resource scheduling, a range of scheduling approaches and techniques has emerged. Yang et al. presented a multi-population competition-cooperation-based scheduling of field service resources in cloud manufacturing ^[4]. Meanwhile, Sun et al. put forth an efficient, cost-effective and energy-saving multiple workflow scheduling in hybrid clouds ^[5]. Zhou et al. introduced a scalable genetic algorithm based on heuristic local search for multidimensional resource scheduling in cloud computing ^[6]. Agarwal et al. analyzed the scheduling of multiprocessor tasks in fog cloud computing using a multi-objective hybrid genetic algorithm ^[7]. Zhang et al. utilized an extended multi-factor evolutionary algorithm to propose personalized demand-driven multi-task scheduling in cloud manufacturing ^[8]. Xiong et al. presented a successful adaptive adjustment model for scheduling tasks and allocating resources in cloud manufacturing that is based on the interests of various stakeholders ^[9]. Zhang et al. examined scheduling of multiple tasks in cloud remanufacturing systems that incorporate reuse, reprocessing, and replacement while factoring in quality uncertainty ^[10]. Wang et al. proposed decomposition-based multi-objective evolutionary algorithms for the joint scheduling of virtual machines and tasks in cloud computing within the data space ^[11].

A key technology for intelligent scheduling of multi-cloud resources is the knowledge reasoning system, and its associated techniques and methods are continually undergoing innovation. Wu et al. proposed a neural symbol inference method employing dynamic knowledge partitioning technology ^[12], while Wang et al. suggested a set method for the diagnosis of mechanical faults under unbalanced conditions using DenseNet and evidence reasoning rules ^[13]. Gao et al. conducted an analysis of a method for managing the health of mechanical equipment. The method is based on improved intuitive fuzzy entropy and case-based reasoning technology ^[14]. Fard et al. introduced a hybrid method of geographic information systems and evidence reasoning for selecting sustainable waste power plant sites ^[15]. Xu et al. proposed a new online optimization method for boiler combustion systems based on data-driven technology and case-based reasoning principles ^[16]. Kalhori et al. introduced a novel fuzzy inference method of interval 2 type for classification systems. This method is grounded on the normal form of possibility-based fuzzy metrics ^[17]. Wang et al. proposed a task recommendation technique that integrates multi-perspective social relationship learning and reasoning in mobile crowd perception systems ^[18]. Xu et al. evaluated a new approach to online combustion optimization for boilers. The technique combines dynamic modeling, multi-objective optimization, and improved case-based reasoning ^[19]. Yadav et al. presented a hybrid method using behavioral reasoning theory ^[20]. Zhang et al. proposed a warning method based on fuzzy evidence reasoning, which takes into account heterogeneous information ^[21]. Zhao et al. have introduced a method of spatial case-based reasoning for the assessment of regional landslide risk. In a similar vein, Long et al. have proposed a parameterized extended case-based reasoning approach based on functional foundations to enable automatic experiential reasoning in mechanical product design ^[22,23]. Finally, Chen et al. have analyzed a decision-making method using logical reasoning to handle qualitative knowledge ^[24]. Wang and Gao proposed a method for evidence localization in digital forensics, utilizing a case-based variable scale reasoning approach ^[25]. Cercone et al. highlighted the advantages of a hybrid architecture incorporating rule induction and case-based reasoning ^[26]. Sottara et al. introduced a configurable Rete-OO engine able to infer various types of incomplete information ^[27]. Cao et al. conducted an analysis of the interpretability of expert systems employing belief rules ^[28]. Guo et al. formulated a multi-layer case-based reasoning approach for intricate product systems ^[29].

In brief, the prevailing method for knowledge reasoning is the expert system. It utilizes a set of rules to quantify the information obtained from expert interviews. The logical reasoning mechanism then processes this rule set, generating prompts, predictions or advice as appropriate. Several approaches exist to logical reasoning mechanisms, including fuzzy logic. The technique providing the most apt response to data is termed case-based reasoning. It is a learning and reasoning strategy that stores events and compares them.

An alternative knowledge reasoning methodology lies in machine learning, deep learning, and reinforcement learning applied to various types of neural networks. Ouache et al. have introduced a framework founded on evidence reasoning and machine learning for assessing and forecasting human-caused fire accidents ^[30]. Kierner et al. presented a classification system for hybrid architectures, which integrate rule-based reasoning and machine learning in clinical decision-making systems ^[31]. Bride et al. examined the fundamentals of high-performance machine learning for logical reasoning and verification ^[32]. Chen et al. introduced recursive inference based on training time for machine learning ^[33]. Bellomarini et al. proposed a knowledge graph with machine learning and reasoning capabilities ^[34]. Namvar et al. introduced a method for integrating intelligent reasoning into machine learning development ^[35]. Krüger et al. put forth an interpretable machine learning method for predicting student dropout rates ^[36], while Gao et al. presented a deep learning method based on evidence reasoning rules ^[37]. Liu and Qian have conducted an investigation of knowledge graph inference through reinforcement learning in the context of aluminum alloy applications ^[38]. Meanwhile, Muslim et al. have introduced a reinforcement learning-based framework for offloading computing services in both edge and core cloud environments ^[39].

2.2. Endogenous security issues and theory

Endogeneity means that one or more explanatory variables in the system model are correlated with the random perturbation term. If all the explanatory variables are ${X}_{i}$ and the random perturbation ${u}_{i}$ , endogeneity can be expressed as: $cov\left({u}_{i}, {X}_{i}\right)\ne 0$ .

Endogenous problems prevail in the field of cyberspace security ^[3]. Endogenous security (ES) issues of multi-cloud resource scheduling and reasoning systems include functional security issues and information security issues. Functional security concerns pertain to any impairment of a system's expected function (EF), while information security concerns include incidents of information leakage, tampering, and similar occurrences. In addition to knowledge reasoning EFs, knowledge reasoning systems have associated visible side effect functions (VSEFs) or invisible dark functions (IDFs). When a VSEF is detected, a security patch is used. However, security patches themselves can have new security problems. In summary, whenever a knowledge-based reasoning EF is performed, there must be a VSEF or an IDF.

