Differential equation modelling was earlier used to discover better and understand various biological phenomena and social problems. We hope to understand the stability of the system and the Hopf bifurcation based on the characteristic roots of the linear system. Because group competitive sports require participants to have certain competitive skills, those who do not have sports skills but want to develop into activities must receive training and specific training. Therefore, based on the research background, the article proposes a time-lag group competitive martial arts activity model with a time lag effect. Through delay differential equation theory and Hopf bifurcation theory, the stability of the equilibrium point and the existence of periodic solutions generated by the Hopf bifurcation caused by the ‘instability’ of the equilibrium point are discussed. Finally, the theoretical results are simulated and verified with the help of MATLAB software.

#### Keywords

- group competitive martial arts activities
- balance point
- stability
- time lag
- Hopf bifurcation
- numerical simulation

#### MSC 2010

- 34A08

Differential equation modelling can be traced back to the early population model of Malthus and the predator–prey model of Lotka and Volterra. These models were once used to discover and understand various biological phenomena and social problems [1]. However, an overly simple model is difficult to accurately reflect the observed complex dynamic behaviours, such as periodic solutions. To this end, the model needs to be continuously improved. One method increases the equation's dimensionality continuously, but the cost is that it is difficult to estimate the parameters multiplied by actual data. Another way is to consider the time lag effect. The time lag effect is common in real problems. Time lag can correspond to the incubation period, delivery delay and response delay of the disease.

Moreover, simple time-delay differential systems often contain a wealth of complex dynamic behaviours. Some scholars have studied the gene regulation model of the time lag effect. Some scholars have conducted Hopf bifurcation research on Volterra predator–predator system with time delay [2]. Until the 1960s, the research on delayed differential systems mainly focused on stability, boundedness, asymptotics and equilibrium, periodic solutions, and oscillations of almost periodic solutions. Compared with ordinary differential systems, there is relatively little research on branch theory.

Hall was the first to study the local bifurcation of time-delay differential systems. He studied the existence of the central flow pattern of time-delay differential systems and the Hopf bifurcation theorem. However, Hall's theory is difficult to apply to practical problems. For time-delay differential systems with finite time delays, we hope to understand the stability of the system and the Hopf bifurcation based on the characteristic roots of the linear system. Since the characteristic equation of a linear system is a function of time delay, the characteristic root is also a function of time delay [3].

Moreover, the stability of the singularity will change as the time delay changes so that Hopf branches will occur near some critical values. Part of the time-delay differential equations will alternately appear from stable to unstable and then to stable as the singularity of the time-delay changes. This phenomenon is the so-called stable switching phenomenon.

Some scholars have proposed an ordinary differential equation model for group competitive sports activities [4]. Competitive activities here mean that participants must have certain skills rather than professional competitive sports activities. The article divides the human population into three categories. He set the total number of three types of people to 1, then the ordinary differential equation model of group competitive sports activities is as follows:
_{i}

Because group competitive sports require participants to have certain competitive skills, those who do not have sports skills but want to develop into activities must receive training and specific training. It must take a certain amount of time [5]. The above model does not consider the time lag effect. This paper discusses the stability of the equilibrium point and the existence of periodic solutions generated by the Hopf bifurcation caused by the ‘instability’ of the equilibrium point through the theory of delay differential equations and the Hopf bifurcation theory. Finally, the theoretical results are simulated and verified with the help of MATLAB software.

Considering that it takes a certain amount of time for an individual who participates in sports skills to transform into an individual who can participate in sports competitions, it is set as in Ref. [6]. Then the improved delay differential equation model can be expressed as:
_{i}

First calculate the balance point of the system. We make the right end of the system (1) zero, that is,
_{*}(^{*}, ^{*}), where

_{0}

Boundary equilibrium point
_{1}: _{1} > 0, the boundary equilibrium point is a non-negative equilibrium point:
_{1} = _{1} − _{2} = _{2} − _{1}, _{1} = _{1} − _{2} − _{3}^{−λτ} of the characteristic Eq. (4).

Hypothesis
_{3} > −_{2} is true, you can get:
_{0}(_{0}, 0) is unstable. If _{3} < −_{2} is true, when

_{3} < −_{2}

Suppose _{1}, the conclusions that can be obtained for any _{3} > −_{2}, _{0}(_{0}, 0) is unstable; b) If _{3} < −_{2}, _{0}(_{0}, 0) is locally asymptotically stable.

_{*}

^{*},

^{*})

_{2}_{2} + _{1} − _{2} + _{*}(^{*}, ^{*}): _{*} (^{*}, ^{*})

According to the above formula, the linearisation system at the positive equilibrium point _{*} (^{*}, ^{*}) can be obtained as:

_{2} _{3}_{1} − _{1} = _{1} − _{2} + _{0}

First, prove that the characteristic equation has a pair of purely imaginary roots _{0}, _{0} > 0 at the positive equilibrium point _{*}.

When _{*} is asymptotically stable [10]. The case of _{3}: _{1} − _{1} = _{1} − _{2} + _{0}, then Eq. (8) has a pair of purely imaginary roots _{0} that are conjugate to each other when _{j}

According to Eq. (10), we can get:
_{j}_{0}) satisfies Eq. (8), with the definition
_{j}_{*}. Substituting _{0} cos _{0} sin _{0} cos _{1}_{0}^{2} sin _{1}_{0}, _{j}

The stability of the time-delay system at _{j}_{*}(^{*}, ^{*}) [11].

We select a set of parameters as follows:
_{0} and obtains the waveform diagram and the plane phase diagram of the numerical solution of the system (Figures 1 and 2). The numerical simulation results at time lag _{0} are the same as above (see Figures 3 and 4).

MATLAB numerical simulation shows that the stability of the positive equilibrium has changed with the change of time delay [12].

There are certain critical points in the process of the time delay increasing from 0. At these critical points, the stability of the system equilibrium point always changes from stable to unstable; from zero to increase every time the critical value is passed from the positive equilibrium point. Expenditure periodic solution. Second, a numerical simulation of the time-lag group competitive sports activity model was carried out by MATLAB software. The theoretical results of the above model were verified by selecting a set of parameters.

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