This article examines a dynamic model of a computer information security system, taking into account factors that affect its stability. The stability of the protection system is analyzed, and the dependence of its behavior on parameters that affect the level of security is considered. The protection system is presented as a dynamic system in a mathematical sense, the state of which changes according to the given laws of evolution. Theoretical research was carried out using mathematical models formulated in the form of differential equations, with their subsequent implementation in the MATLAB/Multisim environment. The phase portraits of the system indicate its stability in the working range of parameters even under conditions of maximum external influence. The scientific novelty of the work lies in the presentation of a computer security assessment system in the form of a nonlinear dynamic model that takes into account the relationships between key security parameters. Unlike existing studies that focus on individual aspects (antivirus protection, cryptographic algorithms, access policies), the proposed approach allows for a quantitative assessment of the impact of a set of internal and external factors on the stability of the system, as well as determining the dynamics of its behavior in real time.
The problem of protecting information from unauthorized access and unwanted influence has existed for as long as the concept of valuable information itself. With the development of automated control systems, network technologies, and personal computers, the issue of data protection has taken on new relevance.
The problem of ensuring computer security has become increasingly complex in recent decades, as information technologies have deeply penetrated into all spheres of human activity. Modern information systems are exposed to a wide spectrum of threats, ranging from hardware and software failures to sophisticated cyberattacks aimed at disrupting the confidentiality, integrity, and availability of information. According to recent reports by ENISA and ISO/IEC 27001 standards, the number and diversity of threats are steadily increasing, requiring more advanced and adaptive protection mechanisms.
Traditional approaches to computer security mainly rely on separate tools such as antivirus software, firewalls, cryptographic protocols, and access control policies. While these methods remain effective for mitigating specific types of threats, they do not provide a holistic view of the system’s resilience, particularly
when complex interdependencies between parameters are considered. In this context, mathematical modeling of computer security systems as dynamic nonlinear systems has gained importance, since it allows for the quantitative evaluation of
0000-0002-0086-9131 (V. Akhramovych); 0009-0003-2787-8745 (V. Akhramovych); 0000-0001-8565-1117 (A. Ilyenko);
0009-0003-8518-6915 (O. Kryvokulska); 0000-0003-4051-1190 (V. Zhebka)
challenges, the present study aims to develop and analyze a nonlinear differential model of a
computer protection system, focusing on its stability and the dynamic impact of both internal and
external parameters. Such an approach not only complements existing studies but also provides a
universal framework for quantitative assessment of information security at the level of an
individual computer [
Factors contributing to increased information security requirements include: growth in data
volumes and their concentration in shared databases; multitasking and real-time operation;
automation of data exchange over long distances; the development of global networks and financial
systems, which have become targets of cybercrime [
Therefore, studying the impact of computer system parameters on the effectiveness of information protection is a task of both practical and theoretical importance.
Recent studies on computer and information security can be divided into two major directions. The
first group focuses on practice-oriented solutions, including protection against hardware and
software failures, recovery of lost data, and methods of countering malicious software and fraud
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Article [
Articles [
Work [
Further analysis [
The study [
The practical effectiveness of antivirus solutions is assessed in [
Article [
The growth in cyber threats has necessitated mathematical modeling of the behavior of
malicious objects, as emphasized in article [
In [
In [
Article [26] creates a model for assessing cyber threats in microcontrollers, which takes into account various factors of information influence and optimizes threat parameters. This approach is relevant because it expands the application of mathematical models to internal hardware components, which is related to your topic (hardware failures and malfunctions).
The work [27–31] proposes a security model for socio-cyber-physical systems that takes into
account not only technical but also social factors, including the influence of social engineering. The
mathematical basis is based on a modification of the Lotka–Volterra equations. This confirms the
trend toward using nonlinear system models to describe security in conditions of complex
interrelationships. The study in [28] presents a systematic review of modern ML/DL methods for
ensuring security in cloud environments. It emphasizes the importance of adaptive protection
systems capable of operating in dynamic environments, which has parallels with the approach of
modeling computer resilience as a nonlinear system.
