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Abstract
In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.
Highlights
In the past few decades, chaos has been established as an important branch of modern physics and mathematics due to its objectivity and universality as well as its wide application, for example, in cryptography [1], economics [2], electronic communication [3], and even in the popular field of neural network [4, 5] in recent years
In 2006, Lu and Liu analyzed the dynamics of the fractional Liu system and realized the circuit based on the approximation theory of the fractional operator [13]
We propose a new fourth-order commensurate fractional system based on an extension of the three-order Liu system, which is simpler than the system in [16]
Summary
In the past few decades, chaos has been established as an important branch of modern physics and mathematics due to its objectivity and universality as well as its wide application, for example, in cryptography [1], economics [2], electronic communication [3], and even in the popular field of neural network [4, 5] in recent years. We propose a new fourth-order commensurate fractional system based on an extension of the three-order Liu system, which is simpler than the system in [16]. It is a particular system with many interesting features that are not available in other systems that have been proposed: (1) it has nine terms, including two quadratic nonlinearities, a weak feedback term, and a constant.
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