Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both "superelements" are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

Optimizing the combination of data-driven and model-based elements in hybrid reservoir computing

Hybrid reservoir computing combines purely data-driven machine learning predictions with a physical model to improve the forecasting of complex systems. In this study, we investigate in detail the predictive capabilities of three different architectures for hybrid reservoir computing: the input hybrid (IH), output hybrid (OH), and full hybrid (FH), which combines IH and OH. By using nine different three-dimensional chaotic model systems and the high-dimensional spatiotemporal chaotic Kuramoto-Sivashinsky system, we demonstrate that all hybrid reservoir computing approaches significantly improve the prediction results, provided that the model is sufficiently accurate. For accurate models, we find that the OH and FH results are equivalent and significantly outperform the IH results, especially for smaller reservoir sizes. For totally inaccurate models, the predictive capabilities of IH and FH may decrease drastically, while the OH architecture remains as accurate as the purely data-driven results. Furthermore, OH allows for the separation of the reservoir and the model contributions to the output predictions. This enables an interpretation of the roles played by the data-driven and model-based elements in output hybrid reservoir computing, resulting in higher explainability of the prediction results. Overall, our findings suggest that the OH approach is the most favorable architecture for hybrid reservoir computing, when taking accuracy, interpretability, robustness to model error, and simplicity into account.

Chaotic and Hyperchaotic Self-Oscillations of Lambda Diode Composed by Generalized Bipolar Transistors

This paper is focused on the investigation of self-oscillation regimes associated with very simple structure of lambda diode. This building block is constructed by using coupled generalized bipolar transistors. In the stage of mathematical modeling, each transistor is considered as two-port described by full admittance matrix with scalar polynomial forward trans-conductance and linear backward trans-conductance. Thorough numerical analysis including routines of dynamical flow quantification indicate the existence of self-excited dense strange attractors. Plots showing first two Lyapunov exponents as functions of adjustable parameters, signal entropy calculated from generated time sequence, sensitivity analysis, and other results are provided in this paper. By the construction of a flow-equivalent chaotic oscillator, robustness and long-time geometrical stability of the generated chaotic attractors is documented by the experimental measurement, namely by showing captured oscilloscope screenshots.

A simple no-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation

This paper proposes a simple no-equilibrium chaotic system with only one signum function as compared with the existing no-equilibrium chaotic ones with at least one quadratic or higher nonlinearity. The system has the offset boosting of three variables through adjusting the corresponding controlled constants. The resulting hidden attractors can be distributed in a 1D line, a 2D lattice, a 3D grid, and even in an arbitrary location of the phase space. Particularly, a hidden chaotic bursting oscillation is also observed in this system, which is an uncommon phenomenon. In addition, complex hidden dynamics is investigated via phase portraits, time series, Kaplan-Yorke dimensions, bifurcation diagrams, Lyapunov exponents, and two-parameter bifurcation diagrams. Then, a very simple hardware circuit without any multiplier is fabricated, and the experimental results are presented to demonstrate theoretical analyses and numerical simulations. Furthermore, the randomness test of the chaotic pseudo-random sequence generated by the system is tested by the National Institute of Standards and Technology test suite. The tested results show that the proposed system has good randomness, thus being suitable for chaos-based applications such as secure communication and image encryption.

New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields

This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement.

Synchronization of different fractional order chaotic systems with time-varying parameter and orders

In this paper, synchronization of time-varying fractional order chaotic systems, is introduced. Parameters of system play an important role in chaotic systems. A time-varying parameter is selected for chaotic systems, also orders of systems are considered as time-varying orders. A reliable controller is designed to synchronize two different fractional order chaotic systems based on Lyapunov stability theory on fractional order systems. Numerical simulations and practical implementation of proposed method are presented to verify the results.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

Dynamical tangles in third-order oscillator with single jump function

This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

n-scroll chaotic attractors from a first-order time-delay differential equation

In this paper, a novel first-order delay differential equation capable of generating n-scroll chaotic attractor is presented. Hopf bifurcation of the introduced n-scroll chaotic system is analytically and numerically determined. The bifurcation diagram and Lyapunov spectrum of the system are calculated and the results show that the system has a chaotic regime in a wider parameter range. Furthermore, period-3 behavior has been observed on the system. Circuit realizations of two-, three-, four-, and five-scroll chaotic attractors are also presented.

A simple Lorenz circuit and its radio frequency implementation

A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations. The simplicity of the circuit makes it suitable for radio frequency applications. The power spectrum of the observed oscillations displays a peak frequency as high as 930 kHz and significant power beyond 1 MHz.

Integrator-based circuit-independent chaotic oscillator structure

An integrator-based chaotic oscillator structure composed of three cascaded inverting, noninverting, and differential integrators is presented. The nonlinearity responsible for folding the trajectories is introduced by a single switching diode which is controlled by the output of the first integrator in the cascade. Chaotic behavior is verified on the functional level of the structure rendering it circuit-independent. A possible circuit realization is given and a canonical single-parameter-controlled ordinary differential equation capturing the qualitative dynamics of similar fourth-order integrator-based chaotic oscillators is proposed.