Current research confirms that nonlinear mathematical models and machine learning methods are
the dominant approaches to assessing cybersecurity. Summarizing the results of the analysis of
literary sources [
The analysis of existing studies highlights both the achievements and the limitations of current approaches in assessing computer security. While practice-oriented works provide valuable recommendations for addressing software failures, malware protection, and access control, and mathematical models demonstrate the potential of nonlinear and stochastic methods, the literature still lacks an integrated approach that considers the computer as a dynamic nonlinear system with interdependent parameters. This gap defines the direction of the present research.
The aim of this work is to develop a methodology for assessing the level of security of computer systems, taking into account both external and internal influencing factors. To achieve this aim, it is necessary to: model a nonlinear system of computer protection; investigate its stability under conditions of both absence and presence of destabilizing factors.
In this study, the apparatus of nonlinear differential equations, fuzzy cognitive modeling, and fuzzy set theory was applied to describe weakly formalized processes of information security. The mathematical models were implemented in MATLAB and Multisim environments, which enabled the construction of block diagrams of the protection system, the simulation of external attacks, and the evaluation of the dynamic behavior of the system.
This methodological framework provides the foundation for the subsequent sections, where the proposed models are analyzed in detail, and their effectiveness in capturing the dynamics of computer security is evaluated.
The suggested model considers the impact of the following factors: data integrity control, access control, firewall operation, antivirus protection, auditing, backup, as well as factors related to software and hardware failures.
The system of nonlinear differential equations, which allows us to determine dynamics of the protection indicator depending on the parameters, has been obtained. Analytical solutions and numerical modeling demonstrated that even under weak nonlinearity it is possible to evaluate quantitative characteristics of the influence. Let’s imagine a linear model of the computer.
A dynamic system is considered to be specified if the coordinates defining its state and the
operator describing the evolution of the initial state over time are given. The mathematical
representation of such systems can be implemented in the form of discrete models, systems of
differential equations, partial differential equations, integral and integro-differential equations,
systems with impulse effects, hybrid systems, or Markov processes. To describe the level of
computer security, let us consider a linear model [
= I d Ac ( R + Ad )− (C d 2 + C d 1) I dt (1) where Z p—coefficient reflecting the impact of additional information protection measures (e.g., combating electromagnetic interference, physical protection, etc.); Q—coefficient reflecting data integrity and authenticity control (value 0…1); W—coefficient reflecting the separation of access to information (value 0…1); F—coefficient reflecting the operation of a firewall (packet filter)—used to control incoming and outgoing traffic (value 0…1); V—coefficient reflecting antivirus software (value 0…1) Z p 1—coefficient reflecting software component failures and malfunctions (value 0 or 1); Z k—coefficient reflecting hardware component failures and malfunctions (value 0 or 1) C v — coefficient reflecting the impact of personal data leakage speed; C k—coefficient reflecting the impact of the amount of personal data on its leakage; I d—coefficient reflecting user identification (value 0 or 1); A—coefficient reflecting user authentication (value 0 or 1); R—coefficient reflecting data backup (value 0…1); Ad—coefficient reflecting auditing (used for monitoring, logging) (value 0…1);. C d 1—coefficient reflecting the impact of security on information leakage. C d 2—coefficient reflecting the impact of system size on security.
Since the linear model does not take into account the actual relationships between parameters, the system (1) is supplemented with nonlinear components: (2): { dt dI − Z ( Z p + Q + W + F + V + Z p 1+ Z k )+(C v+ C k ) I + L2 I 2 + L3 I 3+ … dZ dt
= I d Ac ( R + Ad )− (C d 2 + C d 1) I + K 2 Z 2 + K 3 Z 3 … where L2, L3, etc. K2, K3, etc. are some linear operators.
We consider the nonlinearity of the system to be weak, which allowed us to find solutions for each equation of system (2) using the method of successive approximation, setting: I = I 1+ I 2 + I 3+ … , dI = 0 ; Z = Z 1+ Z 2 + Z 3+ … , dI = 0 ; (2) (3) (4) dI = 0 , dI = 0 and dZ = 0 , dZ = 0 , dt dt
I = I 0 sinωt , Z = Z 0 sinωt .