Ahmad et al. conducted an overview of the security challenges and solutions in 5G ^[40], while Hu et al. presented a meticulously designed framework for network security defense ^[41]. Nevertheless, the inherent security issue of intelligently scheduling multi-cloud resources remains unresolved.

The proposed theoretical model by Wu for endogenous security assurance in cyberspace not only provides protection from identified security risks, but also shields against unknown threats and attacks ^[3]. The essence of endogenous security lies in the DHR mechanism, which selects groups of actuators dynamically from heterogeneous and redundant ones. This not only fulfils the expected system functions, but also adds uncertainty for adversaries. The system's structure and operation mechanism guarantee its reliability and make it more difficult for attackers to carry out effective attacks.

The knowledge reasoning system program's evaluation method can draw reference from both the system reliability theory and engineering applications' practical experience ^[3]. The varied forms of information knowledge representation, including meta-knowledge, rule knowledge, case knowledge, and artificial intelligence model knowledge, combined with the diverse knowledge reasoning methods and redundancy of parallel reasoning architecture design, constitute the fundamental characteristics of multi-channel knowledge reasoning systems with endogenous safety. These features form the basis for calculating system reliability.

To address the endogenous security issue of the decision-making system used in scheduling multi-cloud resources, the intelligent reasoning system architecture for multi-cloud resource scheduling has been designed with the application of endogenous security theory.

3. Endogenous security model and reliability of reasoning systems in multi-cloud resource scheduling

3.1. Causes of endogenous security issues of reasoning systems in multi-cloud resource scheduling

The EF of a multi-cloud resource scheduling intelligent system is to safely and reliably process and transmit cloud network resource scheduling information and cloud network data streams to realize the knowledge reasoning function. However, as shown in Figure 1, there are different VSEFs in addition to the existence of IDFs. Due to the existence of meta-knowledge, rule-knowledge, instance-knowledge, AI model-knowledge, etc., natural polymorphic knowledge reasoning represented by information knowledge is characterized by time-varying, heterogeneity, and redundancy in cloud network resource scheduling intelligence. At the same time, the knowledge polymorphism of reasoning system intelligent resource scheduling in cloud network data stream security and network transmission of cloud network resource scheduling information also leads to predictable VSEFs such as difficulty in knowledge representation of complex information data, inappropriate matching of knowledge representation with inference engine type, and abnormal results when inference engine handles unknown boundary conditions.

Figure 1. Causes of reasoning system ES issues.

DownLoad: Full-Size Img PowerPoint

There are two reasons for the endogenous security problem of reasoning systems in data flow security, as shown in Figure 2. On one hand, it refers to the polymorphism, naturalness and human interference in the knowledge representation of the information data to be processed. Since the multi-path effect of intelligent scheduling of cloud network resources starts from the knowledge representation of scheduling information, it will reach different knowledge inference systems through many different paths such as meta-knowledge, rule-based knowledge, case-based knowledge, and AI model-based knowledge. We expect to build a multi-level inference system where the raw information data to be processed will converge in a more accurate and reasonable way in the multi-level inference system through a number of accessible paths. However, the information processing paths with different accessibility are different, and the knowledge representation of the information data to be processed may be uncertain. Therefore, this VSEF can also be called the predictable side effect function of path selection. On the other hand, an attacker who receives a signal from an unknown path eavesdrops on it or launches an attack. This is an artificial security attack on the IDF in the integrated resource scheduling path of the cloud network. The occurrence of the above VSEFs and IDFs will surely bring endogenous security problems to the multi-cloud resource scheduling inference system.

Figure 2. The ES issues of a reasoning system in multi-cloud resource scheduling.

DownLoad: Full-Size Img PowerPoint

3.2. Endogenous security model for reasoning systems

The endogenous security model of the inference system, demonstrated in Figure 3, comprises four layers of distinct inference engines.

Figure 3. Endogenous security model for reasoning systems in multi-cloud resource scheduling.

DownLoad: Full-Size Img PowerPoint

● Heterogeneous 1. Principle-based reasoning system based on meta knowledge includes meta knowledge data representation: raw data, data elements, element parameters; and principle reasoning system: axiom of business, business formula, business rules.

● Heterogeneous 2. Rule-based reasoning system based on expert knowledge includes expert knowledge data representation: business rule characteristic, rule characteristic parameters; and rule reasoning system: business rules, rule-based reasoning.

● Heterogeneous 3. Case-based reasoning system includes business case data representation: case structure characteristic, structural characteristic parameters; and case-based reasoning system: successful instances, case-based reasoning.

● Heterogeneous 4. AI algorithm reasoning system based on machine learning includes data representation of machine learning model: AI model characteristic, model characteristic parameters; and AI algorithm reasoning: AI algorithm, AI algorithm reasoning.

The principle-based reasoning system based on meta knowledge is the most basic reasoning method and can be used in both rule-based reasoning and case-based reasoning, while rule-based reasoning can be used in case-based reasoning. The feature characteristic parameters representation of business knowledge in all the above reasoning systems is uniform.

The elements of the heterogeneous redundancy reasoning systems set L with 4 reconfigurable equivalent multi-clod resource scheduling function F are equal to 15 and are the sum of the combinations in ${C}_{4}^{n}$ , where $n\in (\mathrm{1, 2}, \mathrm{3, 4})$ .

For reasoning scenario elements j (j = 1, 2, 3, 4), i.e., meta-knowledge-based reasoning, rule-knowledge-based reasoning, instance-knowledge-based reasoning, and AI algorithm-based reasoning, design flaws or loopholes are allowed to exist that are different in nature (patterns of differences) from the other elements in the set L.

