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Combined application of blockchain technology in fractional calculus model of supply chain financial system
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Combined application of blockchain technology in fractional calculus model of supply chain financial system Ting Chen∗, Derong Wang School of Logistics Management and Engineering, Nanning Normal University, China
a r t i c l e
i n f o
Article history: Received 25 July 2019 Revised 22 September 2019 Accepted 25 September 2019 Available online xxx MSC: 34C15 37D45
a b s t r a c t This paper mainly studies the application of fractional calculus system in the financial system of blockchain supply chain. Firstly, based on the blockchain supply chain game model, the paper introduces the theory of fractional calculus, establishes a three-dimensional supply chain fractional calculus game model which is closer to the actual situation, and analyzes its dynamics by using the theory of nonlinear dynamics. Learning characteristics, controlling the behavior of chaos. Finally, the dissertation, distributed storage, traceability, tamper-proof and intelligent contract features of blockchain technology are applied to the bank credit system to verify the feasibility of this system. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Blockchain technology Supply chain finance Fractional calculus system Credit banking system
1. Introduction Supply chain finance is a comprehensive roll-out of financial services throughout the supply chain. Banks rely on the trust of core companies in the supply chain to provide upstream suppliers with accounts receivable financing and downstream dealers for accounts payable financing and other related Financial Services. The credit granted by upstream and downstream enterprises in the supply chain is realized through the strong credit conditions of the core enterprises and the strong information integration capability. This kind of financial service in the entire supply chain is an important part of commercial banking and product innovation. direction. Supply chain finance is intrinsically more inclusive and open to SMEs. Therefore, it provides a good idea for solving the financing problems of SMEs. Since the introduction of supply chain finance, it has been optimistic about the risks from all parties. In terms of control, it integrates the core enterprise and the supporting upstream and downstream enterprises as a whole, and transforms the uncontrollable risks of the single enterprise into the overall controllable risks. Through the innovation of this risk control method, it not only increases the business of commercial banks. The scale
∗
Corresponding author. E-mail addresses: [email protected] (T. Chen), [email protected] (D. Wang).
has also solved the liquidity needs of SMEs and has implemented the purpose of financial services to the real economy [1]. The background of blockchain technology and related knowledge have a certain relationship with the system of fractional calculus equations. If based on the blockchain supply chain game model, the theory of fractional calculus is introduced to establish a three-dimensional closer to the actual situation. The fractionalorder calculus game model of supply chain is analyzed by using the theory of nonlinear dynamics. Then the fractional-order calculus model of supply chain financial system based on blockchain technology is established, and the financial credit banking system is constructed. Applied analysis studies were conducted to control the behavior of chaos. At this point, the decentralization, distributed storage, traceability, tamper-proof and intelligent contract features of blockchain technology are applied to the bank credit system to verify the feasibility of this system. A corresponding analysis of the combined application of blockchain technology in the fractional calculus model of supply chain financial system was found. This can effectively find the connection between each other. This paper introduces fractional-order calculus modeling and analysis of blockchain technology supply chain financial system. This is because in the actual supply chain system, the “bullwhip effect” caused by a large number of random and unpredictable factors makes the manufacturer unable to be precise. Predicting market demand, by establishing a fractional-order calculus game model, we can get rid of the past volume to predict the future, which can effectively reduce the “bullwhip effect.”
https://doi.org/10.1016/j.chaos.2019.109461 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: T. Chen and D. Wang, Combined application of blockchain technology in fractional calculus model of supply chain financial system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109461
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2. Basic concept description
the accounting rights; the private chain specifies the specific node for accounting; the alliance chain generally uses the POS mechanism or the DPOS mechanism for accounting, that is, each node passes the size of the acquired equity or The voting rights are granted to a representative, and the representatives with the highest number of votes are booked in turn according to the established schedule. Contract layer: Apply programmable scripts and algorithms to design smart contracts to implement intelligent transactions at the application layer. Application layer: applied to a specific scenario [5]. The node accesses the blockchain network to obtain user functions and service functions and accept node management. Cross-chain management: In order to cope with the problem of excessive storage of information in the main chain, the side chain is optimized to operate the main chain, and the information exchange and sharing between the main chain and the side chain is realized through cross-chain management. Fig. 1 is an example diagram of a blockchain system.
2.1. Blockchain technology (1) Blockchain technology concept. Blockchain technology is a reliable database technology that is collectively maintained through decentralization and de-trusting. The core advantage of blockchain technology is to solve the centralization and realize the decentralized transaction between transaction entities. By using asymmetric encryption algorithms, timestamp technology and distributed consensus, peer-topeer transactions without trust are achieved [2]. (2) Technical characteristics of blockchain. Blockchain technology has the characteristics of decentering, distributed storage, traceability, non-tamperability and smart contract [3]. The first is to go to the center. The blockchain network consists of a single P2P node, with no independent thirdparty control centers, and each node has equal rights and obligations. The second is distributed storage. Each node in the P2P network has a complete data backup of the blockchain system, and any node data corruption has no effect on the entire network operation. The third is traceable. The data storage in the blockchain network is permanent, and the timestamp technology can be used to track the information transmission path and realize data traceability. Fourth, it cannot be tampered with. Form a continuous, complete database through timestamp technology to prevent data tampering and forgery. The fifth is a smart contract. The execution conditions are programmed by the script code and the contract execution conditions are automatically executed when the conditions are met. (3) Blockchain type. According to the scope of application, the blockchain is divided into a public chain, a coalition chain and a private chain. The public chain is completely open, the data on the chain is open and transparent, all nodes can join, and participate in the information recording and verification process; the alliance chain is partially decentered, applicable to a certain industry, and the participants can directly determine or select the database to read the permissions. It may be public, or limited to network participants; the private chain generally exists within the company, the node has only read and write rights, and the accounting rights are determined by the enterprise [4]. According to the degree of independence, the blockchain is divided into a main chain and a side chain. The main chain is a formal, independent blockchain network that preserves the complete data of the blockchain network; the sidechain is a supplement to the main chain, anchoring a node of the main chain, and alleviating excessive problems in the main chain information storage space. The authenticity is guaranteed by the main chain. (4) Blockchain operation process. The use of blockchain technology involves the interaction of data layer, core layer and application layer and even cross-chain. The core layer includes network layer, consensus layer and intelligent contract layer. Data layer: Each block is connected in chronological order through a chain structure to form a main chain, and the blockchain data is stored. Network layer: It consists of a single node, and obtains the accounting right through the “mining” method. After the accounting is completed, the block information is broadcasted to the whole network, and a new round of accounting is started after the verification is correct. Consensus layer: The ”mining” billing rights need to be determined by the consensus layer. Generally, there are POW mechanism, POS mechanism and DPOS mechanism. The public chain uses the POW mechanism based on the computing power of the node device to determine
2.2. Fractional calculus equation system Let ε be an infinitesimal parameter, ξ0 , ξs , ηsα be an infinitesimal transform generator, introduce an infinitesimal transformation of time, generalized coordinates and fractional generalized fractional momentum:
t ∗ = t + εξ0 (t, qs , pαs ), q∗s = qs + εξs (t, qs , pαs ),
pαs ∗ = pαs + εηs (t, qs , pαs )
(1)
where ξ0 , ξs , ηsα represents the quantity, amount, and scale of the variable under the blockchain technique, t is a time variable, q represents a constant variable, p represents a probability factor, and α represents a dependent variable.
