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Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching

Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of... 1IntroductionBig data and cloud computing technologies have realised extensive storage of knowledge and efficient processing of data. The Internet provides an environment for knowledge-sharing and efficient intercommunication. Artificial intelligence technology allows people to discover the commonalities behind things more and better as well as realise the personalisation of services. Information technology has a far-reaching impact on all walks of life, not only as an efficient tool but also as a natural ‘lifestyle’ and ‘work method’ to integrate people in all fields.The research on education ecology mainly uses ecological theories and methods to study education and teaching problems. It uses ecological systems and functions, ecological balance, system evolution and other viewpoints and principles to study education laws and teach development [1]. Ecological research methods have the characteristics of highlighting systematicity, openness, synergy and nonlinear influence. Therefore, it can better explain the relationship between elements in education informatisation, the mutual influence between elements and the path of system development. This research uses Prigogine's dissipative system theory to explore the evolutionary laws. The thesis uses this theory to help the ecological deduction and innovative development of education informatisation in the new stage.2Theoretical basis2.1Dissipative system theoryDissipative structure theory was first used to study non-equilibrium thermodynamics. Then, non-equilibrium statistical physics is used to reveal the relevant characteristics of complex systems with the in-depth study of its scientific and philosophical significance by scholars of dissipative system theory. The theory of dissipative system has gradually developed into a theoretical system of universal significance and applied to various fields of natural and social research. An essential point of this theory is the self-organising nature of the system [2]. The entire system may undergo abrupt changes through fluctuations. This new system structure needs to continuously exchange material and energy with the outside world to maintain its orderly structure and, at the same time, have a particular ability to resist external interference. The structure with this property is named ‘dissipative structure’ by Prigogine.The dissipative stochastic nonlinear Schrodinger equation is as follows: Consider the following one-dimensional dissipative stochastic nonlinear Schrodinger equation with additive noise (1){du−i(Δu+iau+λ|u|2u)dt=εdWu(t,0)=u(t,1)=0, t≥0, x∈[0,1]⊂Ru(0,x)=u0(x)\left\{ {\matrix{ {du - i(\Delta u + iau + \lambda |u{|^2}u)dt = \varepsilon dW} \hfill \cr {u(t,0) = u(t,1) = 0,\quad t \ge 0,\quad x \in [0,1] \subset Ru(0,x) = {u_0}(x)} \hfill \cr } } \right.Here λ = ±1, the dissipation factor a > 0,ɛ is the noise scale. The complex-valued Wiener process W = W1 + iW2 is defined in the filter probability space (Ω,F,{Ft}t≥0,P) and W has the following Karhunen-Loève expansion (2)W(t,x)=∑k=0∞Qek(x)βk(t)=∑k=0∞ηkek(x)βk(t), t≥0, x∈[0,1]W(t,x) = \sum\limits_{k = 0}^\infty Q{e_k}(x){\beta _k}(t) = \sum\limits_{k = 0}^\infty \sqrt {{\eta _k}} {e_k}(x){\beta _k}(t),\quad t \ge 0,\quad x \in [0,1]Among them Q is the linear positive definite operator on L2 = L2(0,1) and it is commutative with the Δ operator. It satisfies that Qek=ηkek,{ek}k≥1Q{e_k} = \sqrt {{\eta _k}} {e_k},\{ {e_k}{\} _{k \ge 1}}is the eigenvector of the homogeneous Δ operator. βk=βk1+iβk2{\beta _k} = \beta _k^1 + i\beta _k^2and {βki}k≥1,i=1,2{\{ \beta _k^i\} _{k \ge 1,i = 1,2}}is a family of independent and identically distributed real-valued standard Brownian motions. We modify Equation (1) as follows. Define space-time white noise χ˙=dWdt\dot \chi = {{dW} \over {dt}}, set u = p + iq, χ˙=χ˙1+iχ˙2\dot \chi = {\dot \chi _1} + i{\dot \chi _2}, and p, q, χ˙1=dW1dt{\dot \chi _1} = {{d{W_1}} \over {dt}}and χ˙2=dW2dt{\dot \chi _2} = {{d{W_2}} \over {dt}}are real-valued random processes, then Equation (1) can be rewritten as (3){pt+qxx+ap+λ(p2+q2)q=εχ˙1−qt+pxx−aq+λ(p2+q2)p=εχ˙2\left\{ {\matrix{ {{p_t} + {q_{xx}} + ap + \lambda ({p^2} + {q^2})q = \varepsilon {{\dot \chi }_1}} \hfill \cr { - {q_t} + {p_{xx}} - aq + \lambda ({p^2} + {q^2})p = \varepsilon {{\dot \chi }_2}} \hfill \cr } } \right.Then using v = px, w = qx, z = (p,q,v,w)T the above equation can be transformed into a compact form (4)Mdtz+K∂xzdt=−aMzdt+∇S0(z)dt+∇S1(z)∘dW1+∇S2(z)∘dW2M{d_t}z + K{\partial _x}zdt = - aMzdt + \nabla {S_0}(z)dt + \nabla {S_1}(z) \circ d{W_1} + \nabla {S_2}(z) \circ d{W_2}Among them (5)M=(0−100100000000000), K=(00100001−10000−100)M = \left( {\matrix{ 0 & { - 1} & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr } } \right),\quad K = \left( {\matrix{ 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \cr { - 1} & 0 & 0 & 0 \cr 0 & { - 1} & 0 & 0 \cr } } \right)(6)S0(z)=−λ4(p2+q2)2−12(v2+w2), S1(z)=εq, S2(z)=−εp{S_0}(z) = - {\lambda \over 4}{({p^2} + {q^2})^2} - {1 \over 2}({v^2} + {w^2}),\quad {S_1}(z) = \varepsilon q,\quad {S_2}(z) = - \varepsilon pSi(z) means that the equation is established under the meaning of Stratonovich-type stochastic integral.2.2Relevant research on the strategy of informatisation teaching ecology constructionInformatisation teaching ecology is defined as the sum of the processes [3]. The research process of information-based teaching ecology and the development process of teaching application under an information technology environment reciprocate to each other. To develop teaching ecology with the help of information technology, researchers study the renewal and iteration of information teaching ecology from the physical environment and humanistic environment of classroom teaching. According to the change of teaching equipment, four stages of the development of the teaching environment are proposed: the first stage represented by