A novel conservative system with hidden flows evolved from the simplest memristive circuit

Abstract

Over the past few decades, the research of dissipative chaotic systems has yielded many achievements in both theory and application. However, attractors in dissipative systems are easily reconstructed by the attacker, which leads to information security problems. Compared with dissipative systems, conservative ones can effectively avoid these reconstructing attacks due to the absence of attractors. Therefore, conservative systems have advantages in chaos-based applications. Currently, there are still relatively few studies on conservative systems. For this purpose, based on the simplest memristor circuit in this paper, a non-Hamiltonian 3D conservative system without equilibria is proposed. The phase volume conservatism is analyzed by calculating the divergence of the system. Furthermore, a Kolmogorov-type transformation suggests that the Hamiltonian energy is not conservative. The most prominent property in the conservative system is that it exhibits quasi-periodic 3D tori with heterogeneous coexisting and different amplitude rescaling trajectories triggered by initial values. In addition, the results of Spectral Entropy analysis and NIST test show that the system can produce pseudo-random numbers with high randomness. To the best of our knowledge, there is no 3D conservative system with such complex dynamics, especially in a memristive conservative system. Finally, the analog circuit of the system is designed and implemented to test its feasibility as well.

A 4D conservative chaotic system: dynamics and realization

“…If the divergence of a chaotic system equals zero, the system is conservative. If the divergence of a chaotic system is negative, the system is dissipative [13]. When the 4D CCS is substituted into the equation ∇•F(x) = divF(x) = ∂f 1 /∂x+∂f 2 /∂y+∂f 3 /∂z+∂f 4 /∂w, the obtained divergence is zero.…”

“…Therefore, DCSs have poor ergodicity which makes them vulnerable to both reconstruction and decryption when DCSs are used for encryption. However, conservative chaotic systems (CCSs) do not possess these disadvantages because the phase volume of CCSs is conserved and cannot form attractors in the phase space [13].…”

A conservative system based on a triangular wave memristor and its application in image encryption

“…( 16) shows that system (15) is conservative in the phase volume, now we discuss whether the system is HCS. According to [13], the system can be expressed as ( ẋ, ẏ, ż, ω)…”

“…However, one major issue has concerned us. In chaotic cryptosystems, the system itself is usually used as an important part of the secret key, and the chaotic attractors generated by dissipative systems can be reconstructed using the time-delay method [12], which in turn enables to crack the key and attack the cryptosys-tem [13]. In contrast, the conservative chaotic system is sensitive to the initial conditions and has the characteristics of random motion trajectory, but does not form chaotic attractors [14,15], thus having the excellent quality of resisting the reconstruction of the key.…”

A Conservative Memristive Chaotic System with Extreme Multistability and Its Application in Image Encryption

“…At present, a large number of dissipative memristive chaotic systems have been applied to image encryption and show good encryption performance. In a chaotic encryption scheme, chaotic attractors generated by dissipative systems can be reconstructed using time delay methods [15] to crack keys and encryption systems [16]. Conservative chaotic systems are more random to motion and do not form chaotic attractors, so conservative systems have the advantage of resisting key reconstruction [17].…”