This article proposes a nonlinear mental health evaluation model after combining the nonlinear finite element equation with the mental health education of teachers in colleges and universities. We take the “self-concept” in psychology as an order parameter to judge mental health. At the same time, logical thinking is used to guide college teachers to rethink the process of their psychological activities. Research has found that the nature of human psychology is nonlinear. It is feasible to use nonlinear analysis methods to improve the status quo of original data analysis in psychological and behavioral control research.

#### Keywords

- Nonlinear
- Finite element equation
- Colleges and universities
- Psychological education
- Response model

#### MSC 2010

- 34A34

The destruction of college teachers’ psychological contract is the recognition and evaluation of college teachers’ failure to fulfill the organization’s responsibility in the psychological contract. Research has shown that psychological contract destruction will harm college teachers’ emotions, attitudes, and behaviors. It, therefore, has become an important topic in the field of psychological contract research.

The research object of Stochastic Catastrophe Theory (SCT) is the system represented by stochastic differential equations. It describes and explains the sudden change in a mode that reflects a random disturbance system’s “equilibrium nature” [1]. In practice, changes in university teachers’ psychological contract level will always be disturbed by uncertainty, so this article uses SCT to analyze the internal mutation mechanism of psychological contract establishment and destruction. This provides a theoretical basis for predicting the occurrence of damage.

The dynamic equation of the system in the catastrophe model is shown in equation (1)

Here

Equation (3) represents the probability density function of a random process _{0} as its initial value [4]. This function has a limit _{0}) = ^{*} when

In the formula, _{a}^{*} does not depend on time

It is the limit probability density function ^{*} that reflects the statistical evolution characteristics of the random process in SCT, and the tool for reflecting the “equilibrium state” of the process is the differentiable common and anti-mode of ^{*}. They represent the points with the largest and smallest probability of random variables. Stochastic differential equations establish a connection between the potential function of a deterministic system and the static probability density function of a stochastic process [5]. Therefore, we can study the discrete changes of the “equilibrium state” in the statistical sense of the random process by studying the discrete changes of the mode and anti-mode of the deterministic function ^{*}.

Hysteresis, bimodality, and sudden jumps have their existence in the process of psychological contract destruction [6]. This provides a sufficient theoretical basis for using the catastrophe model to describe the dynamics of psychological contract establishment and destruction. The catastrophe theory can effectively analyze the occurrence of psychological contract destruction as a mutation phenomenon.

Among the traditional seven classic catastrophe models, the cusp model has been widely used due to its simple structure and rich content. In the cusp catastrophe model, we can typically find abrupt characteristics such as hysteresis, bimodality, and sudden jump, which correspond to some phenomena that appear in the process of psychological contract destruction [7]. Therefore, it is reasonable to use the cusp model to describe the dynamics of the psychological contract. In this way, the dynamic equation of the evolution of the psychological contract level of college teachers can be described by the cusp model shown in equation (6)

The psychological contract level _{1}, _{2},… _{k}

The parameter _{0}, _{1}, _{2},…_{k}_{0}, _{1}, _{1},…_{k}_{1}, _{2},… _{k}

Equation (6) is a deterministic ordinary differential equation. It does not consider the interference caused by unknowable random factors. However, in reality, the fluctuating process of university teachers’ psychological contract is a complex system. Its changes will be affected by personality and organizational atmosphere and many unforeseen internal and external factors [9]. In some cases, this influence cannot be ignored. Therefore, a reasonable Brownian motion disturbance term is introduced based on formula (6) to describe these random disturbances, and the following equation is obtained

Among them,

The first is to select the independent control variable _{1}, _{2},… _{k}_{0}, _{1}, _{2},…_{k}_{0}, _{1}, _{1},…_{k}

Existing empirical studies have shown that college teachers’ personality and organizational climate variables can effectively predict the level of the psychological contract. For example, extraversion, conscientiousness, and neuroticism in the Big Five personality can affect psychological contracts. A scholar’s longitudinal study of 106 new university teachers shows that due diligence can effectively regulate the relationship between psychological contract destruction and turnover intention, organizational loyalty, job satisfaction, and job performance. Some scholars have found through research that consistency, conscientiousness, and neuroticism have similar regulatory effects. On the other hand, the fair atmosphere can effectively regulate the degree of psychological contract destruction. For example, it is positively related to job satisfaction, role performance, and organizational citizenship behavior but negatively related to turnover intention [10]. The leader-member relationship in the interpersonal relationship atmosphere has also been proved to regulate the degree of psychological contract destruction effectively. Therefore, in this paper, personality variables (denoted as _{1}) and organizational climate variables (denoted as _{2}) as independent control variables are reflected in the model of this paper. Then there is

