# Mathematical Approaches Transform Cybersecurity from Protoscience to Science
## 4 Mathematical Approaches to Cybersecurity
### Catastrophe Theory  
Catastrophe theory, in mathematics, is a set of methods that are used to study and classify how a system can undergo sudden significant changes in behavior as one or more variables that control it constantly change. Catastrophe theory is generally considered a branch of geometry, as variables and the resulting behavior are depicted as curves or surfaces, and the official development of the theory is attributed mainly to the French topologist [PERSON]. Many physical systems can be described as potentially dependent on control parameters. If we do not include the time factor, the system must be static, and we must work with the Elementary Catastrophic theory. Otherwise, we must describe a system of dynamic equations that uses gradient methods from the bifurcation theory. According to [PERSON]'s book [73], 'The Change in the Phase Space of a System', the following variants may arise:  
A. When the parameter changes from the equilibrium position, a limit cycle is born (with a radius of the order \(\sqrt{\varepsilon}\) when the parameter's value differs from the bifurcation one by \(\epsilon\)). Equilibrium stability proceeds into a cycle, while equilibrium itself becomes unstable (Figure 4). B. In equilibrium, an unstable limit cycle dies; the area of attraction of the equilibrium position decreases to zero, after which the cycle disappears, and its instability is transferred to equilibrium. A. In equilibrium, an unstable limit cycle dies; the area of attraction that [PERSON] noticed, alongside A. [PERSON] and his students, even before the war (in 1939), proved that positions merge with the unstable, as described above, in addition to the loss of stability in the equilibrium. The methods of loss of stability are just as described in type A or B in the ordinary one-parameter families of systems with two-dimensional phase space; no other types of loss in stability occur. It was later proven that in phase space systems with a larger dimension, the loss of stability of the equilibrium positions with a change in one parameter occurred in one of the ways described. In the directions of all additional coordinate axes, the balance remains attractive with a change in the parameter [73]. If the equilibrium position is in a steady state or is a steady state in the real system, then when the parameter changes in cases A and B, the following phenomena can be observed. B. After the loss of equilibrium stability, the steady state is the oscillatory periodic regime; the amplitude of the vibrations is proportional to the square root of the super-criticality. The difference between the parameter and the critical value is that at which the equilibrium loses its stability [55]. Figure 4: Visualization of Swallowtail catastrophe (**a**,**b**) two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear; (**c**)—Butterfly catastrophe. Catastrophe theory is a mathematical approach to studying discontinuities or sudden shifts in behavior in complex systems. One of the most basic models in catastrophe theory is the cusp catastrophe model, represented by the equation \(V(x,a,b)=-ax-bx^{2}+\frac{x^{4}}{4}\). \(V(x,a,b)\) represents the potential function, \(x\) is the system's state variable, and \(a\) and \(b\) are control parameters. The potential function exhibits a local minimum, indicating the system's stable equilibrium state. When control parameters change, the equilibrium state can shift suddenly, leading to catastrophe. In the context of cybersecurity, \(x\) could represent the security level of a system, \(a\) could represent the attacker's capabilities, and \(b\) could represent the defender's efforts to protect the system. By analyzing the behavior of the potential function and its derivatives, it is possible to identify the conditions under which a sudden shift in the security state might occur, potentially leading to a catastrophic security breach.