blackboard or writing whiteboard, the second stage is the computer combined with projection template, the third stage is the computer combined with interactive electronic whiteboard and the fourth is the computer combined with multiple Tap of the touch screen. Scholars took the star C teaching application to state the transformation of teaching resources, teaching processes and teaching methods under the support of the fourth-generation teaching environment.3Self-organised information teaching ecological modelThe teaching ecosystem has the dual functions of student development and teacher professional growth. It can be seen from the self-organising development law of informatisation teaching ecology that teachers’ innovation and community development are the keys to the transformation of informatisation ecology [4]. Based on this principle, this paper combines the UGBS concept of the integration of information technology and education and teaching proposed by Yang Zongkai based on the ecological circle structure characteristics. It proposes a dual ecological model of information-based teaching ecology that represents the self-organised form. It is shown in Figure 1.Fig. 1The structure and external environment of the informatisation teaching ecology3.1Ecological structure of informatisation teachingInformatisation classroom is the venue for the development of informatisation teaching. It is the principal place for teacher-student interaction and student-student interaction. In the information classroom space, teachers use students as the main object to transform the object. The cultivation of students in the information classroom is an obvious function of the information teaching ecology. Its essence is a process of objectification of the subject. In the classroom teaching ecological layer, teachers transform the material flow, energy flow and information flow in the internal and external environment into the knowledge flow, ability flow and value flow of students through multiple teaching activities [5]. The teacher community teaching is based on a particular organisational environment and organisational activities and comprises teachers with different knowledge structures, teaching abilities, thinking styles and behaviour styles. Community is the group resonance of teachers. This is an effective form of professional learning, communication, reflection and a way for teachers’ collective emotions to exist.3.2Interaction between binary subsystemsThe intelligent classroom and the teacher community are mutually interdependent and affect each other. Moreover, the dual development of subject and object in the process of ecological practice is realised through the dual integration of ‘teaching’ and ‘research’. A smart classroom is the basis of activities of the teacher community. Teachers mainly use teaching activities to understand and promote students’ cognition, thinking and physical and mental development. Informatised classroom is the primary manifestation of teachers’ professional values, teaching ideas and teaching skills, and it is also the source of teachers’ professional reflections [6]. The teacher community outside the classroom provides teachers with a flow of information and energy for professional growth.On the other hand, teachers in the classroom provide a steady stream of negative entropy for the growth and development of students. Students are the main targets of teachers’ practical activities and provide primary problem streams for teachers’ professional development and provide spiritual support for professional growth. It can be seen that the cultivation of students and the development of teachers are mutually the input and output, forming a dual power system.3.3The external environment of informatisation teaching ecology and its supporting mechanismEach organisation unit represents different professional forces in the cooperation and division of labour to form a community to construct an informatisation teaching ecology. As the supporting system of the informatisation teaching ecology, the external ecological environment has a complex and overlapping influence on the development and evolution of the ecology [7]. This is considered as the information technology material environment, information teaching operation and maintenance environment, policy and system environment for ecological development and the professional information and energy for the ecological input of teaching behaviour evaluation and teaching research through professional exchanges. Thus, the internal and external environments of the information-based teaching ecology form a broader material and humanistic ecological circle and professionals from various systems integrate to form a mutually beneficial and symbiotic ecological relationship.4Ecological diagnosis model and applicationThe thesis establishes a dual ecological model of information-based teaching ecology in the form of self-organisation. This article analyses the core elements of informatisation teaching ecology from the target level, element level, variable level and state-level [8]. The information teaching ecology's positive entropy and negative entropy index system is constructed by combining divergence and convergence. This research applies it to the informatisation teaching ecology measurement of H Province. It can diagnose the development of the informatisation teaching ecology in the province. It provides essential and suggestions for the further development of educational informatisation.4.1Establishment of positive and negative entropy index systemThe positive entropy index system of information teaching ecology mainly includes three elements of mutual influence among smart classrooms, teacher community and subsystems, as shown in Figure 2.Fig. 2The positive entropy index system of information teaching ecologyThe negative entropy index system of information teaching ecology is constructed from the three elements of information technology material environment, information teaching operation and maintenance environment and