∂ ∂ ∂ + ξs + ηs α ∂t ∂ qs ∂ ps ∂ ∂ ∂ X∗ = + gs + hs α ∂t ∂ qs ∂ ps
V∗ =
ξ0
(2)
Proposition 1. Under the transformation of infinitesimal generator ξ0 , ξs , ηsα , the regular equation of the fractional Hamiltonian system remains invariant, then the system is Lie symmetry, and the infinitesimal transform generator must satisfy the following equation:
[X ∗ , V ∗ ] =
d ξ0 ∗ X dt
(3)
where [X ∗ , V ∗ ] = X ∗V ∗ − V ∗ X ∗ is the Lie bracket operation. Proof. Obtained by the differential Eq. (10) under the infinitesimal transformation (11): .∗
.α
qs = gs (t ∗ , q∗s , pαs ∗ ), ps ∗ = hs (t ∗ , q∗s , pαs ∗ ), (s = 1, 2, . . . , n ) The series expansion skips the high-order term
. ∗ . . ξ s − qs ξ 0 = V ∗ ∂∂ Hpαs ∗ .α . . ηs − ps ξ 0 = −V ∗ ∂∂Hqs
O(ε 2 )
(4)
to get:
(5)
The Noether symmetry of the mechanical system directly derives the Noether conserved quantity, and the Lie symmetry and Mei symmetry do not necessarily lead to a conserved quantity. The following proposition gives the form of the condition and conserved quantity of the Noether conserved quantity resulting from the Lie symmetry of the fractional Hamiltonian system in phase space [6]. Proposition 2. If the infinitesimal generator ξ0 (t ,qs , pαs ), ξs (t, qs , pαs ), ηs (t, qs , pαs ). Satisfies the Lie symmetry determining Eq. (15) of the fractional Hamiltonian system, and the existence of
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Fig. 1. Blockchain system example diagram.
the canonical function G(t, q, pαs ) satisfies the following Lie structural equation:
pαs X ∗ (ξs ) −
∂ H∗ ∂ H∗ ξ0 − ξ − H ∗ X ∗ ( ξ0 ) + X ∗ ( G ) = 0 ∂t ∂ qs s
(6)
Then the Noether conserved quantity caused by the Lie symmetry of the fractional Hamiltonian system is:
I = pαs ξs − H ∗ ξ0 + G = const
(7)
Proof. The derivative of Eq. (7) can be obtained:
dI =− dt
. . . .α ∂ H∗ . α ∂ H∗ ∂ H∗ . ∗ α + qs + p s ξ0 − H ξ 0 + ps ξs + ps ξ s + G α ∂t ∂ qs ∂ ps (8)
L=
. 1 α 2 w2 2 (D q ) − q , Q s = (1 − α )e−(1−α )t q 2 2
(10)
The property of fractional derivatives: Let α ∈ R+, λ ∈ C, and satisfy the case where the fractional order and both are constant 0 < α < 1. ant. Then the fractional generalized fractional order momentum and the generalized fractional Hamiltonian energy function of the system are:
pα =
∂L
= Dα q, H ∗ = e(1−α )t
∂ Dα q
1 α 2 w2 2 (p ) + q 2 2
(11)
The differential equation of motion of the system obtained by Eq. (10) is:
.
Brought by structural Eq. (6) and simultaneous (4):
.α dI ∂H . α ∂H ∂H . =− qs + p ξ ξ + ps ξs + dt ∂ qs ∂ pαs s 0 ∂ qs s .α. . .α .α ∂H = (− ps qs + qs ps )ξ0 + ps + ξ =0 ∂ qs s
The Lagrange function and the generalized non-potential force of a single degree of freedom fractional order system are:
∗ q = ∂∂ Hpα == e(1−α )t pα
.α
p = − ∂∂Hq = −w2 e(1−α )t q
(9)
∗
(12)
The Eq. (12) is determined by Lie symmetry to obtain an infinitesimal generator that satisfies the following differential equation:
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. . . . α
ξ − qξ 0 = (1 − α )e(1−α )t pα + e(1−α )t p ξ0 + e(1−α )t η .α . .
. η − p ξ 0 = − (1 − α )e(1−α )t w2 q + e(1−α )t w2 q ξ0 − e(1−α )t w2 ξ (13) It is easy to know that the infinitesimal generator of the equation can take the following solution:
ξ0 = 0, ξ = q, η = pα
(14)
Substituting the Lie structural Eq. (16) to find the display specification function:
G(t, q, pα ) =
w2 e(1−α )t q2 − e(1−α )t ( pα ) dt 2
(15)
The corresponding Lie–Noether conserved quantity is:
I = pα q +
w2 e(1−α )t q2 − e(1−α )t ( pα ) dt = const 2
(16)
3. Establishment of fractional calculus model for supply chain financial system based on blockchain technology 3.1. Wolf method to establish fractional calculus of supply chain financial system
Fig. 2. Bifurcation diagram of parameter μ and maximum Lyapunov exponent diagram.