We determine the intrinsic quantitative relationship between the independent control parameters personality variable _{1} and organizational climate variable _{2} and the state variable

This article adopts the way of college teachers’ perception to measure the psychological contract. The questionnaire includes two dimensions: “The responsibility of the organization to university teachers” and “The responsibility of university teachers to the organization.” The internal consistency coefficients of the two dimensions in the study are 0.69 and 0.74, respectively. The internal consistency coefficient of the total questionnaire is 0.89. The personality measurement adopts the Big Five short personality questionnaire compiled by Goldberg. The content includes five sub-questions for neuroticism, responsibility, extroversion, pleasantness, and openness. Each sub-questionnaire has 5 items, a total of 25 items. The internal consistency coefficients of each sub-questionnaire were 0.59, 0.76, 0.49, 0.73, and 0.55, respectively [11]. The internal consistency coefficient of the total questionnaire is 0.69. The organizational climate questionnaire is formed by referring to the Litwin and Stringer classic scales and the organizational climate scale compiled by Xie Hefeng, conducting interviews with college teachers, and making semantic revisions. The questionnaire includes five dimensions: innovation atmosphere, fairness atmosphere, support atmosphere, interpersonal relationship atmosphere, and university teacher identity atmosphere. The internal consistency coefficients of each sub-questionnaire were 0.83, 0.82, 0.72, 0.80, and 0.76, respectively. The internal consistency coefficient of the total questionnaire is 0.94. For all the questionnaires, strict two-way translation and a Likert 5-point scoring method were used.

The research object is a teacher from a university in Wuhan, China. A total of 310 questionnaires were sent out, and 283 were returned. The return rate was 91.29%. Among them, there are 269 valid questionnaires, and the effective recovery rate is 86.77%. Among them, there are 166 men and 103 women. One hundred thirty-nine people are married, and 130 people are unmarried. There are 124 people with a bachelor’s degree and 145 people with a master’s degree or above. The average working life is 3.63 years. The average age is 29.3 years.

The appropriate estimation method is Cuspfit fitting software. The basic idea of this method is to find the best fitting value of the parameter _{0}, _{1}, _{2},…_{k}_{0}, _{1}, _{2},…_{k}

After collecting the above-mentioned valid data (_{0}, _{1}, _{2}, _{0}, _{1}, _{2}. Since the contribution levels of the two types of independent observation variables to the regularization factor and the divergence factor are not known in advance, the latter may be the binary function of the former or the unary function of the former [12]. The Cuspfit software itself also allows certain values _{1}, _{2}, _{1}, _{2} to be zero. For example, if the organizational climate does not affect the common factor _{2} = 0, and other factors can be inferred similarly. In this way, 16 groups of optional cusp fitting models are obtained, as shown in Table 1. The 11th group model (

In

And

List of cusp model fitting results.