policy and institutional environment. The details are shown in Figure 3.Fig. 3Ecological negative entropy index system of informatisation teaching4.2Ecological diagnosis model based on the dissipative structureThe thesis uses the Brussels model to quantitatively analyse the dynamic evolution of the dissipative structure of the information teaching ecology. Usually, the mathematical conditions are solved to get the dynamic critical value condition for forming the dissipative structure: the negative entropy B of the system is more significant than 1 + A2. This article escapes the model based on the characteristics of the information teaching ecology [9]. We set the positive entropy of the information teaching ecology to be equal to A and the negative entropy to be equal to B. It can be confirmed that the basis for discriminating the dissipative structure of the information-based teaching ecology is:(7)|B|={<1+A2,Non−dissipative structure=1+A2,Critical structure>1+A2,Dissipative structure|B| = \left\{ {\matrix{ { < 1 + {A^2},{\rm{Non}} - {\rm{dissipative\;structure}}} \hfill \cr { = 1 + {A^2},{\rm{Critical\;structure}}} \hfill \cr { > 1 + {A^2},{\rm{Dissipative\;structure}}} \hfill \cr } } \right.Equation (7) shows that only when the negative entropy of the information-based teaching ecology is large, and the positive entropy is small, the dissipative structure will be formed to promote the continuous evolution of the information-based teaching ecology.4.3Calculation formula of positive and negative entropyFirst, we can calculate the entropy value of an index system with m variable layers and n evaluation levels according to Boltzmann's formula. The formula for calculating the entropy value of the i evaluation index is as follows:(8)ei=k∑j=1npijlnpij{e_i} = k\sum\limits_{j = 1}^n {p_{ij}}\ln {p_{ij}}Among them, k is a constant, usually kA = −1/ln n is used for the calculation of positive entropy and kB = 2/ln n ○ pij is used for calculation of negative entropy. From formula (8), it can be concluded that the value range of eAi is usually [0,1]. eAi The closer it is to 0, the more stable the internal structure of the ecology and the opposite, the more disorderly. The value range of eBi is [−2,0]. When eBi is closer to −2, it indicates that the external ecological environment contributes to the ordering of the ecological interior and vice versa. Secondly, the entropy weight of the i evaluation index can be calculated according to the entropy value of the index. The entropy weight calculation formulas of the i evaluation index in positive and negative entropy are shown in formulas (9) and (10), respectively.(9)ωAi=1−eAim−∑i=1meAi\omega {A_i} = {{1 - {e_{Ai}}} \over {m - \sum\limits_{i = 1}^m {e_{Ai}}}}(10)ωBi=2+eBi2m+∑i=1meBi\omega {B_i} = {{2 + {e_{Bi}}} \over {2m + \sum\limits_{i = 1}^m {e_{Bi}}}}Further, the positive entropy value EA and negative entropy value EB of the information teaching ecology can be obtained based on the overall entropy value calculation formula (10) of the index system.(11)E=∑i=1meiωiE = \sum\limits_{i = 1}^m {e_i}{\omega _i}4.4Analysis of empirical resultsThis article selects 41 digital demonstration schools in H Province as samples for empirical analysis. Based on the informatisation teaching ecological indicator system, two sets of questionnaires for schools and teachers are compiled. It is further divided into five grades nA = nB = 5 based on the results of the questionnaire. At the same time, mA = 12, mB = 16 can be known from the evaluation index system. Based on formulas (8)–(11), the information teaching ecology's positive and negative entropy values are obtained. The details are shown in Tables 1 and 2.Table 1Ecological positive entropy of informatisation teachingTarget layerVariableEntropyEntropy weightThe positive entropy of information teaching ecology 0.4402Digital Resource Management0.49070.0840Informationised teaching methods0.53260.077lTeaching model innovation0.7060.0485Learning test and evaluation0.4080.0976Information Exchange Platform0.54890,0744Online collaborative teaching and research0.49730.0829Online information sharing0.66730,0549Teaching demonstration and training0.7220.0458Teacher information awareness0.07120.1532Teacher information knowledge0.44150.0921Teacher development0.4080.0976Student literacy0.4420.092Table 2Ecological negative entropy of informatisation teachingTarget layerVariableEntropyEntropy weightInformatisation teaching ecological negative entropy −0.8811Campus network construction−0.61330.1214Data centre management−0.79460.1055Teacher and student electronic terminal−0.79690.1053Digital teaching space−1.61680.0336Innovation space−1.70510.0258Digital Resource Library−1.82750.0151Teacher Teaching System−1.60980.0342Teacher Teaching and Research System−1.54950.0394School Management System−1.5140.0426School Evaluation System−1.89760.009Home-school exchange platform−1.65860.0299Information team structure−1.04440.0837Guarantee and incentive system−0.24220.1539Informatisation funding investment−0.77070.1076Campus network construction−1.31910.0596Data centre management−1.61820.0334It can be seen from the table that the total entropy of the informatisation teaching ecology of 41 digital demonstration schools in H province indicates that the overall entropy is in a relatively orderly state. Thus, the existing external environment of the information teaching ecology promotes the continuous ordering of smart classrooms, teacher communities and their influence. At the same time, |B| = 0.8811 < 1 + A2 = 1.1938 can be known according to the discriminant formula of the dissipative structure. This shows that the current information teaching ecology is not in a state of the dissipative structure. That is, it has failed to cause a change in the information teaching ecology.From the above results, it can be seen that the average entropy value of the positive entropy index of the informatisation teaching ecology is 0.4946 [10]. On the other hand, the relatively high entropy values of teaching model innovation (0.7060), teaching