Keywords: Nonlinear Time Series and Identification, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Theory and Applications of Complex Dynamical Networks
Abstract: Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Fréchet discrete gradient method, regardless of the choice of the finite learning rate.

Keywords: Providing A Discussion Forum for Physics, Chaos and Control System Communities
Abstract: In this paper, a Model Predictive Control (MPC) scheme incorporating Integral Sliding Mode (ISM) is presented to deal with a class of uncertain nonlinear systems. Classical design of ISM control components requires the knowledge of the system nominal dynamics. In the present paper, it is assumed that the nominal dynamics is unknown, and that matched disturbances affect the system state evolution. In the proposed approach, two Deep Neural Networks (DNNs) are employed to estimate the drift dynamics and control effectiveness matrix of the system. The approximation errors introduced by the DNNs, along with the matched disturbances acting on the system, are compensated by the DNN-based ISM control. The weights of the DNNs are tuned relying on adaptation laws derived from Lyapunov analysis, providing theoretical guarantees. A solution to the considered control problem based on the joint use of the DNN-based ISM control and MPC is discussed. The proposal is assessed in simulation on the classical Duffing oscillator with satisfactorily results.

Keywords: Synchronization of Delay Systems, Chaotic Delay Systems, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Recent Advances in Control and Anti-Control of Chaotic or Complex Systems
Abstract: In this research, we explore the utilization of the time-delayed feedback method to stabilize unstable steady states and aperiodic orbits within the chaotic fractional-order Rössler Oscillator. Employing the time-delay feedback control algorithm, we identify specific parameter ranges enabling the successful stabilization of unstable equilibria, considering variations in both feedback gain and time delay. The method proves ineffective in stabilizing equilibria with an odd number of positive real eigenvalues. Nevertheless, strategic choices of feedback gain and time delay facilitate the stabilization of unstable periodic orbits, especially when the delay aligns with the target orbit's period, utilizing the Grünwald-Letnikov (GL) Characterization of fractional-order derivative and Matignon criterion. Unlike previous research works where Caputo and Riemann-Liouville characterization of fractional derivatives are used, we are using GL characterization because of its simplicity and ease of implementation and demonstrating the stability using the analytical and numerical analysis and plots of eigenvalues. Additionally, our analysis highlight the effectiveness of a sinusoidally modulated time delay, significantly expanding the stability region of steady states beyond the capabilities of the traditional time-delayed feedback scheme with a constant delay. Furthermore, the analysis of eigenvalues before and after applying the control strategy offers tangible insights into the system's stability dynamics.

Keywords: Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems
Abstract: In this paper, we develop a model reduction method for cooperative nonlinear systems via their variational systems. Inspired by scalable stability analysis of cooperative systems based on separable functions, we provide novel L1 differential observability and L∞ differential reachability functions and study them by using vector-valued functions. These vector-valued functions characterize not-so-important state variables. By truncating them, model reduction is performed, which preserves cooperativity and stability properties.

Keywords: Nonlinear Time Series and Identification
Abstract: This work brings together the moment matching approach based on Loewner functions and the classical Loewner framework based on the Loewner pencil in the case of bilinear systems. New Loewner functions are defined based on the bilinear Loewner framework, and a Loewner equivalent model is produced using these functions. This model is composed of infinite series that needs to be truncated in order to be implemented in practice. In this context, a new notion of approximate Loewner equivalence is introduced. In the end, it is shown that the moment matching procedure based on the proposed Loewner functions and the classical interpolatory bilinear Loewner framework both result in kappa-Loewner equivalent models, the main difference being that the latter preserves bilinearity at the expense of a higher order.

Keywords: Theory and Applications of Complex Dynamical Networks
Abstract: We propose a two-sided interconnection-based method for model order reduction of quadratic-bilinear systems. Exploiting the solution of an infinite-dimensional system of Sylvester-like equations we define the ``swapped'' nonlinear moment by a power series representation. We then consider together the ``direct'' moment and the ``swapped'' moment and propose two families of reduced-order models that achieve two-sided approximate moment matching. We illustrate the results of the paper on the problem of reducing the 1-D Burgers' equation.

Keywords: Theory and Applications of Complex Dynamical Networks, Nonlinear Time Series and Identification
Abstract: This paper studies the construction of nonlinear interpolants consisting of second-order equations in the Loewner framework. First, conditions are given for which a system of second-order equations interpolates sets of right and left tangential data mappings, which results in second-order tangential generalized controllability and observability mappings allowing for the direct treatment of systems in second-order form. Following this, a family of interpolants with second-order equation structure is given. The results are demonstrated by constructing a reduced order model of a two link robotic manipulator in second-order equation form.

Keywords: Recent Advances in Synchronization (and Observer-Design) for Chaotic or Complex Systems, Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Theory and Applications of Complex Dynamical Networks
Abstract: In this paper, we employ a backstepping approach to address the multi-agent flocking/consensus problem. We present a general procedure for coordinating a group of agents, which may exhibit non-linear and heterogeneous behaviors. Additionally, we derive closed-form expressions for the control inputs, accommodating systems with relative degrees up to 4. We also introduce a method to incorporate reference tracking into the framework. The validity of our proposed approach is demonstrated through simulation results.