The workflow of the intelligent system for multi-cloud resource scheduling is as follows: the requirement for multi-cloud resource scheduling is entered from the input sequence of the system. The input agent then uniformly analyzes and consolidates the data to form representations of meta knowledge data, expert knowledge data, business instance data, and machine learning model data. According to the endogenous security mechanism, the default selection strategy of the dynamic heterogeneous reasoning system is utilized to invoke the relevant dynamic heterogeneous reasoning system. The output agent of the system generates the reasoning results of the output sequence and completes the self-learning case of the knowledge system.

When faced with a sudden security threat, the immune response system is triggered automatically. This, alongside the activation of other heterogeneous reasoning systems, is designed to maintain the normal functioning of the system and achieve a redundant backup immune effect.

3.3. Reliability assessment methods for endogenous security reasoning systems in multi-cloud resource scheduling

The following is a quantitative analysis and comparison of the reliability of single channel modules and multi-redundant channel modules.

Due to uncertain internal resource depletion or external threat attacks, we assume that the failure events of the channel units included in the system follow an exponential distribution.

Fault density function:

$f\left(t\right) = \lambda {e}^{-\lambda t}$

(1)

Unreliability:

$F\left(t\right) = 1-{e}^{-\lambda t}$

(2)

Reliability:

$R\left(t\right) = 1-F\left(t\right) = {e}^{-\lambda t}$

(3)

Failure rate:

$\lambda \left(t\right) = \frac{f\left(t\right)}{R\left(t\right)} = \lambda$

(4)

Mean time between failures:

$MTBF = \frac{1}{\mathit{\lambda }}$

(5)

According to the definition and reliability logic diagram of a parallel system, its unreliability mathematical model is:

${F}_{s}\left(t\right) = \prod _{i = 1}^{n}{F}_{i}\left(t\right)$

(6)

where ${F}_{s}\left(t\right)$ represents the system's unreliability, and ${F}_{i}\left(t\right)$ represents the ith unit's unreliability.

Set the failure rates of the two channel modules as ${\lambda }_{1}$ and ${\lambda }_{2}$ . The unreliability can be obtained by using the following formulas:

${F}_{1}\left(t\right) = 1-{e}^{-{\lambda }_{1}t}$

${F}_{2}\left(t\right) = 1-{e}^{-{\lambda }_{2}t}$

According to Eq (6), the unreliability of the dual redundant module composed of two channel modules is:

${F}_{s}\left(t\right) = \prod _{i = 1}^{2}{F}_{i}\left(t\right) = {F}_{1}\left(t\right)*{F}_{2}\left(t\right) = \left(1-{e}^{-{\lambda }_{1}t}\right)*\left(1-{e}^{-{\lambda }_{2}t}\right) = 1-{e}^{-{\lambda }_{1}t}-{e}^{-{\lambda }_{2}t}+{e}^{-({\lambda }_{1}t+{\lambda }_{2}t)}$

(7)

According to Eq (3), the reliability of dual redundancy module is:

${R}_{s}\left(t\right) = 1-{F}_{s}\left(t\right) = {e}^{-{\lambda }_{1}t}+{e}^{-{\lambda }_{2}t}-{e}^{-({\lambda }_{1}t+{\lambda }_{2}t)}$

(8)

In particular, when the control module and task management module adopt the same redundancy in the dual redundancy design, namely ${\lambda }_{1} = {\lambda }_{2} = \lambda$ . According to Eq (8), the reliability of dual redundancy module is:

${R}_{s}\left(t\right) = 2{e}^{-\lambda t}-{e}^{-2\lambda t}$

(9)

Assuming that the mean time between failures (MTBF) of the single channel module is 20000 h, it can be seen from Eq (5) that the failure rate of the single channel module is

${\lambda }_{p1} = \frac{1}{MTBF} = 0.00005 .$

According to Eq (3), the reliability of the single channel module for 2000 hours is

${R}_{p1}\left(t\right) = {e}^{-{\lambda }_{p1}t} = 0.9048 .$

According to Eq (9), the reliability of the dual redundant channel module for 2000 hours is

${R}_{p2}\left(t\right) = {2e}^{-{\lambda }_{p1}t}-{e}^{-2{\lambda }_{p1}t} = 0.9909 .$

Table 1 lists the reliability of endogenous security reasoning systems in different redundant channels. It can be seen that the reliability of the system can be greatly improved after the multi-redundant channel design is adopted, that is, the task reliability of the system has been greatly improved.

Table 1. List of the reliability of endogenous security reasoning systems in different redundant channels.

Working hours	Single channel	Dual channels	Three channels	Four channels
20,000	0.3678	0.6004	0.7474	0.8403
16,000	0.4493	0.6968	0.8330	0.9080
12,000	0.5488	0.7964	0.9082	0.9586
8000	0.6703	0.8913	0.9642	0.9882
4000	0.8187	0.9671	0.9940	0.9989
2000	0.9848	0.9909	0.9991	0.9999

| Show Table

DownLoad: CSV

4. Designing an endogenous security reasoning system in multi-cloud resource scheduling

4.1. Data flow security protection system

4.1.1. Security capability component classification and grading

Based on the security industry experience and with reference to the information security technology related standards, the security components in the cloud network system can be categorized and graded. The system security protection level, security layering and security capability components correspond to the following Table 2.

Table 2. Security component classification hierarchy list.