demonstration and training (0.7220) and online information sharing (0.6673) indicate that there is still an imbalance in the development of teaching model innovation, demonstration and guidance between schools and teachers. This requires the further improvement and promotion of teaching innovation model exploration and demonstration leading practice. This is also a vital issue in the realisation of educational informatisation from 1.0 to 2.0.On the other hand, in the negative entropy index of the information technology teaching ecology, the absolute values of the negative entropy of the campus network construction (−0.6133), data centre management (−0.7946) and teacher-student electronic terminal (−0.7969) are low in the material environment elements of information technology, while the values of digital teaching space (−1.6168) and innovation space (−1.7051) are relatively high. This is because the development of essential equipment such as networks, data centres and terminals in each school is more consistent and has reached the requirements of digital campus construction. However, there are differences in constructing a more prosperous digital environment, and each school can make breakthroughs in the information infrastructure. This provides a particular foundation for the reform of the information teaching ecology [11]. The relatively high absolute value of negative entropy of related indicators of information-based teaching operation and maintenance environment elements can effectively inject entropy flow into the information-based teaching ecology. On the other hand, the absolute value of negative entropy of the overall indicators is low in terms of policy and institutional environmental elements, especially the guarantee and incentive system (−0.2422) indicator. At present, each unit has the same level of school management rules and regulations, teaching incentive policies and resource co-construction and sharing mechanisms. Be assured. However, formulating an interconnected system for the overall development of the school that is suitable for education informatisation 2.0 needs to be innovated and explored.5Related policies and recommendations5.1Change the organisational structure and lay a solid foundation for innovationThe reshaping of the teaching ecology is a comprehensive adjustment process of human resources, materials, systems and information. Note that the positioning of the value of organisational philosophy is first reflected in the organisational structure. So implement decentralisation and proper distributed management on management decision-making power. Exploratory problems often require group decision-making and the participation of all teachers can significantly atimulate and enhance the initiative and enthusiasm of teachers [12]. The transformation of management from administrative order to service management requires corresponding structural changes. Build intercommunication channels conducive to the efficient transmission of information while maintaining the flat and simplified organisational structure and enhancing the organisation's collaboration characteristics and decision-making benefits. This is conducive to the rapid deployment of school human resources and materials and promotes the flow and effective conversion of information and energy in the ecology.5.2Create a research atmosphere and enhance the sense of innovationInnovative organisations are organisations that can guide and encourage innovative behaviours. First of all, cultivating an organisational atmosphere for expressing different opinions and tolerating new ideas and new ideas should be done. It requires a relatively loose mechanism environment and a democratic and equal decision-making mechanism to ensure the participation of teachers in learning and research. Secondly, it takes both operational guidance and active encouragement into consideration [13]. Finally, while launching a wealth of information-based teaching and research activities, it encourages innovative ideas and behaviours of informal organisations and pays attention to the psychological achievements of teachers.Strengthen school-to-school exchanges, promote mutual learning of advanced educational management experience and explore suitable operating mechanisms between schools to form a cooperative model that can effectively implement, evaluate, feedback and adjust. Provide opportunities for in-depth exchanges between teachers through foreign cooperation, including class management, education and teaching, teaching, research and scientific research. Provide opportunities for in-depth exchanges between students through foreign cooperation, including mutual learning and mutual exchanges. It will bring more resources and opportunities to the respective schools through foreign cooperation, bringing advantages in school management, education, teaching, campus culture, research and scientific research, exchanges, and learning.6ConclusionThe information-based teaching ecosystem must achieve a self-organising state. The key to maintaining the sustained vitality and innovation of the dual ecosystem is to enhance teachers’ research awareness. Education informatisation 2.0 stage requires teachers to pursue teaching innovation based on informatisation teaching competence. The innovation of education often starts from organisational management and leadership reform and should take the formation of corresponding organisational culture as a sign of maturity. Based on the theory of dissipative systems, this research explores the laws of ecological development. It proposes to develop new teaching relationships from the following three aspects; develop a digital teacher community and create a cultural atmosphere so that the environment stimulates everyone to innovate and explore. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Nonlinear Sciences de Gruyter

Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching

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Publisher
de Gruyter
Copyright
