Keywords: Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Bifurcations in Chaotic or Complex Systems, Nonlinear Behavior of Electronic Systems
Abstract: This paper presents a novel method for calculating the positive largest Lyapunov exponent (LLE) using only SPICE-like programs. The LLE is a measure of the rate of divergence of nearby trajectories in phase space, and is an important indicator of chaotic behaviour. The proposed method calculates the LLE directly from systems represented in schematic circuits, inspired by interval extensions that exploit divergent trajectories to calculate the lower bound error. The exponent is obtained from the slope of the line derived from the lower bound error, using a differential amplifier to quantify it, along with a straightforward linear fit to the logarithm of the divergence of two circuits’ trajectories. This approach is unique as it requires only user skills in SPICE-like programs, unlike most other approaches, providing a simple, efficient, and accurate way to analyse and characterize chaotic dynamics. Results demonstrate that the suggested method successfully estimated the LLE for two well-known systems.

Keywords: Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Hybrid Systems and Chaos, Grazing Bifurcations
Abstract: The propagation of ellipsoids is a tool that can be used to study the stability and the convergence of non-linear systems with time. Recent works have presented a method to enclose the propagation of ellipsoids via nonlinear functions. While this method is effective in high dimensions, it requires the Jacobian matrix of the function to be invertible. In this paper, we propose to generalize this method to include non-invertible Jacobians, using degenerate ellipsoids. The generalized method is then applied to a nonlinear and/or synchronous hybrid system.

Keywords: Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Recent Advances in Control and Anti-Control of Chaotic or Complex Systems
Abstract: This paper addresses the state-feedback control problem for the class of state-polynomial discrete-time systems. The continuous-time polynomial nonlinear model is discretized by the second-order Runge-Kutta method. The Lyapunov theory and the exponential stability were employed to derive the conditions. The sum of squares formulation was used to check the constraints. Two approaches are presented, the first makes use of the Lyapunov function to recover the gain matrices. While the second formulation allows the design of rational state feedback control gains. We evaluated the impact of the step size used in the discretization process in the results. Numerical experiments were used to illustrate the potential of the proposed technique.

Keywords: Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Recent Advances in Synchronization (and Observer-Design) for Chaotic or Complex Systems
Abstract: Sylvester equations, their duals, and their nonlinear generalizations arise in a wide variety of systems and control problems. Here we provide an overview of classical and recent applications of these equations in output regulation, stabilization of cascaded systems, observer design, and model order reduction, for both linear and nonlinear systems. Similarities and differences between the linear and nonlinear cases are highlighted, with an emphasis on how design formulations and associated dual designs for linear systems may generalize to the nonlinear setting.

Keywords: Experimental Chaos and Synchronization, Chaos/Bifurcation Control in Physics, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems
Abstract: Electronic analog circuits are valuable platforms to experimentally address physical systems whose implementation would otherwise be unwieldy. Besides validating numerical predictions, electronic analogs of nonlinear systems are also ideal tools to study the effect of noise injection in realistic conditions and are often more scalable than numerical approaches. In this work, we address the effect of noise injection in an electronic analog of the Burridge-Knopoff mechanical model of earthquake faults. Noise is injected into the friction term of the mechanical system: interestingly, the presence of a sufficient level of noise can trigger the onset of chaos in otherwise periodic regimes.

Keywords: Limit Cycles in Networks of Oscillators, Experimental Chaos and Synchronization
Abstract: It is well-known that coupled oscillators may exhibit different types of collective behavior, like for example full synchronization, coordinated motion, cooperative behavior, among others. This is indeed the topic addressed in this paper. In particular, we briefly report on a special coordinated motion observed in a triplet of mechanical oscillators with Huygens' coupling: two oscillators achieve complete synchronization between them but they oscillate in anti-phase and with different amplitude with respect to the third oscillator. Each oscillator is equipped with a van der Pol term, in order to have self-sustained oscillations and furthermore, it is assumed that all the parameters of the oscillators are identical except for the parameter of the van der Pol term, which is different for one of the oscillators, ultimately resulting in oscillations with different amplitudes. Then, by using a transformation, the model of the system is reduced and the resulting model is analyzed using the Poincaré method of perturbation. It is analytically demonstrated that the coordinated motion above described exists and is asymptotically stable provided that the parameter discrepancy between the oscillators satisfies an specific condition. These results are further illustrated by numerical simulations and are experimentally validated.