Security level	Security layer	Security capability component
2	Terminal	Terminal virus defense
2	Data	Data identification
2	Data	Document encryption
2	Application	Website content monitoring
2	Application	Web page tamper proof (extranet)
2	Cloud	WAF (internet outlet)
2	Cloud	VPN
2	Cloud	Firewall (FW)
2	Cloud	Fortress machine
2	Cloud	Vulnerability scanning
2	Network	Mobile malicious program
2	Network	Network abnormal traffic monitoring
2	Network	Flow direction monitoring
2	Network	Online log retention
2	Network	Stiff wood creep detection
2	Network	Unrecorded website detection
2	Network	Domain name information security management
2	Network	Spam message interception
2	Network	IDC/ISP
3	Terminal	Terminal access management
3	Terminal	Terminal data leakage prevention
3	Data	Data desensitization
3	Data	Database audit
3	Data	Network DLP
3	Data	Data encryption
3	Application	Unified access (4A)
3	Application	Mobile app shell
3	Application	Code audit
3	Cloud	IPS (internet outlet)
3	Cloud	WAF (intranet outlet)
3	Cloud	Anti-Virus Gateway
3	Cloud	Honeypot system (extranet)
3	Cloud	Full flow (extranet)
3	Network	DNS
4	Terminal	Enterprise terminal leakage prevention
4	Data	Data destruction
4	Application	Web page tamper proof (intranet)
4	Application	Mimicry defense
4	Cloud	Host protection
4	Cloud	Honeypot system (intranet)
4	Cloud	Full flow (intranet)
4	Network	Attack traceability

| Show Table

DownLoad: CSV

4.1.2. Changes in security components for data flowing through different levels of protection

The cloud network data flow environment needs to focus on the security components that need to be changed as the data flows through different levels of protection.

A function of data flow security protection level adjustment related components can be designed. Referring to Table 2.

● Adjustments must be made to security level 3 components when there is data flow between protection level 2 and protection level 3.

● Adjustments to security level 4 components are necessary when transferring data between protection level 3 and protection level 4.

● When data flows between protection level 2 and protection level 4, it is necessary to adjust security level 3 and 4 components.

4.2. Data flow security for multi-cloud resource scheduling

The main strategies are:

● Private cloud resources offer better security value compared to public and industry clouds.

● When deciding between industry and public cloud options, consider the economic impact of billing costs.

● It is important to note that different clouds offer varying degrees of data security. Choose the most appropriate security assurance capabilities according to your needs.

● The varying cloud usage methods have different reliability and resource utilization benefits, thus it's crucial to opt for the one that offers optimal reliability and resource utilization benefits.

These principles are also the basis for reasoning rules and also for selecting the feature space when using AI algorithms as shown in Table 3.

Table 3. Attribute characteristics of multi-cloud resource scheduling.

Symbols	Attribute Name	Supplementary note
${C}_{1},{C}_{2},{C}_{pub}$ , ${C}_{pri}$	Name of cloud	Indicate the first and second public cloud, public cloud, and private cloud, respectively.
${C}_{maxvol}$	Maximum cloud storage capacity
${C}_{used}$	Cloud storage space assigned
${C}_{rem}$	Remaining cloud storage space
${C}_{occratio}$	Cloud storage space utilization
${C}_{cat}$	Classification of clouds	Common cloud classifications include public, private and industry clouds.
${C}_{eco}$	Cloud economics metrics
${C}_{sec}$	Security capability levels for the Cloud	For example, information security assurance level protection level 2, level 3 and level 4.
${C}_{perf}$	Integrated performance of cloud usage	Cloud usage performance and reliability
${C}_{dem}$	Cloud space demanded	Cloud space to be allocated
${C}_{demsec}$	Cloud space demanded security level	Security level of cloud space to be allocated
${C}_{sch}$	Cloud selected for scheduling

| Show Table

DownLoad: CSV

Main reasoning rules are shown in Table 4.

Table 4. Main reasoning rules.

No.	Name of rule	Rule application process
Rule 1	Private cloud first	In multi-cloud resource scheduling, if ${C}_{pub}$ and ${C}_{pri}$ are public and private clouds respectively, then private cloud C1 is selected first. ${C}_{sch}=$ ${C}_{pri}$
Rule 2	Choose a cloud with lower operating costs first	In multi-cloud resource scheduling, if ${C}_{1}$ , ${C}_{2}$ are both public clouds, and ${C}_{1}$ is more low-cost than ${C}_{2}$ , then C1 is selected first. ${C}_{sch}=$ ${C}_{1}$
Rule 3	Cloud with higher security level first	In multi-cloud resource scheduling, if ${C}_{1}$ , ${C}_{2}$ are both public clouds, and ${C}_{1}$ is higher security level than ${C}_{2}$ , then C1 is selected first. ${C}_{sch}=$ ${C}_{1}$ cc
Rule 4	First choose the cloud with higher overall performance	In multi-cloud resource scheduling, if ${C}_{1}$ , ${C}_{2}$ are both public clouds, and ${C}_{1}$ is higher overall performance than ${C}_{2}$ , then C1 is selected first. ${C}_{sch}=$ ${C}_{1}$

| Show Table

DownLoad: CSV

Based on the size of data storage space and data flow security protection level division of existing multiple cloud networks, with the help of cloud network security components, the intelligent scheduling of multiple cloud network resources is carried out according to the user's cloud network data space size and security protection level requirements.

4.3. Endogenous security models for reasoning systems tailored to application scenarios

We customize the endogenous security model of the inference system based on the application scenario of cloud network resource scheduling in data flow security. Since the number of heterogeneous redundant knowledge reasoning functions is 2, the endogenous secure dual-channel multi-cloud resource scheduling intelligent reasoning system model, as shown in Figure 4, has a set of three different reasoning system scenarios with three different reasoning elements: rule-based reasoning scenarios, AI algorithmic reasoning scenarios, and scenarios with both rule-based reasoning and AI algorithmic reasoning.

Figure 4. Dual channel intelligent system model with endogenous security in multi-cloud resource scheduling.

DownLoad: Full-Size Img PowerPoint

4.3.1. Input agent

Supporting polymorphic knowledge representation.

Input sequence:

● Multi-cloud resource scheduling requirements.

● Demanded cloud capacity, protection level requirements, etc.

Polymorphic knowledge representation:

● AI algorithm feature knowledge extraction: the attribute features are shown in Table 3.

● Rule knowledge representation: as Table 4 parses the premise of the rule used for reasoning.

4.3.2. Heterogeneous knowledge reasoning system engine functions

Two heterogeneous functions:

● Rule reasoning system based on expert knowledge (Heterogeneous 1)

The rule-based reasoning algorithm, as shown in Algorithm 1, is given below.