© 2021 Pingping Wang et al., published by Sciendo
eISSN
2444-8656
DOI
10.2478/amns.2021.2.00098
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Abstract

1IntroductionBig data and cloud computing technologies have realised extensive storage of knowledge and efficient processing of data. The Internet provides an environment for knowledge-sharing and efficient intercommunication. Artificial intelligence technology allows people to discover the commonalities behind things more and better as well as realise the personalisation of services. Information technology has a far-reaching impact on all walks of life, not only as an efficient tool but also as a natural ‘lifestyle’ and ‘work method’ to integrate people in all fields.The research on education ecology mainly uses ecological theories and methods to study education and teaching problems. It uses ecological systems and functions, ecological balance, system evolution and other viewpoints and principles to study education laws and teach development [1]. Ecological research methods have the characteristics of highlighting systematicity, openness, synergy and nonlinear influence. Therefore, it can better explain the relationship between elements in education informatisation, the mutual influence between elements and the path of system development. This research uses Prigogine's dissipative system theory to explore the evolutionary laws. The thesis uses this theory to help the ecological deduction and innovative development of education informatisation in the new stage.2Theoretical basis2.1Dissipative system theoryDissipative structure theory was first used to study non-equilibrium thermodynamics. Then, non-equilibrium statistical physics is used to reveal the relevant characteristics of complex systems with the in-depth study of its scientific and philosophical significance by scholars of dissipative system theory. The theory of dissipative system has gradually developed into a theoretical system of universal significance and applied to various fields of natural and social research. An essential point of this theory is the self-organising nature of the system [2]. The entire system may undergo abrupt changes through fluctuations. This new system structure needs to continuously exchange material and energy with the outside world to maintain its orderly structure and, at the same time, have a particular ability to resist external interference. The structure with this property is named ‘dissipative structure’ by Prigogine.The dissipative stochastic nonlinear Schrodinger equation is as follows: Consider the following one-dimensional dissipative stochastic nonlinear Schrodinger equation with additive noise (1){du−i(Δu+iau+λ|u|2u)dt=εdWu(t,0)=u(t,1)=0, t≥0, x∈[0,1]⊂Ru(0,x)=u0(x)\left\{ {\matrix{ {du - i(\Delta u + iau + \lambda |u{|^2}u)dt = \varepsilon dW} \hfill \cr {u(t,0) = u(t,1) = 0,\quad t \ge 0,\quad x \in [0,1] \subset Ru(0,x) = {u_0}(x)} \hfill \cr } } \right.Here λ = ±1, the dissipation factor a > 0,ɛ is the noise scale. The complex-valued Wiener process W = W1 + iW2 is defined in the filter probability space (Ω,F,{Ft}t≥0,P) and W has the following Karhunen-Loève expansion (2)W(t,x)=∑k=0∞Qek(x)βk(t)=∑k=0∞ηkek(x)βk(t), t≥0, x∈[0,1]W(t,x) = \sum\limits_{k = 0}^\infty Q{e_k}(x){\beta _k}(t) = \sum\limits_{k = 0}^\infty \sqrt {{\eta _k}} {e_k}(x){\beta _k}(t),\quad t \ge 0,\quad x \in [0,1]Among them Q is the linear positive definite operator on L2 = L2(0,1) and it is commutative with the Δ operator. It satisfies that Qek=ηkek,{ek}k≥1Q{e_k} = \sqrt {{\eta _k}} {e_k},\{ {e_k}{\} _{k \ge 1}}is the eigenvector of the homogeneous Δ operator. βk=βk1+iβk2{\beta _k} = \beta _k^1 + i\beta _k^2and {βki}k≥1,i=1,2{\{ \beta _k^i\} _{k \ge 1,i = 1,2}}is a family of independent and identically distributed real-valued standard Brownian motions. We modify Equation (1) as follows. Define space-time white noise χ˙=dWdt\dot \chi = {{dW} \over {dt}}, set u = p + iq, χ˙=χ˙1+iχ˙2\dot \chi = {\dot \chi _1} + i{\dot \chi _2}, and p, q, χ˙1=dW1dt{\dot \chi _1} = {{d{W_1}} \over {dt}}and χ˙2=dW2dt{\dot \chi _2} = {{d{W_2}} \over {dt}}are real-valued random processes, then Equation (1) can be rewritten as (3){pt+qxx+ap+λ(p2+q2)q=εχ˙1−qt+pxx−aq+λ(p2+q2)p=εχ˙2\left\{ {\matrix{ {{p_t} + {q_{xx}} + ap + \lambda ({p^2} + {q^2})q = \varepsilon {{\dot \chi }_1}} \hfill \cr { - {q_t} + {p_{xx}} - aq + \lambda ({p^2} + {q^2})p = \varepsilon {{\dot \chi }_2}} \hfill \cr } } \right.Then using v = px, w = qx, z = (p,q,v,w)T the above equation can be transformed into a compact form (4)Mdtz+K∂xzdt=−aMzdt+∇S0(z)dt+∇S1(z)∘dW1+∇S2(z)∘dW2M{d_t}z + K{\partial _x}zdt = - aMzdt + \nabla {S_0}(z)dt + \nabla {S_1}(z) \circ d{W_1} + \nabla {S_2}(z) \circ d{W_2}Among them (5)M=(0−100100000000000), K=(00100001−10000−100)M = \left( {\matrix{ 0 & { - 1} & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr } } \right),\quad K = \left( {\matrix{ 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \cr { - 1} & 0 & 0 & 0 \cr 0 & { - 1} & 0 & 0 \cr } } \right)(6)S0(z)=−λ4(p2+q2)2−12(v2+w2), S1(z)=εq, S2(z)=−εp{S_0}(z) = - {\lambda \over 4}{({p^2} + {q^2})^2} - {1 \over 2}({v^2} + {w^2}),\quad {S_1}(z) = \varepsilon q,\quad {S_2}(z) = - \varepsilon pSi(z) means that the equation is established under the meaning of Stratonovich-type stochastic integral.2.2Relevant research on the strategy of informatisation teaching ecology constructionInformatisation teaching ecology is defined as the sum of the processes [3]. The research process of information-based teaching ecology and the development process of teaching application under an information technology environment reciprocate to each other. To develop teaching ecology with the help of information technology, researchers study the renewal and iteration of information teaching ecology from the physical environment and humanistic environment of classroom teaching. According to the change of teaching equipment, four stages of the development of the teaching environment are proposed: the first stage represented by blackboard or writing whiteboard, the second