Due to the complexity of the fractional-order system itself, the definition method and the Jacobian matrix method are not applicable, and the Wolf method and the small data method can directly analyze the time series generated by the system, and the most widely used, for this reason, The paper will focus on the Wolf method [7]. Let x1 , x2 , . . . , xk , . . . be a chaotic time series, time delay τ , embedding dimension m, then reconstruct phase space
Y (ti ) = [x(ti ), x(ti + τ ), . . . , x(ti + (m − 1 )τ )]
(17)
Where i = 1, 2, 3, . . . , N, the steps to calculate the maximum Lyapunov exponent (LLE) are as follows: (1) Let the initial point be Y(t0 ), search for the nearest neighbor Y0 (t0 ) of Y(t0 ), and record L0 = |Y (t0 ) − Y0 (t0 )|; (2) Time evolution of Y(t0 ) and Y0 (t0 ), if |Y (t1 ) − Y0 (t1 )|ε at time t1 , then retain Y(t1 ), where ε 0 is the given distance threshold; (3) Find the neighboring point Y1 (t1 ) of Y(t1 ), satisfy |Y (t1 ) − Y1 (t1 )|ε, and the angle with Y(t1 ) is as small as possible, remember L1 = |Y (t1 ) − Y1 (t1 )|; (4) Continue to repeat the above process until Y(t) reaches the end point N of the time series. At this time, the total number of times of evolution is M, then LLE is
σ=
1 tm − t0
M i=0
ln
Li
(18)
Li
3.2. Simulation of supply chain financial fractional calculus system In the system, the producer’s output adjustment speed has different effects on the market. As the production adjustment speed becomes larger, the system will change from stable to tailoring until it finally becomes chaotic. When the system enters chaos, it will give Enterprises bring harm, which has a negative impact on the production and sales of products, and the market will become chaotic. Therefore, it is necessary to take corresponding measures to control the chaotic behavior, so that the supply chain enterprises maintain the Nash equilibrium state. Control the effect of the adjustment speed k1 on the system, so the resulting control system is:
c (v1 ) xt = (1 − μ )k1 xt −xt2 + 0.2xt + 4.5 + μ sin (wxt ) 2
Fig. 3. Yield distribution of X at μ = 0.1.
2
2
c (v2 ) yt = (1 − μ )k2 yt −yt2 + 0.4yt + 4.6 + μ sin (wyt ) c (v3 ) zt = (1 − μ )k3 zt −zt2 + 0.6zt + 4.7 + μ sin (wzt )
(19)
When μ = 1, the control system degenerates into the original system. The ordering decisions of distributors and retailers are in a stable state, so no research is done. Make the following parameter settings for the control system. The order of the control system is set to v1 = 0.95, v2 = 0.91, v3 = 0.9, and the production adjustment speed k1 = 0.16, k2 = 0.02, k3 = 0.3. The initial value x0 = 1.5, y0 = 1.5, z0 = 1.5. The disturbance coefficient w = 0.2, and the setting control parameter μ has a value range of μ ∈ (0.05, 0.5). Figs. 1–4 shows the bifurcation diagram of the control system’s bifurcation diagram, the maximum Lyapunov exponent plot and the yield adjustment factor of 1k when μ is taken at different values [8]. As can be seen from Fig. 2, the system is unstable at μ ∈ (0.05, 0.11), LLE is greater than zero, and is equal to zero at the branching point. When μ ∈ (0.26, 0.5), the system is stable, and LLE is less than zero. The above shows that with the increase of control parameters, the system under chaotic state gradually experiences 8 cycles, 4 cycles, 2 cycles, and finally stabilizes at the Nash
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the bifurcation diagram seen is presented as a straight line, which proves that the system is stable under this parameter. This is consistent with the bifurcation diagram of Fig. 2 control parameter μ. From this we can get the following conclusions: (1) When the control parameter μ=0.1, the system is in the chaotic zone; (2) When the control parameter μ=0.2, the system is in the branching area; (3) When the control parameter μ=0.3, the system is in the stable zone. When the control parameter μ=0.1, although the system is in the chaotic region, the stable region is increased. When the control parameter μ=0.2, the chaotic phenomenon is suppressed; when the control parameter μ=0.3, the chaotic phenomenon is completely suppressed. Therefore, after adding the control parameter μ, the purpose of control is achieved. We then performed a simulation analysis of the method on a computer and recorded the analysis curve. As shown in Fig. 6. Fig. 4. The yield of X at μ = 0.2.
4. Blockchain supply chain finance credits banking system construction There are five levels in the blockchain as a whole. The infrastructure is shown in Table 1. The data layer, the network layer, the consensus layer (incentive layer), the contract layer, and the application layer. Among them, the data layer, the consensus layer (incentive layer), and the contract layer are the core parts of the blockchain. As can be seen from Table 1, the blockchain selects a P2P network transmission protocol with a completely distributed and tolerable single point of failure at the lowest layer of the network layer, while monitoring broadcast data in real time to verify new data credibility, which makes the area The validity of the blockchain data is guaranteed [13], and there is no centralized node in the entire database, avoiding excessive hierarchical structure and the use of invalid data [14]. The blockchain consensus layer enables each network node to process block data more efficiently based on the network layer decentralized system. This is one of the core advantages of blockchain technology [15]. The simulation of the common blockchain financial system (CBFS) and the fractional blockchain financial system (FBFS) compares the relationship between response time and actual effect. As shown in Fig. 7.
Fig. 5. Yield breakdown of X at μ = 0.3.
equilibrium point. The chaotic behavior is controlled and the chaotic phenomenon disappears [9]. Figs. 3 and 5 are bifurcation diagrams of the yield adjustment factor k1 when μ is taken at different values. When μ=0.1, chaos is generated, but compared with the bifurcation diagram without control parameter μ in Fig. 3-1, it is found that the stable region is expanded at this time, and the first branching lags; when the control parameter μ = 0.2 There is only one bifurcation; when μ = 0.3,
4.1. Software architecture (1) Blockchain base platform. Use the alliance chain as the underlying blockchain infrastructure, maintain a multi-node P2P distributed network, provide data chain and contract chain support, receive contract call commands from the credit management system, start and run smart contracts
Table 1 Blockchain infrastructure. Financial application script data structure PoW P2P protocol
Credit bank Application layer algorithm Contract layer Data mode Data layer PoS Consensus layer Broadcast mechanism Network layer
Enterprise blockchain application Sand table data storage DPoS Verification mechanism
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Fig. 6. Simulation Analysis of Supply Chain Financial Fractional Calculus System.