Algorithm 1 Rule-Based Reasoning Algorithm (Heterogeneous 1)
1.	Input Data: multi-cloud resource scheduling operations
2.	Initialize Security Policy Parameters: SP0(ST0, Zone0 = {Clouds, Networks, Security-levels})
3.	Configure Cloud Network Data Flow Security Policy: SP (ST, Zone)
4.	Call rules in order of priority:
	a. Rule 1: Choose a private cloud first
	b. Rule 2: Choose a cloud with lower operating costs first
	c. Rule 3: Choose a cloud with a high level of security first
	d. Rule 4: Choose a cloud that has high overall performance first
5.	Allocate multi-cloud resources: SP = SP + SP0 = {MaxST, MinZone0} = {Max (ST, ST0), Min (Zone, Zone0)}
6.	Call function of data flow security protection level adjustment related components
7.	Output operation scheme of resource scheduling

● AI algorithm inference reasoning system based on machine learning (Heterogeneous 2)

Feature selection:

In each multi-cloud resource scheduling operation, feature combinations with high relevance are selected to form a vector representation and stored as a training set in the database.

In the multi-cloud resource scheduling operation, the ${C}_{rem}$ , ${C}_{sec}$ , ${C}_{eco}$ , ${C}_{perf}$ feature parameters are the key parameters that reflect the business operation strategy. Among them, ${C}_{sec}$ , ${C}_{eco}$ feature parameters are relatively fixed in the scheduling operation process, so we should focus on ${C}_{rem}, \; {C}_{perf}$ feature parameters. For example, in the scenario with two clouds, if we focus only on the remaining storage space of the clouds ${C1.C}_{rem}350GB$ , ${C2.C}_{rem}200GB$ and the allocated space ${C}_{dem}100GB$ each time, the feature space vector can be represented as follows:

$( {C}_{dem} , {C1.C}_{rem} , {C2.C}_{rem} ) = (100,350,200).$

Selection of KNN algorithm:

The KNN algorithm handles the classification problem with high immunity and reliability. The above features ensure that the KNN algorithm works well in intelligent processing of running logs in multi-cloud resource scheduling scenarios. When using the KNN algorithm, the model parameter k is set to 3, and we can take the three nearest neighbor samples each time.

4.3.3. Iterative judging criteria and self-learning of AI algorithms

Design of endogenous security immunization mechanisms:

The selection of a particular heterogeneous redundant reasoning system to participate in the reasoning task should be uncertainty specific.

● Running state monitoring of each inference system in case of failure.

● Formulation of selection strategy, such as selection triggered by failure of any of the reasoning systems, random selection, specific selection, and priority selection.

Self-learning of AI algorithms involves saving the output sequences of each resource scheduling operation policy execution into the training sample library, while using the updated training sample library for learning and training of AI algorithms to obtain the latest model parameters.

4.3.4. Feedback control scheduler

Databases and AI models:

Databases: Datasets of scheduling operation logs for multi-cloud resources, cloud feature database, multi-cloud resource Security Policy Library (SPL), list of layered and graded security capability components and resource scheduling operation database.

AI models: Business rule and AI algorithm bases.

4.3.5. Output agent

Multi-cloud resource size allocation and data security protection level in the resulting sequence of inference system outputs are compared with the input user requirements to fine-tune the security components needed for the data flow security protection level to form the final sequence of outputs that satisfy the user requirements.

5. Experiment on endogenous security DHR features of the constructed intelligent reasoning system in multi-cloud resource scheduling

The environment used for the experiment, see Figure 3, consists of two parts: a multi-cloud resource scheduling intelligent system with endogenous security and dual-channel redundancy and an external attacker, in which the multi-cloud resource scheduling intelligent system with endogenous security and dual-channel redundancy consists of a multi-cloud resource scheduling demand input, a multi-cloud resource scheduling intelligent system, and a scheduling operation scheme output; the external attack part is responsible for various attacks on the resource scheduling system and for interfering with the reasoning channel that destroys the current working state. The objectives and contents of the experiment are detailed in Table 5.

Table 5. The objectives and contents of the experiment of endogenous security DHR features.

No.	Experiment name	Experimental contents
Experiment 1	Dynamic feature	● Setting the dynamic channel selection policy. ● External attack is applied to the current working channel so that the channel cannot work normally. ● Verifying whether the system can automatically switch to another channel so that the system can continue to maintain the working state, and loading the security components to repair the reasoning channel damaged by the attack.
Experiment 2	Heterogeneous feature	● Selecting a rule-based reasoning channel and an AI algorithm KNN-based reasoning channel. ● Accomplishing the task of multi-cloud resource scheduling operation ● Verify whether the results of the output scheduling operation scheme are consistent.
Experiment 3	Redundant feature	● Calculating and comparing the reliability of a single rule-based reasoning channel, a single AI algorithm-based reasoning channel, and a dual-channel redundant reasoning system working continuously for a period of time. ● Verify the practical effect of reliability enhancement in redundant channel mode.

| Show Table

DownLoad: CSV

5.1. Dynamic feature

The dynamic feature is shown by the uncertainty of heterogeneous reasoning systems selected to participate in reasoning tasks. The following measures will ensure the uncertainty of the reasoning task involved in the selection of heterogeneous redundant reasoning systems:

● Formulation of selection strategies, such as random selection, specific selection and priority selection. The Table 6 shows the situation of using random selection strategy to select an inference engine for each task. It can be seen from the table that the uncertainty of inference engine selection is guaranteed. According to the parity of random integers, they correspond to different heterogeneous inference systems.

Table 6. The situation of using random selection strategy to select the inference engine.

Task name	Random number	Selected inference engine
Task1	23	Rule Based (Heterogeneous 1)
Task2	68	AI Algorithm (Heterogeneous 2)
Task3	54	AI Algorithm (Heterogeneous 2)
Task4	17	Rule Based (Heterogeneous 1)
Task5	6	AI Algorithm (Heterogeneous 2)
Task6	85	Rule Based (Heterogeneous 1)
Task7	39	Rule Based (Heterogeneous 1)

| Show Table

DownLoad: CSV

● External attack is applied to the current working channel so that the channel cannot work normally.