stage is the computer combined with projection template, the third stage is the computer combined with interactive electronic whiteboard and the fourth is the computer combined with multiple Tap of the touch screen. Scholars took the star C teaching application to state the transformation of teaching resources, teaching processes and teaching methods under the support of the fourth-generation teaching environment.3Self-organised information teaching ecological modelThe teaching ecosystem has the dual functions of student development and teacher professional growth. It can be seen from the self-organising development law of informatisation teaching ecology that teachers’ innovation and community development are the keys to the transformation of informatisation ecology [4]. Based on this principle, this paper combines the UGBS concept of the integration of information technology and education and teaching proposed by Yang Zongkai based on the ecological circle structure characteristics. It proposes a dual ecological model of information-based teaching ecology that represents the self-organised form. It is shown in Figure 1.Fig. 1The structure and external environment of the informatisation teaching ecology3.1Ecological structure of informatisation teachingInformatisation classroom is the venue for the development of informatisation teaching. It is the principal place for teacher-student interaction and student-student interaction. In the information classroom space, teachers use students as the main object to transform the object. The cultivation of students in the information classroom is an obvious function of the information teaching ecology. Its essence is a process of objectification of the subject. In the classroom teaching ecological layer, teachers transform the material flow, energy flow and information flow in the internal and external environment into the knowledge flow, ability flow and value flow of students through multiple teaching activities [5]. The teacher community teaching is based on a particular organisational environment and organisational activities and comprises teachers with different knowledge structures, teaching abilities, thinking styles and behaviour styles. Community is the group resonance of teachers. This is an effective form of professional learning, communication, reflection and a way for teachers’ collective emotions to exist.3.2Interaction between binary subsystemsThe intelligent classroom and the teacher community are mutually interdependent and affect each other. Moreover, the dual development of subject and object in the process of ecological practice is realised through the dual integration of ‘teaching’ and ‘research’. A smart classroom is the basis of activities of the teacher community. Teachers mainly use teaching activities to understand and promote students’ cognition, thinking and physical and mental development. Informatised classroom is the primary manifestation of teachers’ professional values, teaching ideas and teaching skills, and it is also the source of teachers’ professional reflections [6]. The teacher community outside the classroom provides teachers with a flow of information and energy for professional growth.On the other hand, teachers in the classroom provide a steady stream of negative entropy for the growth and development of students. Students are the main targets of teachers’ practical activities and provide primary problem streams for teachers’ professional development and provide spiritual support for professional growth. It can be seen that the cultivation of students and the development of teachers are mutually the input and output, forming a dual power system.3.3The external environment of informatisation teaching ecology and its supporting mechanismEach organisation unit represents different professional forces in the cooperation and division of labour to form a community to construct an informatisation teaching ecology. As the supporting system of the informatisation teaching ecology, the external ecological environment has a complex and overlapping influence on the development and evolution of the ecology [7]. This is considered as the information technology material environment, information teaching operation and maintenance environment, policy and system environment for ecological development and the professional information and energy for the ecological input of teaching behaviour evaluation and teaching research through professional exchanges. Thus, the internal and external environments of the information-based teaching ecology form a broader material and humanistic ecological circle and professionals from various systems integrate to form a mutually beneficial and symbiotic ecological relationship.4Ecological diagnosis model and applicationThe thesis establishes a dual ecological model of information-based teaching ecology in the form of self-organisation. This article analyses the core elements of informatisation teaching ecology from the target level, element level, variable level and state-level [8]. The information teaching ecology's positive entropy and negative entropy index system is constructed by combining divergence and convergence. This research applies it to the informatisation teaching ecology measurement of H Province. It can diagnose the development of the informatisation teaching ecology in the province. It provides essential and suggestions for the further development of educational informatisation.4.1Establishment of positive and negative entropy index systemThe positive entropy index system of information teaching ecology mainly includes three elements of mutual influence among smart classrooms, teacher community and subsystems, as shown in Figure 2.Fig. 2The positive entropy index system of information teaching ecologyThe negative entropy index system of information teaching ecology is constructed from the three elements of information technology material environment, information teaching operation and maintenance environment and policy and institutional environment. The