Fig. 7. Comparison map of common blockchain financial system (CBFS) and fractional blockchain financial system (FBFS).
from the chain, and the operation result hash value is stored in consensus to achieve trust. (2) The support layer contains shared databases and smart contracts. The shared database here refers to the user and credit information database in the chain, and stores the business plaintext data corresponding to the hash value on the blockchain. To meet traceability characteristics, shared databases are designed with data as the center, and smart contracts perform business-related validation or other transactions that require automation [10]. (3) Credit management system. The credit management system is connected with the blockchain basic platform through the support layer, and the users and their credits are managed in the blockchain, realizing the credit card trust and trust sharing, and the mutual recognition and interchange between the alliances.
4.2. Network architecture The blockchain of the credit banking system consists of nodes provided by the main institutions of the alliance. All nodes are equal in status, and the user’s credit-to-credit operations are verified to form a trusted record and stored. From a security perspective, the Credit Union Alliance has set up a management system that is responsible for reviewing the identity of the organization’s nodes, but as an operator of the blockchain network, it does not participate in the identification and recording of credits. 4.3. System function The function of the blockchain-based credit banking system is mainly designed around user management and credit management. The credit bank blockchain users include educational
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Fig. 8. Process of credit exchange.
institution users and student users, and different types of users have different permissions. Educational institution users are inhouse educational institutions that are accredited by relevant departments and credit-banking associations and are qualified to award credits, and can upload credit information issued by them to the blockchain platform. Student users are learners who are enrolled in one or more resource agencies who can query their credit information and decide how to use these credits. Credit management includes operations such as winding, querying, and swapping credits. Participants will first study at a qualified educational institution within the Alliance and earn credits. The educational institution records the credits obtained by the students in their respective information systems, and then calls the “credit-winding” smart contract through the credit management system to initiate the credit information uplink request. The smart contract verifies the legality of the uplink request, and the verification passes, and the corresponding information is stored securely and reliably on the blockchain platform; if the verification fails, the corresponding error information is returned. Users with credit-checking privileges can query the credited information by logging out the “credit query” smart contract to collect, analyze or verify the information. Participants can use the “credit exchange” smart contract provided by the system to convert the credits obtained from one institution to another, in order to obtain the corresponding certificates and certificates. Since credit management operations have more commonalities, Fig. 8 illustrates the process of credit exchange [11,12].
5. Conclusion Firstly, the thesis introduces the blockchain technology and the theoretical knowledge of fractional calculus. Based on the fractional calculus game model, a three-dimensional blockchain supply chain financial fractional calculus game model is proposed. Then the predictor-correction algorithm is used to solve the fractional difference equation, and the complex dynamics of the model are analyzed by numerical simulation. Finally, the control method for the chaotic phenomenon appearing in the model is given. The SCFDGM constructed in this paper solves the limitations of the first-order differential game model in the modeling and analysis of the supply chain system, and can well predict future market changes, providing a new solution for the research and management of the supply chain. Finally, the paper applies the blockchain
technology supply chain financial fractional calculus system to the credit banking system, and has achieved good application results. Declaration of Competing Interest This article has no conflict of interest. Acknowledgement Project to Enhance the Basic Ability of Young and Middle-aged Teachers in Guangxi Universities in 2016—-“Research on Risk Prevention of Logistics and Financial Business of Guangxi Logistics Enterprises” (No. KY2016YB265). References [1] Smirnov V, Volchenkov D. Five years of phase space dynamics of the standard & poor’s 500. Appl Math Nonlinear Sci 2019;4(1):203–16. [2] González JLR, López JAV, Martínez MF. Raptor’s “right hunger” characterization to develop sustainable exclusion areas for wildlife at civil & military airports. Appl Math Nonlinear Sci 2018;1(2):335–44. [3] Baskonus HM, Bulut H, Sulaiman TA. New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Appl Math Nonlinear Sci 2019;4(1):141–50. [4] Bagley, Ronald L. Power law and fractional calculus model of viscoelasticity. Aiaa J 2012;27(10):1412–17. [5] Jen-Hung T, Yen-Chih L, Bin C, Shih-wei L. Governance on the drug supply chain via gcoin blockchain. Int J Environ Res Public Health 2018;15(6):1055. [6] O’Leary DE. Configuring blockchain architectures for transaction information in blockchain consortiums: the case of accounting and supply chain systems. Intell Syst Account Finance Manage 2017;24(4):138–47. [7] Treiblmaier H. The impact of the blockchain on the supply chain: a theory-based research framework and a call for action. Supply Chain Manage 2018;23(6):545–59. [8] Rinaudo CL, Lee S, Hali K. How securitization can benefit from blockchain technology. J Struct Finance 2017;23(2):51–4. [9] Yang Q, Chen D, Zhao T, Chen Y. Fractional calculus in image processing: a review. Fract Calc Appl Anal 2016;19(5):1222–49. [10] Tavares D, Almeida R, Torres DFM. Constrained fractional variational problems of variable order. IEEE/CAA J Autom Sin 2017;4(1):80–8. [11] Sopasakis P, Sarimveis H, Macheras P, Dokoumetzidis A. Fractional calculus in pharmacokinetics. J Pharmacokinet Pharmacodyn 2017;45(1):1–19. [12] Almeida R, Guzowska M, Odzijewicz T. A remark on local fractional calculus and ordinary derivatives. Open Math 2016;14(1):1122–4. [13] Aidara S, Sagna Y. BSDES driven by two mutually independent fractional brownian motions with stochastic lipschitz coefficients. Appl Math Nonlinear Sci 2019;4(1):139–50. ´ [14] Brzezinski DW. Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the riemann-liouville/caputo formulas. Appl Math Nonlinear Sci 2016;1(1):23–40. [15] Burgos C, Cortés JC, Villafuerte L, Villanueva RJ. Mean square calculus and random linear fractional differential equations: theory and applications. Appl Math Nonlinear Sci 2017;2(2):317–28.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Combined application of blockchain technology in fractional calculus model of supply chain financial system Ting Chen∗, Derong Wang School of Logistics Management and Engineering, Nanning Normal University, China
a r t i c l e
i n f o
Article history: Received 25 July 2019 Revised 22 September 2019 Accepted 25 September 2019 Available online xxx MSC: 34C15 37D45
a b s t r a c t This paper mainly studies the application of fractional calculus system in the financial system of blockchain supply chain. Firstly, based on the blockchain supply chain game model, the paper introduces the theory of fractional calculus, establishes a three-dimensional supply chain fractional calculus game model which is closer to the actual situation, and analyzes its dynamics by using the theory of nonlinear dynamics. Learning characteristics, controlling the behavior of chaos. Finally, the dissertation, distributed storage, traceability, tamper-proof and intelligent contract features of blockchain technology are applied to the bank credit system to verify the feasibility of this system. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Blockchain technology Supply chain finance Fractional calculus system Credit banking system