● Running state monitoring of each inference system, the application program of each inference system can ensure that when an exception occurs, another inference system can turn to the formal working mode by providing whether the running state is normal.

5.2. Heterogeneous feature

The heterogeneous feature is shown by the consistency of results achieved by heterogeneous reasoning systems. The same reasoning task should obtain the same reasoning result in two heterogeneous reasoning systems. The experimental requirements are detailed in Table 7 below.

Table 7. Experimental requirements for the heterogeneous feature of reasoning systems with endogenous security.

Experimental procedure	Experimental content	Detailed description
1. Initial parameter setting	(1) Selecting dual channels	a) Rule-based inference channel b) AI algorithm KNN based reasoning channel
	(2) Initial state of multi-cloud space and security policy	a) Multi-cloud maximum storage, allocated space and remaining space b) Cloud network security protection zones c) Cloud security levels
	(3) Multi-cloud resource scheduling operations task	a) Resource space, security level requirements, multi-cloud residual space
2. Rule-based reasoning system (Heterogeneous 1)	Resource scheduling process	a) Resource scheduling operation arithmetic rules b) Channel 1 output scheme
3. KNN-based reasoning system (Heterogeneous 2)	Resource scheduling process	a) Input parameters and algorithm description of the KNN algorithm b) Channel 2 output scheme
4. Experimental conclusion judgment	Comparing resource scheduling output schemes for two-channel reasoning for consistency	a) If the output schemes of channel 1 and channel 2 are the same, functional consistency is successfully verified. b) Otherwise, the functional consistency is inconsistent and the verification fails.

| Show Table

DownLoad: CSV

5.2.1. Initial parameter setting

1) Selecting dual channels

Here dual redundant inference channels are chosen, rule-based inference channel and KNN-based inference channel.

2) Initial state of multi-cloud space and security policy

First, security policy (SP) includes security tokens (ST) and security area (Zone):

$\mathit{SP (ST, Zone)}$

Then, security policy library (SPL):

$\mathit{SPL = ｛SP1, SP2, SP3 …｝}$

Initial SP parameters:

$\mathit{SP0(ST0, Zone0)}$

Initial cloud C1:

$C1.{C}_{used} = 1650GB \\ C1.{C}_{rem} = 350GB$

Initial cloud C2:

${C2.C}_{used} = 800GB \\ C2.{C}_{rem} = 200GB$

Cloud C1 and C2 initial states are shown in Table 8.

Table 8. Cloud C1 and C2 initial states.

ID	Name of Cloud	Maximum storage space	Cloud space used	Cloud space remaining	Boundary	Security level
ID0	C1	2000 GB	1650 GB	350 GB	N1, N2	Level 2
ID0	C2	1000 GB	800 GB	200 GB	N3, N4	Level 2

| Show Table

DownLoad: CSV

(ⅰ) Cloud network data flow security policy configuration:

$\mathit{SP (ST, Zone).}$

(ⅱ) Implementation of security policies at all layers of data flow security protection system:

Calling Algorithm 1: Rule-based reasoning algorithm to MinZone and MaxST.

$SP = SP + SP\mathit{0} \\ = \mathit{｛MaxST, MinZon0｝} \\ = \mathit{｛Max (ST, ST0), Min (Zone, Zone0)｝}$

(10)

3) Multi-cloud resource scheduling task

The multi-cloud resource scheduling intelligent reasoning system training sample set in is a repository of historical operations. Now only the features ${C1.C}_{rem}$ , ${C2.C}_{rem}$ in the Table 3 are taken, and they can be serialized as follows:

$( {C}_{dem} , {C1.C}_{rem} , {C2.C}_{rem} ) = (100,350,200).$

Table 9. Resource scheduling intelligent reasoning system training sample set.

ID	${C1.C}_{eco}$	${C1.C}_{rem}$	${C1.C}_{eco}$	${C1.C}_{rem}$	Operation scheme (C1, C2)	Class
1	6	200	5.5	700	(100, 0)	1
2	6	250	5.5	600	(50, 50)	2
3	6	750	5.5	100	(100, 0)	1
4	6	300	5.5	800	(0,100)	3
5	6	400	5.5	200	(50, 50)	2
6	6	350	5.5	250	(50, 50)	2
7	6	450	5.5	100	(100, 0)	1
8	6	500	5.5	850	(0,100)	3
……	……	……	……	……	……	……
N	6	350	5.5	200	(50, 50)	2
N + 1	6	300	5.5	150	(50, 50)	2

| Show Table

DownLoad: CSV

The resource scheduling results for the two heterogeneous inference systems should adhere to the same operation-scheme (50, 50), with cloud C1 and C2 assigned to schedule 50 GB of resource space each.

5.2.2. Rule-based reasoning system (Heterogeneous 1)

As you can see from Table 8, the current resource scheduling task is to allocate 100 GB of security level 3 storage and the associated application deployment. It can be serialized as follows:

$( {C}_{dem} , {C1.C}_{rem} , {C2.C}_{rem} ) = (100,350,200).$

Calling Algorithm 1, multi-cloud resource calling rules 1–4 and Eq (10), the operation-scheme assigns 50 GB of resource space to clouds C1 and C2, resulting in corresponding resource scheduling results.

5.2.3. KNN-based reasoning system (Heterogeneous 2)

Now only the features ${C1.C}_{rem}$ , ${C2.C}_{rem}$ in the Table 3 are taken, and multi-cloud resource scheduling demand can be serialized as follows:

$( {C}_{dem} , {C1.C}_{rem} , {\mathrm{C}2.\mathrm{C}}_{\mathrm{r}\mathrm{e}\mathrm{m}} ) = (100,350,200).$

When using the KNN algorithm, the model parameter k is set to 3. By querying Table 9, it has been determined that samples with the IDs 5, 6, and 7 are the three nearest neighbor samples. Since IDs 5 and 6 belong to Class 2 and ID 7 belongs to Class 1, it is classified as Class 2, using the operation scheme (50, 50).

