Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay

Abstract

We consider the computational implementation of the algorithm for Lyapunov exponents spectrum numerical estimation for delay differential equations. It is known that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem which allows us to calculate the required parameters very efficiently. Therefore, we can only talk about the estimates of the characteristics in some sense close to the Lyapunov exponents. In this paper, we propose two methods of linearized systems solutions processing. One of them is based on a set of impulse functions, and the other is based on a set of trigonometric functions. We show the usage flexibility of these algorithms in the case of quasi-stable structures when several Lyapunov exponents are close to zero. The developed methods are tested on a logistic equation with a delay, and these tests illustrate the "proximity" of the obtained numerical characteristics and Lyapunov exponents.

Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays

“…The system dynamics of uncertain Rössler systems with unknown time delays are shown in Figure 3. From Figures 3 and 4, the dynamics of the system (31) can be suppressed to the pre-estimated bounds as expected by the proposed controller (32).…”

“…According to ( 16) and ( 19), the controlled dynamics will be bounded by and 4, the dynamics of the system (31) can be suppressed to the pre-estimated bounds as expected by the proposed controller (32).…”

“…Therefore, for applications to show whether a dynamical system is chaotic, it is most important to calculate the largest Lyapunov exponent (further denoted as LLE) in that dynamical system. If the LLE in this system is positive, the system should be chaotic [32,34].…”

“…According to (16) and (19), the controlled dynamics will be bounded by ψ 1 = 0.1324, ψ 2 = 0.1676, ψ 3 = 0.0988 while unmatched disturbances are happening. The adaptive memoryless sliding mode controller is also designed as (32), and the simulation results are shown in Figures 5 and 6. From Figures 5 and 6, the dynamics of the system (31) can be suppressed to the estimated bound by the proposed controller (32).…”

“…In this paper [25], an robust adaptive controller has been developed to asymptotically stabilize a class of uncertain nonlinear systems with the assumed unknown and bounded disturbances. In recent years, many researchers have found that systems are often affected by time delays, resulting in complex behaviors of the systems, such as loss of stability, bifurcation, and chaos [2,[25][26][27][28][29][30][31][32][33]. Among them, the one-dimensional dynamical system with delay will produce high-dimensional chaotic phenomena, which will affect the state responses of the systems [34,35].…”