Memristor and fractional-order derivatives are feasible options for constructing new systems with complex dynamics.
The memristor was postulated theoretically in 1971 by Chua as the fourth element in electronic circuits, but it was until 2008 that HP Labs introduced the first practical implementation.
This research presents a new fractional-order chaotic system based on a fractional- order memristor (fracmemristor). It is worth noting that this chaotic system based on a fracmemristor does not have any equilibrium points but generates infinitely many hidden chaotic attractors and other dynamical behaviors. Systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits, bifurcation diagrams, Lyapunov exponents, and riddled basins of attraction.
The memristor is a two-terminal element where the magnetic flux between its terminals is a function of the electric charge that passes through it. As it has many attractive characteristics (non-volatility, re-programmability, analog storage properties), is being explored for a wide variety of applications in different areas such as neural networks, secure communications, neuromorphic computation, memristor-based circuits, and so forth. Also, the memristor has been thoroughly studied as a nonlinear element to obtain chaotic systems. Those memristive systems can generate both self-excited and hidden attractors. Since a hidden attractor appears when its basin of attraction does not intersect with small neighborhoods of the unstable equilibrium, they are considered very important to understand unexpected and disastrous responses in engineering. Therefore, it is necessary to address new modeling approaches to gain insights about the complex nonlinear phenomena emerging in memristive systems. However, the memristor similar to the resistor, inductor, and capacitor is a real electrical element [2]. As a result, the nonlinear constitutive relation between the device voltage and current has been more efectively explained using the arbitrary-order version of its mathematical model [3,21].
In this manner, the fractional calculus theory can help us to describe the memory (storage) of memristive systems due to the mathematical definitions of the fractional-order derivatives are based on the hereditary and memory properties of their kernels. Therefore, the link between memristors and fractional-order is straightforward.
For details consult:
Muñoz-Pacheco, J. M. (2019). Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor. The European Physical Journal Special Topics,228 (10), 2185-2196.