5.2.4. Consistency of reasoning results

It can be seen from the above that this reasoning result of operation-scheme (50, 50) is the same as that of rule-based reasoning. Therefore, it verifies the consistency of results achieved by heterogeneous reasoning systems.

5.3. Redundant feature

Redundant feature is shown by reliability analysis of multi-channel reasoning in data flow security. The failure rate of the rule-based reasoning system is ${\lambda }_{1}$ , and the failure rate of the AI-algorithm-based reasoning system is ${\lambda }_{2}$ .

Due to uncertain internal resource depletion or external threat attacks, assuming that the mean time between failures (MTBF) of the rule-based reasoning channel of the single-channel module is 10000 h, it can be seen from Eq (5) that the failure rate of the single channel module is:

${\lambda }_{1} = \frac{1}{MTBF} = 0.0001 .$

Assuming that the mean time between failures (MTBF) of the reasoning channel of the single channel module based on the AI algorithm is 8000 h, it can be seen from Eq (5) that the failure rate of the single channel module is:

${\lambda }_{2} = \frac{1}{MTBF} = 0.000125 .$

It can be seen from Eq (3) that the reliability of rule-based reasoning channel of the single channel module working for 2000 h is:

${R}_{1}\left(t\right) = {e}^{-{\lambda }_{1}t} = 0.8187 .$

It can be seen from Eq (3) that the reliability of reasoning channel based on AI algorithm of the single channel module working for 2000 h is:

${R}_{2}\left(t\right) = {e}^{-{\lambda }_{2}t} = 0.7788 .$

According to Eq (8), the reliability of the dual redundant channel module for 2000 hours is:

${R}_{s}\left(t\right) = {e}^{-{\lambda }_{1}t}+{e}^{-{\lambda }_{2}t}-{e}^{-\left({\lambda }_{1}t+{\lambda }_{2}t\right)} = 0.9599 .$

The reliability of the reasoning system with different channel designs is shown in Table 10.

Table 10. List of the reliability of the dual redundant channel reasoning system.

Working hours	AI algorithm-based reasoning	Rule-based reasoning	Dual redundant channel
8000	0.3679	0.4493	0.6519
6000	0.4724	0.5488	0.7619
4000	0.6065	0.6703	0.8703
2000	0.7788	0.8187	0.9599

| Show Table

DownLoad: CSV

Table 10 lists the reliability of the reasoning system in multi-cloud resource scheduling with different channel designs. The reliability of the rule-based channel, the AI algorithm-based reasoning channel, and the dual redundant channel for 2000 hours are 0.7788, 0.8187, and 0.9599, respectively.

So far, the above three experiments validate the endogenous security DHR property of our designed multi-cloud resource scheduling intelligent system.

5.4. Experimental results

The above proves the endogenous DHR property of the dual-channel system, so that when the intelligent reasoning channel in the work process encountered external threats using known or unknown system vulnerabilities to attack the failure, one of the standby reasoning channels will be timely transformed into a normal working state, while loading security components to repair the failed channel, to ensure that the system always has the ability to self-heal. We have used the multi-cloud resource scheduling intelligent run log database to carry out experiments. The security and reliability of the whole reasoning system have improved, as shown in Table 11, and the endogenous security of the system is achieved.

Table 11. Some reasoning results of the endogenous dual-channel system.

Task name	Random number	Selected inference engine	Heterogeneous system 1 Operation scheme (C1, C2)	Heterogeneous system 2 Operation scheme (C1, C2)	Whole system state
Task 1	23	Rule Based (Heterogeneous 1)	(50, 50)	Failure, restored	Running continuously
Task 2	68	AI Algorithm (Heterogeneous 2)	Failure, restored	(100, 0)	Running continuously
Task 3	54	AI Algorithm (Heterogeneous 2)	Failure, restored	(50, 50)	Running continuously
Task 4	17	Rule Based (Heterogeneous 1)	(0, 100)	Failure, restored	Running continuously
Task 5	6	AI Algorithm (Heterogeneous 2)	Failure, restored	(50, 50)	Running continuously
Task 6	85	Rule Based (Heterogeneous 1)	(100, 0)	Failure, restored	Running continuously
Task 7	39	Rule Based (Heterogeneous 1)	(50, 50)	Failure, restored	Running continuously
Task n	……	……	……	……	……

| Show Table

DownLoad: CSV

6. Discussion

Different reasoning systems have different computational efficiencies and time delays, and in order to better support the demand of high-intensity cloud computing for low-latency cloud applications, the more efficient subsystems with rule-based reasoning system should be prioritized as the default setting option when deploying endogenous security reasoning systems for multi-cloud resource scheduling.

7. Conclusions

In this paper, we first apply endogenous security theory to a multi-cloud resource scheduling intelligent reasoning system for the first time. Second, we construct a dual-redundant endogenous security inference system in multi-cloud resource scheduling. Finally, we validate the endogenous security mechanism of the dual-redundant inference system by combining the historical operation data in the cloud resource intelligent scheduling of cloud network systems, and analyze the enhancement of system reliability by the dual-redundant inference system. The results show that our scheme outperforms several representative inference system schemes commonly used in practice.

In conclusion, the endogenous secure inference system incorporated in multi-cloud resource scheduling guarantees system security and significantly enhances system reliability when performing inference tasks. It is suitable for multi-cloud network resource scheduling scenarios with elevated security and reliability demands.

Going forward, we intend to focus on two areas to expand our research work. On one hand, we will conduct reliability tests on various types of inference systems in an organized and methodical manner to gather relevant data. On the other hand, we will strive to optimize and enhance the design of endogenous security mechanisms in inference systems to ensure they meet the new demands and challenges of enterprise digital transformation and ensure the security and reliability of multi-cloud resource invocation intelligent systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

This research has been sponsored by the National Key R & D Program for Privacy Data Protection (Project number: 2021YFB3101300), Topic 5 "Research on Key Technologies for the Protection of Personal Rights and Interests" (Project number: 2021YFB3101305).

