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Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education

Pubblicato online: 15 Jul 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 16 Feb 2022
Accettato: 23 Apr 2022
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Formato
Rivista
eISSN
2444-8656
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01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
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Inglese
Introduction

Professional literacy is the comprehensive quality manifested in the professional process. It is a collection of basic awareness, ability, and knowledge necessary to complete and develop professional activities [1]. Professional literacy is divided into explicit professional literacy and implicit professional literacy. Explicit professional-quality includes job qualifications, professional knowledge, and professional skills. Implicit professional-quality includes professional awareness, ethics, professional style, and professional attitude. Explicit professionalism and implicit professionalism jointly support and promote personal career development.

College students are the main force in the front line of production management. Their professionalism is attracting more and more people's attention. Under the employment-oriented background, many colleges and universities focus on improving the professional skills of college students and ignore the cultivation of college students' hidden professional quality. Implicit professionalism is the basis for developing explicit professionalism. The improvement of explicit professionalism can promote the development of implicit professionalism [2]. The two are interdependent and inseparable. Under this educational model of emphasizing skills and ignoring quality, the professional quality of college students has not been effectively improved. The employment problem of college students is still serious. At present, the professional quality of some college students is not qualified for the job needs of the post, which leads to the phenomenon of “dual difficulties” of difficulty in recruiting employees for enterprises and difficulty in finding employment for college students. We explore how to improve the professional quality of college students and make them adapt to the needs of today's society, which has become an important issue that colleges and universities need to solve [3]. This paper establishes the nonlinear differential equation of professionalism according to the relationship between implicit and explicit professionalism. At the same time, we use the qualitative theory of differential equations to explore the development law of college students' professional quality. The research theory of the article provides some basic suggestions for personnel training in colleges and universities.

The nonlinear dynamic vocational education system

f mathematical operation f is said to be nonlinear if it does not satisfy the additivity and homogeneity requirements [4]. There is essentially only one linear function relationship such as nonlinear relationship, nonlinear transformation, nonlinear operation, nonlinear function, nonlinear functional, nonlinear equation, etc.: y=ax y = ax

Nonlinear characteristics are the main source of infinite diversity, difference, and complexity of the whole vocational education system [5]. The general form of the dynamic equation of the nonlinear continuous vocational education system is the following equations { x1=f1(x1,,xn;c1,,cm)x2=f2(x1,,xn;c1,,cm)xn=fn(x1,,xn;c1,,cm) \left\{ {\matrix{ {x^\prime 1 = {f_1}\left( {{x_1}, \cdots ,\,{x_n};\,{c_1},\, \cdots ,\,{c_m}} \right)} \hfill \cr {x^\prime 2 = {f_2}\left( {{x_1}, \cdots ,\,{x_n};\,{c_1},\, \cdots ,\,{c_m}} \right)} \hfill \cr {x^\prime n = {f_n}\left( {{x_1}, \cdots ,\,{x_n};\,{c_1},\, \cdots ,\,{c_m}} \right)} \hfill \cr } } \right.

At least one of f1, ⋯, fn should be a nonlinear function. Represent control quantities in vector form C=(c1,c2,,cm) C = \left( {{c_1},\,{c_2}, \cdots ,\,{c_m}} \right)

We call this the control vector. We denote F as follows: F(f1,f2,,fn) F\left( {{f_1},\,{f_2}, \cdots ,\,{f_n}} \right)

Then the system of equations (2) can be obtained in the following vector form X=F(X,C) X^\prime = F\left( {X,\,C} \right)

The kinetic equation is the term for kinetics. In the digital vocational education system study, equations (2) and (5) are usually called evolution equations or development equations. We classify vocational education systems according to the evolution equation. The nonlinear evolution equation includes linear and nonlinear systems, centralized parameter systems, and distributed parameter systems.

The free vocational education system and forced vocational education system If the evolution equation is only the dependence of the derivative of the state variable on the state variable, then we call it a free vocational education system [6]. If the evolution equation includes the foreign action term Ψ(t), then we call it a forced vocational education system: X=F(X,C)+Ψ(t) X^\prime = F\left( {X,\,C} \right) + \Psi \left( t \right)

If a new state variable xn+1 (t) is introduced. Assuming xn+1 = Ψ(t), then the forced vocational education system (6) may be transformed into a n + 1 dimensional free vocational education system to deal with. This article therefore mainly discusses the freelance education system. The equation that includes time t in the evolution equation is as follows X=F(X,C,t) X^\prime = F\left( {X,C,t} \right)

The systematic representation of the forcing Ψ(t) as a function of time is shown in Eq. (6). Generally, there is a kind of study on the behavioral characteristics of the nonlinear vocational education system.

Analysis method

According to the theory of differential equations, we find that formula (2) has a unique solution under fairly wide conditions. Since the nonlinear function has infinitely non-interchangeable forms, it reflects the infinitely different qualitative properties of the nonlinear vocational education system [7]. It is impossible to solve the vocational education system described by equation (2) analytically. We cannot explain even the simplest one-dimensional nonlinear vocational education system. x=f(x) x^\prime = f\left( x \right)

Function f (x) still has infinitely many analytical solutions. For example, we can use the separation of variables method to obtain analytical solutions when f (x) has the appropriate form.