Keywords: New Applications in Chaos Control, Chaos-Encrypted Signals, Optical Systems, Biological Systems, Power Converters, Economic Systems, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems
Abstract: When people purchase goods or services, their experiences can meet, exceed, or fall short of their expectations. This process of convergence creates a diverse market, where some customer’s expectations naturally align with the provided quality. The customers whose expectations converge to the quality offered are more likely to remain loyal to the firm in the long run. The customers whose beliefs diverge face a sequence of satisfying and dissatisfying experiences, which makes them likely to resign from using the product. This convergence can be facilitated by the firm’s responsivity, which is costly.

In this paper, we explore how responsivity can present an opportunity for a new market entrant to attract potentially loyal customers in order to deploy a niche left by a first-mover without causing costly competition. Responsivity acts as a niche marketing strategy, and we find how to use it in the optimal way.

Keywords: Theory and Applications of Complex Dynamical Networks, Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems
Abstract: This research formulates an advanced optimization framework for tackling the complex system inherent in two-echelon electric vehicle routing problems. We consider the intricacies of a complex road network, time windows, split deliveries, and battery-swapping stations. By leveraging an integrated approach that determines the optimal locations for intermediate depots and establishes efficient routing, our multi-objective model, solved using the epsilon constraints method in GAMS, adeptly navigates the challenges of urban delivery networks. The model adeptly manages larger goods volumes with split deliveries and enhances urban product distribution logistics using electric vehicles within the sophisticated GAMS/CPLEX environment. This reflects a synergy between sustainability and economic viability, focusing on the utilization of electric vehicles. The results underscore the pivotal role of sophisticated mathematical modeling in improving distribution efficiency and sustainability by optimizing the complex interplay between delivery routes, depot placement, and scheduling in dynamic urban environments.

Keywords: Bifurcations in Chaotic or Complex Systems
Abstract: This paper presents a new contribution in chaos theory using Bouallegue Equation System, this system is built of three blocks, These blocks are connected in cascade, the first block is a system of chaotic attractor,the second block is a system of fractal processes and the third block is a system of neural networks. In this work, we have generated a novel class of hyper chaotic attractors by coupling with neuron containing multi dendrites, each dendrite has its activation function. The activation function contains four parameters form,position of dendrite, parameter of equilibrium and polarity. Each dendrite can take four behaviors by configuration the parameter n , its structure contains four variables (n, p, q, ), so we call this capacity or this potential of activation function, a Variable Structure Model of Neuron (VSMN, for short). The impact of activation function in chaotic attractor generates a new classes of hyperchaos attractors with fluctuation, cutting, folding, amplifying in attractors. Simulations have demonstrated the validity and feasibility of the proposed method. Two results of applications using Bouallegue Equation System in fractal theory are also given to show the similarities in medicine illustrating the fractal chromosome and the fractal lung, and in biology displaying a fractal flower.

Keywords: Bifurcations in Chaotic or Complex Systems
Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected.

Keywords: Bifurcations in Chaotic or Complex Systems, Chaos/Bifurcation Control in Physics
Abstract: The properties of a thermally actuated MEMS sensor with a geometric nonlinearity are investigated. This MEMS sensor is a potential candidate to construct a neuromorphic acoustic sensor to mimic the functionality of the cochlea by controlling an Andronov-Hopf bifurcation with the thermal actuator. The resonance frequency of this sensor becomes tunable by introducing a geometric nonlinearity. With this nonlinearity the frequency response of the neuromorphic acoustic sensor is controllable by assigning a DC-voltage. Moreover, the effects of harmonic excitation on the MEMS sensors are analyzed. Here, the critical points of saddle-node bifurcations are approximated and the movement of the frequency with respect to an harmonic excitation is iscussed. The results are illustrated by numerical simulations.

Keywords: Chaos/Bifurcation Control in Chemical Engineering Reaction, New Applications in Chaos Control, Chaos-Encrypted Signals, Optical Systems, Biological Systems, Power Converters, Economic Systems, Chaos/Bifurcation Control in Physics
Abstract: It is known that periodic forcing of nonlinear flows can result in a chaotic response under certain conditions. Such non-periodic and chaotic solutions have been observed in simulations of heterogeneous gas flow in a pipeline with periodic, time-varying boundary conditions. In this paper, we examine a proportional feedback law for boundary control of a parabolic partial differential equation system that represents the flow of two gases through a pipe. We demonstrate that periodic variation of the mass fraction of the lighter gas at the pipe inlet can result in the chaotic propagation of gas pressure waves, and show that appropriate flow control can suppress this response. We examine phase space solutions for the single pipe system subject to boundary control, and use numerical experiments to characterize conditions for the controller gain to suppress chaos.