Dimension estimates for iterated function systems and repellers. Part II

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

A novel 4 dimensional hyperchaotic system with its control, Synchronization and Implementation

This paper presents a new hyperchaotic system which shows some interesting features, the system is 4-dimensional with 4 nonlinearities. An extensive numerical analysis has showed that the system has some interesting features and strange behaviors. The numerical analysis includes studying the effect of system parameters and initial conditions. Some of the important properties of the system with parameter set, in which the system is hyperchaotic, such as Lyapunov exponents and Lyapunov dimension, dissipation and symmetry are found and discussed. In the next part of our work, a tracking controller for the proposed system is designed and then a synchronization control system for two identical systems is designed. The design procedure uses combination of a simple synergetic control with adaptive updating laws to identify the unknown parameters derived basing on Lyapunov theorem. Hardware implementation based on microcontroller unit (MCU) board is proposed and tested and used to experimentally validate the designed control and synchronization systems. As an application, the designed synchronization system is used as a secure analogue communication system. Using MATLAB, Simulation study for the control and synchronization systems is presented. The simulation and experimental study have been showed excellent results.

An Innovative Way to Generate Hamiltonian Energy of a New Hyperchaotic Complex Nonlinear Model and Its Control

We are implementing a new Rabinovich hyperchaotic structure with complex variables in this research. This modern system is a real, autonomous hyperchaotic, and 8-dimensional continuous structure. Some of the characteristics of this system, as well as for invariance, dissipation, balance, and stability, are technically analyzed. Some other properties are also studied numerically, such as Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic actions. Hamiltonian energy is being studied and applying by using the innovative method. Via active control method, we inhibit our system’s hyperchaotic behavior. The new system’s hyperchaotic solutions are transformed into its unstable, trivial fixed point. The validity of the findings obtained is illustrated by an example of numerics and reenactment. Numerical results are plotted to display the variables of the state after and before the control to demonstrate that the control is being achieved.

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into account the problems of reliable numerical experiments in the context of the shadowing and hyperbolicity theories, experiments are carried out on small time intervals and for trajectories on a grid of initial points in the attractor’s basin of attraction.

Data-Rate Constrained Observers of Nonlinear Systems

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.