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	A. M. Reynolds, C. J. Rhodes, The lévy flight paradigm: Random search patterns and mechanisms, Ecology, 90 (2009), 877–887. doi: 10.1890/08-0153.1. doi: 10.1890/08-0153.1
[2]	T. Kim, S. Kim, Singularity spectra of fractional brownian motions as a multi-fractal, Chaos, Soliton. Fract., 19 (2004), 613–619. doi: 10.1016/S0960-0779(03)00187-5. doi: 10.1016/S0960-0779(03)00187-5
[3]	M. Mignotte, A fractal projection and markovian segmentation-based approach for multimodal change detection, IEEE T. Geosci. Remote, 58 (2020), 8046–8058. doi: 10.1109/TGRS.2020.2986239. doi: 10.1109/TGRS.2020.2986239
[4]	M. O. Cáceres, Non-markovian processes with long-range correlations: Fractal dimension analysis, Braz. J. phys., 29 (1999), 125–135. doi: 10.1590/S0103-97331999000100011. doi: 10.1590/S0103-97331999000100011
[5]	A. Atangana, J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1–7. doi: 10.1177/1687814015613758. doi: 10.1177/1687814015613758
[6]	D. Brockmann, L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Rev. Lett. 98 (2007), 178–301. doi: 10.1103/PhysRevLett.98.178301.
[7]	E. F. D. Goufo, S. Kumar, S. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Soliton. Fract., 130 (2020), 109467. doi: 10.1016/j.chaos.2019.109467. doi: 10.1016/j.chaos.2019.109467
[8]	W. Wang, M. A. Khan, Analysis and numerical simulation of fractional model of bank data with fractal–fractional atangana–baleanu derivative, J. Comput. Appl. Math., 369 (2020), 112646. doi: 10.1016/j.cam.2019.112646. doi: 10.1016/j.cam.2019.112646
[9]	S. Das, Convergence of Riemann-Liouvelli and Caputo Derivative Definitions for Practical Solution of Fractional Order Differential Equation, Int. J. Appl. Math. Stat., 23 (2011), 64–74. doi: 10.1416/i.ijams.2011.03.017. doi: 10.1416/i.ijams.2011.03.017
[10]	A. Atangana, T. Mekkaoui, Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus, Chaos, Soliton. Fract., 128 (2019), 366–381. doi: 10.1016/j.chaos.2019.08.018. doi: 10.1016/j.chaos.2019.08.018
[11]	E. F. D. Goufo, Fractal and fractional dynamics for a 3d autonomous and two-wing smooth chaotic system, Alexandria Engineering Journal, (2020). doi: 10.1016/j.aej.2020.03.011.
[12]	E. F. D. Goufo, Application of the caputo-fabrizio fractional derivative without singular kernel to korteweg-de vries-burgers equation, Math, Model, Anal., 21 (2016), 188–198. doi: 10.3846/13926292.2016.1145607. doi: 10.3846/13926292.2016.1145607
[13]	A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Soliton. Fract., 102 (2017), 396–406. doi: 10.1016/j.chaos.2017.04.027. doi: 10.1016/j.chaos.2017.04.027
[14]	S. İ. ARAZ, Numerical analysis of a new volterra integro-differential equation involving fractal-fractional operators, Chaos, Soliton. Fract., 130 (2020), 109396. doi: 10.1016/j.chaos.2019.109396. doi: 10.1016/j.chaos.2019.109396
[15]	E. F. Doungmo Goufo, The proto-lorenz system in its chaotic fractional and fractal structure, Int. J. Bifurcat. Chaos, 30 (2020), 2050180. doi: 10.1142/S0218127420501801. doi: 10.1142/S0218127420501801
[16]	M. V. Berry, S. Klein, Integer, fractional and fractal talbot effects, J. Mod. Optic. 43 (1996), 2139–2164. doi: 10.1080/09500349608232876.
[17]	A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier Science Limited, 2006). ISBN: 9780444518323 0444518320 0080462073 9780080462073.
[18]	S. Pooseh, H. S. Rodrigues, D. F. Torres, Fractional derivatives in dengue epidemics, In: AIP Conference Proceedings, 1389(1), AIP-2011,739–742. https://arXiv.org/pdf/1108.1683.pdf.
[19]	W. Macek, R. Branco, M. Korpyś, T. Łagoda, Fractal dimension for bending–torsion fatigue fracture characterisation, Measurement, 184 (2021), 109910. doi: 10.1016/j.measurement.2021.109910. doi: 10.1016/j.measurement.2021.109910
[20]	L. R. Carney, J. J. Mecholsky Jr, Relationship between fracture toughness and fracture surface fractal dimension in aisi 4340 steel (2013). doi: 10.4236/msa.2013.44032.
[21]	A. Atangana, S. I. Araz, Atangana-seda numerical scheme for labyrinth attractor with new differ, Geophys. J. Int., 13 (2020), 529–539. doi: 10.1142/S0218348X20400447. doi: 10.1142/S0218348X20400447
[22]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22. doi: 10.1023/A:1016592219341. doi: 10.1023/A:1016592219341

This article has been cited by:

1.	Yaxi Xu, Yi Liu, Ke Shi, Xin Wang, Yi Li, Jizong Chen, An airport apron ground service surveillance algorithm based on improved YOLO network, 2024, 32, 2688-1594, 3569, 10.3934/era.2024164
2.	S. Adlin Jebakumari, Shriya Mahajan, Harshit Raichura, B. Reddy, Zahid Ahmed, Innovative model for security of multi-cloud platform: data integrity perspective, 2024, 0975-6809, 10.1007/s13198-024-02579-2
3.	Danni Liu, Shengda Wang, Chunhui Shi, Jia Li, Xiuhong Jiang, Application and Optimization of Endogenous Security Mechanisms in Photovoltaic Data Transmission and Storage, 2025, 10, 2444-8656, 10.2478/amns-2025-0223

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)