Model | α_{0} |
α_{1} |
α_{2} |
β_{0} |
β_{1} |
---|---|---|---|---|---|

1 | −3.68 | 0 | 0 | −5 | −0.4 |

2 | −5 | 0 | 0 | 2.5 | 0 |

3 | −1.22 | 0.28 | 2.52 | −5 | 0.26 |

4 | −1.94 | 0.25 | 0.25 | −5 | 0 |

5 | −5 | 0.56 | 0 | 2.54 | 0.14 |

6 | −5.18 | 0 | 2.5 | −5 | 0.37 |

7 | −1.47 | 0.25 | 2.52 | −5 | 0 |

8 | −5 | 0 | 0 | 2.53 | −0.14 |

9 | −1.22 | 0 | 2.47 | −5 | 0 |

10 | −5 | 0.29 | 0 | 2.54 | 0 |

11 | −0.57 | 0 | 2.5 | −5 | 0.3 |

12 | −1.1 | 0.26 | 0 | −4.27 | 0 |

13 | −1.4 | 0 | 2.51 | −5 | 0 |

14 | −1.07 | 0.27 | 0 | −4.28 | 5.25 |

15 | −1.1 | 0 | 0 | −4.21 | 0 |

16 | −1.79 | 0.31 | 2.5 | −5 | 0.28 |

Model | β_{2} |
λ | τ | Par | AIC |

1 | 0 | 1.61 | 2.55 | 5 | 727 |

2 | −1.62 | 5 | 2.35 | 5 | 552 |

3 | 0 | 0.34 | 1.71 | 7 | 548 |

4 | −0.14 | 0.57 | 1.73 | 7 | 549 |

5 | −1.62 | 5 | 2.34 | 7 | 553 |

6 | 0.21 | −0.03 | 1.71 | 7 | 551 |

7 | 0 | 0.42 | 1.72 | 6 | 547 |

8 | −1.62 | 5 | 2.34 | 6 | 552 |

9 | −0.23 | 0.66 | 1.75 | 6 | 549 |

10 | −1.62 | 5 | 2.34 | 6 | 552 |

11 | 0 | 0.16 | 1.71 | 6 | 543 |

12 | 0 | 0.5 | 2.21 | 5 | 727 |

13 | 0 | 0.4 | 1.73 | 5 | 547 |

14 | 0 | 0.48 | 2.21 | 6 | 729 |

15 | 0 | 0.5 | 2.22 | 4 | 728 |

16 | −0.18 | 0.53 | 1.73 | 8 | 549 |

At the same time, according to the analysis in the first part, it can be seen that the equilibrium nature of the dynamic evolution of the psychological contract depends on the static probability density function of the model (9). In this way, the nonlinear mutation mechanism of the equilibrium nature of the system depends on the divergence mechanism of ^{*}. Assuming ^{*}′(

This shows that the distribution of roots of ^{*}′(^{3} +

This figure describes the mechanism by which the equilibrium nature of the evolution of the psychological contract changes discretely with the continuous change of parameters. It is based on the traditional classic cusp catastrophe model. The three-leaf surface is the equilibrium surface of the system, and the plane is the system’s control plane. For example, the divergence set of ^{*} in process (9) on the differentiable mode corresponds to the dashed line in the control plane. The acquisition logic of the ^{*}′(^{*}″(

Existing empirical research shows that there are nonlinear characteristics (hysteresis, sudden change, and bimodal phenomena) in the process of psychological contract destruction. Therefore, we need to use a random cusp catastrophe model to describe psychological contract evolution dynamics dynamically. The relationship between the psychological contract of college teachers and the antecedent variables in the evolution process shows both linear correlation and nonlinear relationship. When the data distribution of the current dependent variable is outside the divergent set of the model in this paper, the psychological contract and the antecedent variable are simply linearly correlated. When distributed within the divergent set, it shows a nonlinear mutation relationship.

#### List of cusp model fitting results.

Model | α_{0} |
α_{1} |
α_{2} |
β_{0} |
β_{1} |
---|---|---|---|---|---|

1 | −3.68 | 0 | 0 | −5 | −0.4 |

2 | −5 | 0 | 0 | 2.5 | 0 |

3 | −1.22 | 0.28 | 2.52 | −5 | 0.26 |

4 | −1.94 | 0.25 | 0.25 | −5 | 0 |

5 | −5 | 0.56 | 0 | 2.54 | 0.14 |

6 | −5.18 | 0 | 2.5 | −5 | 0.37 |

7 | −1.47 | 0.25 | 2.52 | −5 | 0 |

8 | −5 | 0 | 0 | 2.53 | −0.14 |

9 | −1.22 | 0 | 2.47 | −5 | 0 |

10 | −5 | 0.29 | 0 | 2.54 | 0 |

11 | −0.57 | 0 | 2.5 | −5 | 0.3 |

12 | −1.1 | 0.26 | 0 | −4.27 | 0 |

13 | −1.4 | 0 | 2.51 | −5 | 0 |

14 | −1.07 | 0.27 | 0 | −4.28 | 5.25 |

15 | −1.1 | 0 | 0 | −4.21 | 0 |

16 | −1.79 | 0.31 | 2.5 | −5 | 0.28 |

Model | β_{2} |
λ | τ | Par | AIC |

1 | 0 | 1.61 | 2.55 | 5 | 727 |

2 | −1.62 | 5 | 2.35 | 5 | 552 |

3 | 0 | 0.34 | 1.71 | 7 | 548 |

4 | −0.14 | 0.57 | 1.73 | 7 | 549 |

5 | −1.62 | 5 | 2.34 | 7 | 553 |

6 | 0.21 | −0.03 | 1.71 | 7 | 551 |

7 | 0 | 0.42 | 1.72 | 6 | 547 |

8 | −1.62 | 5 | 2.34 | 6 | 552 |

9 | −0.23 | 0.66 | 1.75 | 6 | 549 |

10 | −1.62 | 5 | 2.34 | 6 | 552 |

11 | 0 | 0.16 | 1.71 | 6 | 543 |

12 | 0 | 0.5 | 2.21 | 5 | 727 |

13 | 0 | 0.4 | 1.73 | 5 | 547 |

14 | 0 | 0.48 | 2.21 | 6 | 729 |

15 | 0 | 0.5 | 2.22 | 4 | 728 |

16 | −0.18 | 0.53 | 1.73 | 8 | 549 |

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