1. Introduction Supply chain finance is a comprehensive roll-out of financial services throughout the supply chain. Banks rely on the trust of core companies in the supply chain to provide upstream suppliers with accounts receivable financing and downstream dealers for accounts payable financing and other related Financial Services. The credit granted by upstream and downstream enterprises in the supply chain is realized through the strong credit conditions of the core enterprises and the strong information integration capability. This kind of financial service in the entire supply chain is an important part of commercial banking and product innovation. direction. Supply chain finance is intrinsically more inclusive and open to SMEs. Therefore, it provides a good idea for solving the financing problems of SMEs. Since the introduction of supply chain finance, it has been optimistic about the risks from all parties. In terms of control, it integrates the core enterprise and the supporting upstream and downstream enterprises as a whole, and transforms the uncontrollable risks of the single enterprise into the overall controllable risks. Through the innovation of this risk control method, it not only increases the business of commercial banks. The scale
∗
Corresponding author. E-mail addresses: [email protected] (T. Chen), [email protected] (D. Wang).
has also solved the liquidity needs of SMEs and has implemented the purpose of financial services to the real economy [1]. The background of blockchain technology and related knowledge have a certain relationship with the system of fractional calculus equations. If based on the blockchain supply chain game model, the theory of fractional calculus is introduced to establish a three-dimensional closer to the actual situation. The fractionalorder calculus game model of supply chain is analyzed by using the theory of nonlinear dynamics. Then the fractional-order calculus model of supply chain financial system based on blockchain technology is established, and the financial credit banking system is constructed. Applied analysis studies were conducted to control the behavior of chaos. At this point, the decentralization, distributed storage, traceability, tamper-proof and intelligent contract features of blockchain technology are applied to the bank credit system to verify the feasibility of this system. A corresponding analysis of the combined application of blockchain technology in the fractional calculus model of supply chain financial system was found. This can effectively find the connection between each other. This paper introduces fractional-order calculus modeling and analysis of blockchain technology supply chain financial system. This is because in the actual supply chain system, the “bullwhip effect” caused by a large number of random and unpredictable factors makes the manufacturer unable to be precise. Predicting market demand, by establishing a fractional-order calculus game model, we can get rid of the past volume to predict the future, which can effectively reduce the “bullwhip effect.”
https://doi.org/10.1016/j.chaos.2019.109461 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
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2. Basic concept description
the accounting rights; the private chain specifies the specific node for accounting; the alliance chain generally uses the POS mechanism or the DPOS mechanism for accounting, that is, each node passes the size of the acquired equity or The voting rights are granted to a representative, and the representatives with the highest number of votes are booked in turn according to the established schedule. Contract layer: Apply programmable scripts and algorithms to design smart contracts to implement intelligent transactions at the application layer. Application layer: applied to a specific scenario [5]. The node accesses the blockchain network to obtain user functions and service functions and accept node management. Cross-chain management: In order to cope with the problem of excessive storage of information in the main chain, the side chain is optimized to operate the main chain, and the information exchange and sharing between the main chain and the side chain is realized through cross-chain management. Fig. 1 is an example diagram of a blockchain system.
2.1. Blockchain technology (1) Blockchain technology concept. Blockchain technology is a reliable database technology that is collectively maintained through decentralization and de-trusting. The core advantage of blockchain technology is to solve the centralization and realize the decentralized transaction between transaction entities. By using asymmetric encryption algorithms, timestamp technology and distributed consensus, peer-topeer transactions without trust are achieved [2]. (2) Technical characteristics of blockchain. Blockchain technology has the characteristics of decentering, distributed storage, traceability, non-tamperability and smart contract [3]. The first is to go to the center. The blockchain network consists of a single P2P node, with no independent thirdparty control centers, and each node has equal rights and obligations. The second is distributed storage. Each node in the P2P network has a complete data backup of the blockchain system, and any node data corruption has no effect on the entire network operation. The third is traceable. The data storage in the blockchain network is permanent, and the timestamp technology can be used to track the information transmission path and realize data traceability. Fourth, it cannot be tampered with. Form a continuous, complete database through timestamp technology to prevent data tampering and forgery. The fifth is a smart contract. The execution conditions are programmed by the script code and the contract execution conditions are automatically executed when the conditions are met. (3) Blockchain type. According to the scope of application, the blockchain is divided into a public chain, a coalition chain and a private chain. The public chain is completely open, the data on the chain is open and transparent, all nodes can join, and participate in the information recording and verification process; the alliance chain is partially decentered, applicable to a certain industry, and the participants can directly determine or select the database to read the permissions. It may be public, or limited to network participants; the private chain generally exists within the company, the node has only read and write rights, and the accounting rights are determined by the enterprise [4]. According to the degree of independence, the blockchain is divided into a main chain and a side chain. The main chain is a formal, independent blockchain network that preserves the complete data of the blockchain network; the sidechain is a supplement to the main chain, anchoring a node of the main chain, and alleviating excessive problems in the main chain information storage space. The authenticity is guaranteed by the main chain. (4) Blockchain operation process. The use of blockchain technology involves the interaction of data layer, core layer and application layer and even cross-chain. The core layer includes network layer, consensus layer and intelligent contract layer. Data layer: Each block is connected in chronological order through a chain structure to form a main chain, and the blockchain data is stored. Network layer: It consists of a single node, and obtains the accounting right through the “mining” method. After the accounting is completed, the block information is broadcasted to the whole network, and a new round of accounting is started after the verification is correct. Consensus layer: The ”mining” billing rights need to be determined by the consensus layer. Generally, there are POW mechanism, POS mechanism and DPOS mechanism. The public chain uses the POW mechanism based on the computing power of the node device to determine
2.2. Fractional calculus equation system Let ε be an infinitesimal parameter, ξ0 , ξs , ηsα be an infinitesimal transform generator, introduce an infinitesimal transformation of time, generalized coordinates and fractional generalized fractional momentum:
t ∗ = t + εξ0 (t, qs , pαs ), q∗s = qs + εξs (t, qs , pαs ),
pαs ∗ = pαs + εηs (t, qs , pαs )
(1)
where ξ0 , ξs , ηsα represents the quantity, amount, and scale of the variable under the blockchain technique, t is a time variable, q represents a constant variable, p represents a probability factor, and α represents a dependent variable.