details are shown in Figure 3.Fig. 3Ecological negative entropy index system of informatisation teaching4.2Ecological diagnosis model based on the dissipative structureThe thesis uses the Brussels model to quantitatively analyse the dynamic evolution of the dissipative structure of the information teaching ecology. Usually, the mathematical conditions are solved to get the dynamic critical value condition for forming the dissipative structure: the negative entropy B of the system is more significant than 1 + A2. This article escapes the model based on the characteristics of the information teaching ecology [9]. We set the positive entropy of the information teaching ecology to be equal to A and the negative entropy to be equal to B. It can be confirmed that the basis for discriminating the dissipative structure of the information-based teaching ecology is:(7)|B|={<1+A2,Non−dissipative structure=1+A2,Critical structure>1+A2,Dissipative structure|B| = \left\{ {\matrix{ { < 1 + {A^2},{\rm{Non}} - {\rm{dissipative\;structure}}} \hfill \cr { = 1 + {A^2},{\rm{Critical\;structure}}} \hfill \cr { > 1 + {A^2},{\rm{Dissipative\;structure}}} \hfill \cr } } \right.Equation (7) shows that only when the negative entropy of the information-based teaching ecology is large, and the positive entropy is small, the dissipative structure will be formed to promote the continuous evolution of the information-based teaching ecology.4.3Calculation formula of positive and negative entropyFirst, we can calculate the entropy value of an index system with m variable layers and n evaluation levels according to Boltzmann's formula. The formula for calculating the entropy value of the i evaluation index is as follows:(8)ei=k∑j=1npijlnpij{e_i} = k\sum\limits_{j = 1}^n {p_{ij}}\ln {p_{ij}}Among them, k is a constant, usually kA = −1/ln n is used for the calculation of positive entropy and kB = 2/ln n ○ pij is used for calculation of negative entropy. From formula (8), it can be concluded that the value range of eAi is usually [0,1]. eAi The closer it is to 0, the more stable the internal structure of the ecology and the opposite, the more disorderly. The value range of eBi is [−2,0]. When eBi is closer to −2, it indicates that the external ecological environment contributes to the ordering of the ecological interior and vice versa. Secondly, the entropy weight of the i evaluation index can be calculated according to the entropy value of the index. The entropy weight calculation formulas of the i evaluation index in positive and negative entropy are shown in formulas (9) and (10), respectively.(9)ωAi=1−eAim−∑i=1meAi\omega {A_i} = {{1 - {e_{Ai}}} \over {m - \sum\limits_{i = 1}^m {e_{Ai}}}}(10)ωBi=2+eBi2m+∑i=1meBi\omega {B_i} = {{2 + {e_{Bi}}} \over {2m + \sum\limits_{i = 1}^m {e_{Bi}}}}Further, the positive entropy value EA and negative entropy value EB of the information teaching ecology can be obtained based on the overall entropy value calculation formula (10) of the index system.(11)E=∑i=1meiωiE = \sum\limits_{i = 1}^m {e_i}{\omega _i}4.4Analysis of empirical resultsThis article selects 41 digital demonstration schools in H Province as samples for empirical analysis. Based on the informatisation teaching ecological indicator system, two sets of questionnaires for schools and teachers are compiled. It is further divided into five grades nA = nB = 5 based on the results of the questionnaire. At the same time, mA = 12, mB = 16 can be known from the evaluation index system. Based on formulas (8)–(11), the information teaching ecology's positive and negative entropy values are obtained. The details are shown in Tables 1 and 2.Table 1Ecological positive entropy of informatisation teachingTarget layerVariableEntropyEntropy weightThe positive entropy of information teaching ecology 0.4402Digital Resource Management0.49070.0840Informationised teaching methods0.53260.077lTeaching model innovation0.7060.0485Learning test and evaluation0.4080.0976Information Exchange Platform0.54890,0744Online collaborative teaching and research0.49730.0829Online information sharing0.66730,0549Teaching demonstration and training0.7220.0458Teacher information awareness0.07120.1532Teacher information knowledge0.44150.0921Teacher development0.4080.0976Student literacy0.4420.092Table 2Ecological negative entropy of informatisation teachingTarget layerVariableEntropyEntropy weightInformatisation teaching ecological negative entropy −0.8811Campus network construction−0.61330.1214Data centre management−0.79460.1055Teacher and student electronic terminal−0.79690.1053Digital teaching space−1.61680.0336Innovation space−1.70510.0258Digital Resource Library−1.82750.0151Teacher Teaching System−1.60980.0342Teacher Teaching and Research System−1.54950.0394School Management System−1.5140.0426School Evaluation System−1.89760.009Home-school exchange platform−1.65860.0299Information team structure−1.04440.0837Guarantee and incentive system−0.24220.1539Informatisation funding investment−0.77070.1076Campus network construction−1.31910.0596Data centre management−1.61820.0334It can be seen from the table that the total entropy of the informatisation teaching ecology of 41 digital demonstration schools in H province indicates that the overall entropy is in a relatively orderly state. Thus, the existing external environment of the information teaching ecology promotes the continuous ordering of smart classrooms, teacher communities and their influence. At the same time, |B| = 0.8811 < 1 + A2 = 1.1938 can be known according to the discriminant formula of the dissipative structure. This shows that the current information teaching ecology is not in a state of the dissipative structure. That is, it has failed to cause a change in the information teaching ecology.From the above results, it can be seen that the average entropy value of the positive entropy index of the informatisation teaching ecology is 0.4946 [10]. On the other hand, the relatively high entropy values of teaching model innovation (0.7060), teaching demonstration and training (0.7220) and