Geometric methods

Geometric methods are mainly used to analyze the qualitative nature of vocational education systems. It is very convenient and efficient for us to avoid solving equations and directly extract qualitative information about the vocational education system from the structure and parameters of the equations [8]. When the system dimension is low, we can gain a deep understanding of the vocational education system with the help of intuitive images. The qualitative theory of differential equations developed by Poincaré and its subsequent developments provided powerful mathematical tools for this.

Numerical calculation method

We quantitatively study nonlinear vocational education systems (2) using evolution equations. A generally effective method is to use a computer to perform numerical calculations to obtain approximate solutions to the equations. Mainframe computers allow experimental studies of vocational education systems based on mathematical equations. In this way, some intuitions about the non-trivial behavior of nonlinear vocational education systems can be obtained [9]. This method in turn helps us find theoretical breakthroughs.

Weak nonlinear vocational education system adopts linearization method to study the local behavior characteristics of vocational education system Experiments show that this method is very effective [10]. Assuming that f(x) is continuously differentiable at x0, at this time, we expand according to Taylor's formula to obtain the following information: f(x)=f(x0)+f(x0)(xx0)+h(x) f\left( x \right) = f\left( {{x_0}} \right) + f^\prime \left( {{x_0}} \right)\left( {x - {x_0}} \right) + h\left( x \right)

h(x) is a higher-order term. It is the nonlinear remainder. As long as xx0 is small enough, the nonlinear term can be ignored. We substitute equation (9) into equation (8) and omit the high-order term h(x) to obtain the linear approximation expression of the nonlinear vocational education system near point x0 x=f(x0)+f(x0)(xx0)=ax+b x^\prime = f\left( {{x_0}} \right) + f^\prime \left( {{x_0}} \right)\left( {x - {x_0}} \right) = ax + b

Where a = f′(x0), b = f(x0) − x0 f′(x0). Since the system satisfies the requirement of continuous smoothness, the linear vocational education system (10) can characterize the nonlinear vocational education system (8) with sufficient accuracy.

We discuss the nonlinear vocational education system equations (2) for the case of n = 2: x1=f1(x1,x2)x2=f2(x1,x2) \matrix{ {x^\prime 1 = {f_1}\left( {{x_1},{x_2}} \right)} \hfill \cr {x^\prime 2 = {f_2}\left( {{x_1},{x_2}} \right)} \hfill \cr }

We assume that f1, f2 is continuously differentiable around (x10, x20). Equation (11) can be used as a linear model to describe nonlinear systems. As long as the nonlinear system meets the requirements of continuity and smoothness near a certain point, it can be regarded as a weakly nonlinear system by expanding the evolution equation and omitting the nonlinear term. We use linear system theory to analyze the model to obtain an approximate description of the local behavior of nonlinear systems [11]. This method is called the local linearization of nonlinear systems. If the linear model alone cannot meet the requirements, we can use the nonlinear term as a disturbance factor. We make corrections to the linearization analysis results. This is called the linearized perturbation method. It is the main means of linear science to deal with nonlinearity.

Differential equation model of college students' professional quality

Suppose X(t) is an explicit occupational literacy function. X = X0 represents the minimum level of explicit professional quality that college students should achieve. Y(t) is the implicit professional quality function. Y = Y0 represents the minimum level of implicit professional quality that college students should achieve.

We mark x(t) = X(t) − X0, y(t) = Y(t) − Y0, then x(t) ≥ − X0, y(t) ≥ − Y0. According to the relationship between explicit professionalism and implicit professionalism, we can establish a differential equation model { dxdt=ax+bydydt=cy+dxy \left\{ {\matrix{ {{{dx} \over {dt}} = ax + by} \hfill \cr {{{dy} \over {dt}} = cy + dxy} \hfill \cr } } \right.

Where a, b, c, d is an average number. The modeling idea of system (1) is as follows: In the first equation dxdt=ax+by {{dx} \over {dt}} = ax + by , ax represents that the development speed of explicit occupational literacy is proportional to the existing level of explicit occupational literacy. by means that the development speed of explicit professional literacy is also proportional to the existing level of implicit professional literacy [12]. In the second equation dydt=cy+dxy {{dy} \over {dt}} = cy + dxy , cy represents that the development speed of implicit professionalism is proportional to the current level of implicit professionalism. dxy means that in improving explicit professionalism, it interacts with implicit professionalism to promote the development of implicit professionalism.

Qualitative Analysis

Suppose P(x, y) = ax + by, Q(x, y) = cy + dxy. From P(x, y) = 0, Q(x, y) = 0 we get that system (1) has two singularities O (0, 0) and A(x0, y0). where (x0,y0)=(ce,acbd) \left( {{x_0},\,{y_0}} \right) = \left( { - {c \over e},\,{{ac} \over {bd}}} \right) .