∂ ∂ ∂ + ξs + ηs α ∂t ∂ qs ∂ ps ∂ ∂ ∂ X∗ = + gs + hs α ∂t ∂ qs ∂ ps
V∗ =
ξ0
(2)
Proposition 1. Under the transformation of infinitesimal generator ξ0 , ξs , ηsα , the regular equation of the fractional Hamiltonian system remains invariant, then the system is Lie symmetry, and the infinitesimal transform generator must satisfy the following equation:
[X ∗ , V ∗ ] =
d ξ0 ∗ X dt
(3)
where [X ∗ , V ∗ ] = X ∗V ∗ − V ∗ X ∗ is the Lie bracket operation. Proof. Obtained by the differential Eq. (10) under the infinitesimal transformation (11): .∗
.α
qs = gs (t ∗ , q∗s , pαs ∗ ), ps ∗ = hs (t ∗ , q∗s , pαs ∗ ), (s = 1, 2, . . . , n ) The series expansion skips the high-order term
. ∗ . . ξ s − qs ξ 0 = V ∗ ∂∂ Hpαs ∗ .α . . ηs − ps ξ 0 = −V ∗ ∂∂Hqs
O(ε 2 )
(4)
to get:
(5)
The Noether symmetry of the mechanical system directly derives the Noether conserved quantity, and the Lie symmetry and Mei symmetry do not necessarily lead to a conserved quantity. The following proposition gives the form of the condition and conserved quantity of the Noether conserved quantity resulting from the Lie symmetry of the fractional Hamiltonian system in phase space [6]. Proposition 2. If the infinitesimal generator ξ0 (t ,qs , pαs ), ξs (t, qs , pαs ), ηs (t, qs , pαs ). Satisfies the Lie symmetry determining Eq. (15) of the fractional Hamiltonian system, and the existence of
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Fig. 1. Blockchain system example diagram.
the canonical function G(t, q, pαs ) satisfies the following Lie structural equation:
pαs X ∗ (ξs ) −
∂ H∗ ∂ H∗ ξ0 − ξ − H ∗ X ∗ ( ξ0 ) + X ∗ ( G ) = 0 ∂t ∂ qs s
(6)
Then the Noether conserved quantity caused by the Lie symmetry of the fractional Hamiltonian system is:
I = pαs ξs − H ∗ ξ0 + G = const
(7)
Proof. The derivative of Eq. (7) can be obtained:
dI =− dt
. . . .α ∂ H∗ . α ∂ H∗ ∂ H∗ . ∗ α + qs + p s ξ0 − H ξ 0 + ps ξs + ps ξ s + G α ∂t ∂ qs ∂ ps (8)
L=
. 1 α 2 w2 2 (D q ) − q , Q s = (1 − α )e−(1−α )t q 2 2
(10)
The property of fractional derivatives: Let α ∈ R+, λ ∈ C, and satisfy the case where the fractional order and both are constant 0 < α < 1. ant. Then the fractional generalized fractional order momentum and the generalized fractional Hamiltonian energy function of the system are:
pα =
∂L
= Dα q, H ∗ = e(1−α )t
∂ Dα q
1 α 2 w2 2 (p ) + q 2 2
(11)
The differential equation of motion of the system obtained by Eq. (10) is:
.
Brought by structural Eq. (6) and simultaneous (4):
.α dI ∂H . α ∂H ∂H . =− qs + p ξ ξ + ps ξs + dt ∂ qs ∂ pαs s 0 ∂ qs s .α. . .α .α ∂H = (− ps qs + qs ps )ξ0 + ps + ξ =0 ∂ qs s
The Lagrange function and the generalized non-potential force of a single degree of freedom fractional order system are:
∗ q = ∂∂ Hpα == e(1−α )t pα
.α
p = − ∂∂Hq = −w2 e(1−α )t q
(9)
∗
(12)
The Eq. (12) is determined by Lie symmetry to obtain an infinitesimal generator that satisfies the following differential equation:
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. . . . α
ξ − qξ 0 = (1 − α )e(1−α )t pα + e(1−α )t p ξ0 + e(1−α )t η .α . .
. η − p ξ 0 = − (1 − α )e(1−α )t w2 q + e(1−α )t w2 q ξ0 − e(1−α )t w2 ξ (13) It is easy to know that the infinitesimal generator of the equation can take the following solution:
ξ0 = 0, ξ = q, η = pα
(14)
Substituting the Lie structural Eq. (16) to find the display specification function:
G(t, q, pα ) =
w2 e(1−α )t q2 − e(1−α )t ( pα ) dt 2
(15)
The corresponding Lie–Noether conserved quantity is:
I = pα q +
w2 e(1−α )t q2 − e(1−α )t ( pα ) dt = const 2
(16)
3. Establishment of fractional calculus model for supply chain financial system based on blockchain technology 3.1. Wolf method to establish fractional calculus of supply chain financial system
Fig. 2. Bifurcation diagram of parameter μ and maximum Lyapunov exponent diagram.