online information sharing (0.6673) indicate that there is still an imbalance in the development of teaching model innovation, demonstration and guidance between schools and teachers. This requires the further improvement and promotion of teaching innovation model exploration and demonstration leading practice. This is also a vital issue in the realisation of educational informatisation from 1.0 to 2.0.On the other hand, in the negative entropy index of the information technology teaching ecology, the absolute values of the negative entropy of the campus network construction (−0.6133), data centre management (−0.7946) and teacher-student electronic terminal (−0.7969) are low in the material environment elements of information technology, while the values of digital teaching space (−1.6168) and innovation space (−1.7051) are relatively high. This is because the development of essential equipment such as networks, data centres and terminals in each school is more consistent and has reached the requirements of digital campus construction. However, there are differences in constructing a more prosperous digital environment, and each school can make breakthroughs in the information infrastructure. This provides a particular foundation for the reform of the information teaching ecology [11]. The relatively high absolute value of negative entropy of related indicators of information-based teaching operation and maintenance environment elements can effectively inject entropy flow into the information-based teaching ecology. On the other hand, the absolute value of negative entropy of the overall indicators is low in terms of policy and institutional environmental elements, especially the guarantee and incentive system (−0.2422) indicator. At present, each unit has the same level of school management rules and regulations, teaching incentive policies and resource co-construction and sharing mechanisms. Be assured. However, formulating an interconnected system for the overall development of the school that is suitable for education informatisation 2.0 needs to be innovated and explored.5Related policies and recommendations5.1Change the organisational structure and lay a solid foundation for innovationThe reshaping of the teaching ecology is a comprehensive adjustment process of human resources, materials, systems and information. Note that the positioning of the value of organisational philosophy is first reflected in the organisational structure. So implement decentralisation and proper distributed management on management decision-making power. Exploratory problems often require group decision-making and the participation of all teachers can significantly atimulate and enhance the initiative and enthusiasm of teachers [12]. The transformation of management from administrative order to service management requires corresponding structural changes. Build intercommunication channels conducive to the efficient transmission of information while maintaining the flat and simplified organisational structure and enhancing the organisation's collaboration characteristics and decision-making benefits. This is conducive to the rapid deployment of school human resources and materials and promotes the flow and effective conversion of information and energy in the ecology.5.2Create a research atmosphere and enhance the sense of innovationInnovative organisations are organisations that can guide and encourage innovative behaviours. First of all, cultivating an organisational atmosphere for expressing different opinions and tolerating new ideas and new ideas should be done. It requires a relatively loose mechanism environment and a democratic and equal decision-making mechanism to ensure the participation of teachers in learning and research. Secondly, it takes both operational guidance and active encouragement into consideration [13]. Finally, while launching a wealth of information-based teaching and research activities, it encourages innovative ideas and behaviours of informal organisations and pays attention to the psychological achievements of teachers.Strengthen school-to-school exchanges, promote mutual learning of advanced educational management experience and explore suitable operating mechanisms between schools to form a cooperative model that can effectively implement, evaluate, feedback and adjust. Provide opportunities for in-depth exchanges between teachers through foreign cooperation, including class management, education and teaching, teaching, research and scientific research. Provide opportunities for in-depth exchanges between students through foreign cooperation, including mutual learning and mutual exchanges. It will bring more resources and opportunities to the respective schools through foreign cooperation, bringing advantages in school management, education, teaching, campus culture, research and scientific research, exchanges, and learning.6ConclusionThe information-based teaching ecosystem must achieve a self-organising state. The key to maintaining the sustained vitality and innovation of the dual ecosystem is to enhance teachers’ research awareness. Education informatisation 2.0 stage requires teachers to pursue teaching innovation based on informatisation teaching competence. The innovation of education often starts from organisational management and leadership reform and should take the formation of corresponding organisational culture as a sign of maturity. Based on the theory of dissipative systems, this research explores the laws of ecological development. It proposes to develop new teaching relationships from the following three aspects; develop a digital teacher community and create a cultural atmosphere so that the environment stimulates everyone to innovate and explore.

Journal

Applied Mathematics and Nonlinear Sciences – de Gruyter

Published: Jan 1, 2022

Keywords: Nonlinear dissipative system; mathematical equations; teaching ecology; diversified regression model; education informatisation; 46L57

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