The following system of equations was obtained (3): − Z ( Z p + Q + W + F + V + Z p 1+ Z k )+(C v+ C k ) I − L2 ( I 02 sin2 ωt )− L3 ( I 03 sin3 ωt )− … = I d Ac ( R + Ad )− (C d 2 + C d 1) I + K 2 (Z 02 sin2 ωt )− K 3 (Z 30 sin3 ωt )− … The system of equations is presented in the following form (4) (Figure 1): { dt k= 2 dI ∞
= αZ + β 1 I − ∑ L k I 0k sin k ω t , dZ ∞
= β 2 I − ∑ K k Z 0k sin k ω t , dt k= 2 { dt dI dZ dt where α= Z p + Q + W + F + V + Z p 1+ Z k , β 1= C v+ C k ,
β 2= I d ( R + Ad )− (C d 2 + C d 1).
Figure 1 illustrates the results of numerical modeling of the proposed nonlinear protection system under different parameter configurations. To provide a more detailed interpretation, the figure is divided into several subplots (a–f), each highlighting specific aspects of the system’s dynamics. Figure 1 (a-f): Dependencies of the protection indicator on the components of model (4)
The results illustrated in Figure 1 confirm that the proposed nonlinear differential model adequately reflects the resilience of the computer protection system. In all considered cases, including scenarios with destabilizing factors and varying degrees of nonlinearity, the system eventually converges to a stable state. This demonstrates the robustness of the methodology and its suitability for quantitative evaluation of computer security under different operating conditions.
Further transformations using the elimination method led to the following system: dI = 1 ( d 2 Z + 1 ∑∞ (k K k Z 0k sink− 1 ω t cos ω t )) dt β 2 d t2 ω k= 2 (5) All found expressions (5) are substituted into the first equation of system (4): dI = 1 ( d 2 Z + 1 ∑∞ (k K k Z 0k sink− 1 ω t cos ω t )).
dt β 2 d t2 ω k= 2 or: dd2tZ2 − β 1 ddZt − α β 2 Z =
The characteristic equation took the form: λ2− β 1 λ− α β 2= 0. Provided that the discriminant is positive:
N (t )= − 1 ∞
∑ (k K k Z 0k sink− 1 ωtcosωt )+ ω k= 2 ∞ ∞ + β 1 ∑ ( K k Z 0k sink ωt )− β 2 ∑ ( L k I 0k sink ωt )
k= 2 k= 2
The final dependence of the protection indicator takes into account the influence of the main system parameters (Figure 2).
D = β 12 + 4 α β 2 > 0 ⇒ λ1,2= where
Z одн ( t )= c 1 ( t ) e where c '1 ( t ) , c '2 ( t ) found from the system: The solution to a homogeneous equation is defined as: β1+√β12+4 α β2 t 2 β1− √β12+4 α β2 t
2
Z одн ( t )= c 1 e + c 2 e .
Stable oscillatory trajectories were obtained. The general solution of the inhomogeneous equation was found by the method of variation of arbitrary constants: β1+√β12+4 α β2 t β1− √β12+4 α β2 t 2 2 + c 2 ( t ) e .
β + √β 12 + 4 α β 2 c '1 ( t ) 1 2
e c '1 ( t ) e β1+√β12+4 α β2 t
2 β1+√β12+4 α β2 t 2 + c '2 ( t ) e β1− √β12+4 α β2 t 2
= 0 , β − √β 12 + 4 α β 2 + c '2 ( t ) 1 2 e β1− √β12+4 α β2 t 2 = N ( t ) , t Z ( s )= ∫ ( N ( s )− e
t0 t –∫ ( N ( s )− e t0 − β1− √β12+4 α β2 s e
2 − β1− √β12+4 α β2 s e 2 β1+√β12+4 α β2 s
2 √β 12 + 4 α β 2 β1− √β12+4 α β2 s
2 √s β 12 + 4 α β 2
) ds– ) d s (6) (7) (8) (9) (10) (11)
A system of nonlinear differential equations was obtained, which allows determining the dynamics of the protection indicator depending on the parameters. Analytical solutions and numerical modeling showed the stability of the system even with weak nonlinearity. 3.1. Investigation of the stability of the protection system in a computer To assess stability, the analysis of the differential of the protection function was used (Figure 3).