Due to the complexity of the fractional-order system itself, the definition method and the Jacobian matrix method are not applicable, and the Wolf method and the small data method can directly analyze the time series generated by the system, and the most widely used, for this reason, The paper will focus on the Wolf method [7]. Let x1 , x2 , . . . , xk , . . . be a chaotic time series, time delay τ , embedding dimension m, then reconstruct phase space
Y (ti ) = [x(ti ), x(ti + τ ), . . . , x(ti + (m − 1 )τ )]
(17)
Where i = 1, 2, 3, . . . , N, the steps to calculate the maximum Lyapunov exponent (LLE) are as follows: (1) Let the initial point be Y(t0 ), search for the nearest neighbor Y0 (t0 ) of Y(t0 ), and record L0 = |Y (t0 ) − Y0 (t0 )|; (2) Time evolution of Y(t0 ) and Y0 (t0 ), if |Y (t1 ) − Y0 (t1 )|ε at time t1 , then retain Y(t1 ), where ε 0 is the given distance threshold; (3) Find the neighboring point Y1 (t1 ) of Y(t1 ), satisfy |Y (t1 ) − Y1 (t1 )|ε, and the angle with Y(t1 ) is as small as possible, remember L1 = |Y (t1 ) − Y1 (t1 )|; (4) Continue to repeat the above process until Y(t) reaches the end point N of the time series. At this time, the total number of times of evolution is M, then LLE is
σ=
1 tm − t0
M i=0
ln
Li
(18)
Li
3.2. Simulation of supply chain financial fractional calculus system In the system, the producer’s output adjustment speed has different effects on the market. As the production adjustment speed becomes larger, the system will change from stable to tailoring until it finally becomes chaotic. When the system enters chaos, it will give Enterprises bring harm, which has a negative impact on the production and sales of products, and the market will become chaotic. Therefore, it is necessary to take corresponding measures to control the chaotic behavior, so that the supply chain enterprises maintain the Nash equilibrium state. Control the effect of the adjustment speed k1 on the system, so the resulting control system is:
c (v1 ) xt = (1 − μ )k1 xt −xt2 + 0.2xt + 4.5 + μ sin (wxt ) 2
Fig. 3. Yield distribution of X at μ = 0.1.
2
2
c (v2 ) yt = (1 − μ )k2 yt −yt2 + 0.4yt + 4.6 + μ sin (wyt ) c (v3 ) zt = (1 − μ )k3 zt −zt2 + 0.6zt + 4.7 + μ sin (wzt )
(19)
When μ = 1, the control system degenerates into the original system. The ordering decisions of distributors and retailers are in a stable state, so no research is done. Make the following parameter settings for the control system. The order of the control system is set to v1 = 0.95, v2 = 0.91, v3 = 0.9, and the production adjustment speed k1 = 0.16, k2 = 0.02, k3 = 0.3. The initial value x0 = 1.5, y0 = 1.5, z0 = 1.5. The disturbance coefficient w = 0.2, and the setting control parameter μ has a value range of μ ∈ (0.05, 0.5). Figs. 1–4 shows the bifurcation diagram of the control system’s bifurcation diagram, the maximum Lyapunov exponent plot and the yield adjustment factor of 1k when μ is taken at different values [8]. As can be seen from Fig. 2, the system is unstable at μ ∈ (0.05, 0.11), LLE is greater than zero, and is equal to zero at the branching point. When μ ∈ (0.26, 0.5), the system is stable, and LLE is less than zero. The above shows that with the increase of control parameters, the system under chaotic state gradually experiences 8 cycles, 4 cycles, 2 cycles, and finally stabilizes at the Nash
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the bifurcation diagram seen is presented as a straight line, which proves that the system is stable under this parameter. This is consistent with the bifurcation diagram of Fig. 2 control parameter μ. From this we can get the following conclusions: (1) When the control parameter μ=0.1, the system is in the chaotic zone; (2) When the control parameter μ=0.2, the system is in the branching area; (3) When the control parameter μ=0.3, the system is in the stable zone. When the control parameter μ=0.1, although the system is in the chaotic region, the stable region is increased. When the control parameter μ=0.2, the chaotic phenomenon is suppressed; when the control parameter μ=0.3, the chaotic phenomenon is completely suppressed. Therefore, after adding the control parameter μ, the purpose of control is achieved. We then performed a simulation analysis of the method on a computer and recorded the analysis curve. As shown in Fig. 6. Fig. 4. The yield of X at μ = 0.2.
4. Blockchain supply chain finance credits banking system construction There are five levels in the blockchain as a whole. The infrastructure is shown in Table 1. The data layer, the network layer, the consensus layer (incentive layer), the contract layer, and the application layer. Among them, the data layer, the consensus layer (incentive layer), and the contract layer are the core parts of the blockchain. As can be seen from Table 1, the blockchain selects a P2P network transmission protocol with a completely distributed and tolerable single point of failure at the lowest layer of the network layer, while monitoring broadcast data in real time to verify new data credibility, which makes the area The validity of the blockchain data is guaranteed [13], and there is no centralized node in the entire database, avoiding excessive hierarchical structure and the use of invalid data [14]. The blockchain consensus layer enables each network node to process block data more efficiently based on the network layer decentralized system. This is one of the core advantages of blockchain technology [15]. The simulation of the common blockchain financial system (CBFS) and the fractional blockchain financial system (FBFS) compares the relationship between response time and actual effect. As shown in Fig. 7.
Fig. 5. Yield breakdown of X at μ = 0.3.
equilibrium point. The chaotic behavior is controlled and the chaotic phenomenon disappears [9]. Figs. 3 and 5 are bifurcation diagrams of the yield adjustment factor k1 when μ is taken at different values. When μ=0.1, chaos is generated, but compared with the bifurcation diagram without control parameter μ in Fig. 3-1, it is found that the stable region is expanded at this time, and the first branching lags; when the control parameter μ = 0.2 There is only one bifurcation; when μ = 0.3,
4.1. Software architecture (1) Blockchain base platform. Use the alliance chain as the underlying blockchain infrastructure, maintain a multi-node P2P distributed network, provide data chain and contract chain support, receive contract call commands from the credit management system, start and run smart contracts
Table 1 Blockchain infrastructure. Financial application script data structure PoW P2P protocol
Credit bank Application layer algorithm Contract layer Data mode Data layer PoS Consensus layer Broadcast mechanism Network layer
Enterprise blockchain application Sand table data storage DPoS Verification mechanism
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Fig. 6. Simulation Analysis of Supply Chain Financial Fractional Calculus System.