Theorem 1

The singularity O(0, 0) is an unstable system node (13).

Prove from Py|(0,0)=a,Py|(0,0)=b,Qx|(0,0)=0,Qy|(0,0)=c {\left. {{{\partial P} \over {\partial y}}} \right|_{\left( {0,0} \right)}} = a,{\left. {{{\partial P} \over {\partial y}}} \right|_{_{\left( {0,0} \right)}}} = b,{\left. {{{\partial Q} \over {\partial x}}} \right|_{_{\left( {0,0} \right)}}} = 0,{\left. {{{\partial Q} \over {\partial y}}} \right|_{_{\left( {0,0} \right)}}} = c that the linear approximation equation of system (1) at the singular point O(0, 0) is: { ξ1=aξ1+bη1η1=cη1 \left\{ {\matrix{ {\xi _1 = a{\xi _1} + b{\eta _1}} \hfill \cr {\eta _1 = c{\eta _1}} \hfill \cr } } \right.

The characteristic equation of equation (14) is λ2 − (a + c) λ + ac = 0. Solve λ1 = a > 0, λ2 = c > 0. So O(0, 0) is an unstable node of system (14). When the real part of the characteristic root of the system (14) is non-zero, the trajectory of the nonlinear system (13) at the singularity is consistent with the trajectory of the linear approximation equation (14) near the origin. So O(0, 0) is also an unstable node of system (13).

Theorem 2

The singularity A(x0, y0) is the saddle point of the system (13).

Proof It can be seen from Px|(x0,y0)=a,Py|(x0,y0)=b,Qx|(x0,y0)=acb,Qy|(x0,y0)=0 {\left. {{{\partial P} \over {\partial x}}} \right|_{\left( {{x_0},{y_0}} \right)}} = a,{\left. {{{\partial P} \over {\partial y}}} \right|_{\left( {{x_0},{y_0}} \right)}} = b,{\left. {{{\partial Q} \over {\partial x}}} \right|_{\left( {{x_0},{y_0}} \right)}} = {{ac} \over b},{\left. {{{\partial Q} \over {\partial y}}} \right|_{\left( {{x_0},{y_0}} \right)}} = 0 that the linear approximation equation of system (13) at singular point A(x0, y0) is: { ξ˙1=aξ1+bη1η˙1=acbξ1 \left\{ {\matrix{ {{{\dot \xi }_1} = a{\xi _1} + b{\eta _1}} \hfill \cr {{{\dot \eta }_1} = {{ac} \over b}\, \cdot {\xi _1}} \hfill \cr } } \right.

The characteristic equation of equation (15) is λ2ac = 0, and the solution is λ1=a+a2+4ac2>0 {\lambda _1} = {{a + \sqrt {{a^2} + 4ac} } \over 2} > 0 , λ2=aa2+4ac2<0 {\lambda _2} = {{a - \sqrt {{a^2} + 4ac} } \over 2} < 0 . Therefore, the singular point A(x0, y0) is the saddle point of the linear approximation equation (15) and is also a singular point.

Theorem 3

There is no limit cycle in the system (13).

Proof

There is no limit cycle for a quadratic system with only one node and one saddle point, so it can be seen from Theorems 1∼2 that the system (13) does not have a limit cycle.

Phase Diagram

According to Theorems 1∼3, the phase diagram of the differential equation model of college students' professional literacy can be obtained (see Figure 1).

Figure 1

Phase diagram of the differential equation model of college students' professional literacy

From Figure 1, we can analyze the development law of college students' professional quality.

The area below the x axis in Fig. 1 is the collapse area [13]. At this time y(t) = Y(t) − Y0 < 0. At this time, the implicit professional quality of college students cannot reach its minimum standard y = Y0. No matter how high the explicit professional quality of college students is, the overall level of the students' professional quality will become lower and lower with time.

OA above the x axis in Figure 1 is a very meaningful curve [14]. Above the curve OA, with time, the professional quality of college students can enter the state of simultaneous development of recessive professional quality and explicit professional quality. Below the curve OA is now x(t) = X(t) − X0 < 0. The explicit professional quality of college students does not reach its minimum standard X0. Although its implicit professionalism is slightly higher than its minimum standard Y0, the overall professionalism development of students still tends to collapse.

Observe the area enclosed by the upper part of the curve OA and the left side of the y axis. In this area, although the explicit occupational quality of college students is slightly lower than the minimum standard X0, because of the high implicit occupational quality of the students, strong willpower, and the spirit of hard work, the students strive to learn professional skills. Over time, the professional quality of the student enters into the state of simultaneous development of explicit and recessive professional quality.

Conclusion

Through analysis, we can see that implicit professional quality is the basis for developing college students' professional quality. If a person has bad behavior, weak will, and a high skill level, he cannot improve his professional quality. Therefore, colleges and universities should pay more attention to cultivating the hidden professional quality of college students. We can only promote the simultaneous development of recessive professionalism and explicit professionalism in this way.

Figure 1

Phase diagram of the differential equation model of college students' professional literacy
Phase diagram of the differential equation model of college students' professional literacy

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