Since the conditions for the existence and uniqueness of the solution to the Cauchy problem are satisfied, the trajectory of the system is the projection of the integral curve onto the phase space. The intersection of two different trajectories is impossible, which confirms the stability of the solutions.
The initial differential equation of the system: dd2tZ2 − β 1 ddZt − α β 2 Z = The stability of the CM protection system in the presence of influences on it has been investigated.
The behavior of CM resembles the behavior of a biological object. Assumption: the amplitude of
influences is nonlinear in time. Therefore, the following considerations are acceptable. A system
was considered in which the impact of harmful objects on the system and the immune response of
the system were modeled. It was assumed that the dynamics of the harmful object correspond to
the logistic model. The growth of a harmful infection depends on its initial status, the decline
caused by the immune response, and its own density effect, while the change in the immune
response depends on its initial status, natural decline, stimulation leading to an enhanced response,
and damage caused by the harmful object. Finally, the relative characteristic of the damaged organ
depends on the density of the harmful object and its natural degeneration. Then the dynamics of
the system is represented by differential equations [
The results of the program are shown in Figures 4–7.
Phase portraits of the protection system are shown in Figure 7. Phase portraits of the protection system are shown in Figure 8.
The results demonstrate the system’s capability to maintain stability even in the presence of destabilizing factors.
The proposed methodology allowed us to consider the computer protection system as a dynamic nonlinear system, where the security level is described by a system of differential equations. Unlike traditional approaches, which mostly focus on individual aspects (antivirus protection, cryptographic algorithms, or access policies), the proposed model integrates the influence of key factors, including authentication, auditing, firewall operation, backup, and software and hardware failures. The results of analytical and numerical modeling showed that even under conditions of weak nonlinearity, it is possible to obtain quantitative characteristics of the influence of factors on the level of security. The constructed phase portraits confirmed the stability of the system in the working range of parameters, as well as the absence of bifurcations, which indicates the stability of the proposed approach.
A comparison with existing works indicates the novelty of the results. In particular, studies [
The obtained phase portraits confirm the stability of the system and the absence of bifurcations in the working range of parameters (Figures 5, 7). This indicates the stability of the system.
Overall, the proposed methodology not only demonstrates the feasibility of representing computer protection as a dynamic nonlinear system but also opens avenues for extending this approach. The integration of fuzzy cognitive modeling and probabilistic methods makes it possible to incorporate stochastic and weakly formalized factors, including user behavior, social engineering techniques, and random hardware or software failures. This provides a foundation for creating hybrid models capable of adaptive decision-making in real time. Consequently, the presented results can be viewed as a step toward developing universal frameworks for assessing and improving the resilience of information security systems across both standalone computers and complex network infrastructures.
The study proposed and analyzed a methodology for assessing the level of computer security by representing the system as a set of nonlinear differential equations. The developed model integrates both external and internal factors of influence, including user authentication, auditing, firewall operation, antivirus protection, software and hardware failures, and data leakage processes. Numerical modeling carried out in MATLAB and Multisim demonstrated that the system maintains stability within its operational range, even under destabilizing influences.
The main findings of the study can be summarized as follows: 1. A system of nonlinear differential equations has been developed to assess the level of computer security. 2. It has been established that the protection system remains stable even under conditions of maximum external influences within the operating range of parameters. 3. The obtained graphical interpretations (phase portraits and dependencies of the protection index on parameters) allow quantitative assessment of the influence of factors on the level of security. 4. The proposed methodology can be used to monitor and improve the effectiveness of information protection systems in personal computers and network environments.
Unlike traditional approaches that address only isolated aspects of computer security, the methodology presented in this work allows for a holistic evaluation of the computer as a dynamic nonlinear system. This novelty ensures higher diagnostic accuracy and the possibility of identifying critical parameters that most significantly affect the level of protection.
The practical significance of the study lies in its potential application for real-time monitoring of computer security, early detection of cyberattacks, and prediction of their consequences. Moreover, the model can serve as a foundation for developing intelligent decision-support systems in the field of information security.
Future research should focus on extending the model to strongly nonlinear regimes, incorporating hybrid types of threats such as combined DDoS and social engineering attacks, and integrating the proposed methodology with machine learning algorithms to enable adaptive security management.
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