Fig. 7. Comparison map of common blockchain financial system (CBFS) and fractional blockchain financial system (FBFS).
from the chain, and the operation result hash value is stored in consensus to achieve trust. (2) The support layer contains shared databases and smart contracts. The shared database here refers to the user and credit information database in the chain, and stores the business plaintext data corresponding to the hash value on the blockchain. To meet traceability characteristics, shared databases are designed with data as the center, and smart contracts perform business-related validation or other transactions that require automation [10]. (3) Credit management system. The credit management system is connected with the blockchain basic platform through the support layer, and the users and their credits are managed in the blockchain, realizing the credit card trust and trust sharing, and the mutual recognition and interchange between the alliances.
4.2. Network architecture The blockchain of the credit banking system consists of nodes provided by the main institutions of the alliance. All nodes are equal in status, and the user’s credit-to-credit operations are verified to form a trusted record and stored. From a security perspective, the Credit Union Alliance has set up a management system that is responsible for reviewing the identity of the organization’s nodes, but as an operator of the blockchain network, it does not participate in the identification and recording of credits. 4.3. System function The function of the blockchain-based credit banking system is mainly designed around user management and credit management. The credit bank blockchain users include educational
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Fig. 8. Process of credit exchange.
institution users and student users, and different types of users have different permissions. Educational institution users are inhouse educational institutions that are accredited by relevant departments and credit-banking associations and are qualified to award credits, and can upload credit information issued by them to the blockchain platform. Student users are learners who are enrolled in one or more resource agencies who can query their credit information and decide how to use these credits. Credit management includes operations such as winding, querying, and swapping credits. Participants will first study at a qualified educational institution within the Alliance and earn credits. The educational institution records the credits obtained by the students in their respective information systems, and then calls the “credit-winding” smart contract through the credit management system to initiate the credit information uplink request. The smart contract verifies the legality of the uplink request, and the verification passes, and the corresponding information is stored securely and reliably on the blockchain platform; if the verification fails, the corresponding error information is returned. Users with credit-checking privileges can query the credited information by logging out the “credit query” smart contract to collect, analyze or verify the information. Participants can use the “credit exchange” smart contract provided by the system to convert the credits obtained from one institution to another, in order to obtain the corresponding certificates and certificates. Since credit management operations have more commonalities, Fig. 8 illustrates the process of credit exchange [11,12].
5. Conclusion Firstly, the thesis introduces the blockchain technology and the theoretical knowledge of fractional calculus. Based on the fractional calculus game model, a three-dimensional blockchain supply chain financial fractional calculus game model is proposed. Then the predictor-correction algorithm is used to solve the fractional difference equation, and the complex dynamics of the model are analyzed by numerical simulation. Finally, the control method for the chaotic phenomenon appearing in the model is given. The SCFDGM constructed in this paper solves the limitations of the first-order differential game model in the modeling and analysis of the supply chain system, and can well predict future market changes, providing a new solution for the research and management of the supply chain. Finally, the paper applies the blockchain
technology supply chain financial fractional calculus system to the credit banking system, and has achieved good application results. Declaration of Competing Interest This article has no conflict of interest. Acknowledgement Project to Enhance the Basic Ability of Young and Middle-aged Teachers in Guangxi Universities in 2016—-“Research on Risk Prevention of Logistics and Financial Business of Guangxi Logistics Enterprises” (No. KY2016YB265). References [1] Smirnov V, Volchenkov D. Five years of phase space dynamics of the standard & poor’s 500. Appl Math Nonlinear Sci 2019;4(1):203–16. [2] González JLR, López JAV, Martínez MF. Raptor’s “right hunger” characterization to develop sustainable exclusion areas for wildlife at civil & military airports. Appl Math Nonlinear Sci 2018;1(2):335–44. [3] Baskonus HM, Bulut H, Sulaiman TA. New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Appl Math Nonlinear Sci 2019;4(1):141–50. [4] Bagley, Ronald L. Power law and fractional calculus model of viscoelasticity. Aiaa J 2012;27(10):1412–17. [5] Jen-Hung T, Yen-Chih L, Bin C, Shih-wei L. Governance on the drug supply chain via gcoin blockchain. Int J Environ Res Public Health 2018;15(6):1055. [6] O’Leary DE. Configuring blockchain architectures for transaction information in blockchain consortiums: the case of accounting and supply chain systems. Intell Syst Account Finance Manage 2017;24(4):138–47. [7] Treiblmaier H. The impact of the blockchain on the supply chain: a theory-based research framework and a call for action. Supply Chain Manage 2018;23(6):545–59. [8] Rinaudo CL, Lee S, Hali K. How securitization can benefit from blockchain technology. J Struct Finance 2017;23(2):51–4. [9] Yang Q, Chen D, Zhao T, Chen Y. Fractional calculus in image processing: a review. Fract Calc Appl Anal 2016;19(5):1222–49. [10] Tavares D, Almeida R, Torres DFM. Constrained fractional variational problems of variable order. IEEE/CAA J Autom Sin 2017;4(1):80–8. [11] Sopasakis P, Sarimveis H, Macheras P, Dokoumetzidis A. Fractional calculus in pharmacokinetics. J Pharmacokinet Pharmacodyn 2017;45(1):1–19. [12] Almeida R, Guzowska M, Odzijewicz T. A remark on local fractional calculus and ordinary derivatives. Open Math 2016;14(1):1122–4. [13] Aidara S, Sagna Y. BSDES driven by two mutually independent fractional brownian motions with stochastic lipschitz coefficients. Appl Math Nonlinear Sci 2019;4(1):139–50. ´ [14] Brzezinski DW. Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the riemann-liouville/caputo formulas. Appl Math Nonlinear Sci 2016;1(1):23–40. [15] Burgos C, Cortés JC, Villafuerte L, Villanueva RJ. Mean square calculus and random linear fractional differential equations: theory and applications. Appl Math Nonlinear Sci 2017;2(2):317–28.
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