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In this paper a systematic procedure to compute the first integrals of the dynamics of a circuit with an ideal memristor is presented. In this perspective, the state space results in a layered structure of manifolds generated by first integrals, which are associated, via the choice of the initial conditions, to different exhibited behaviors. This feature turns out to be a powerful investigation tool, and it can be used to disclose the coexistence of attractors and the so called "extreme multistability," which are typical of the circuits with ideal memristors. The first integrals can also be exploited to study the energetic behavior of both the circuit and of the memristor itself. How to extend these results to the other ideal memelements and to more complex circuit configurations is shortly mentioned. Moreover, a class of ideal memristive devices capable of inducing the same first integrals layered in the state space is introduced. Finally, a mechanism for the loss of the ideality is conceived in terms of spoiling the first integrals structure, which makes it possible to develop a non-ideal memristive model. Notably, this latter can be interpreted as an ideal memristive device subject to a dynamic nonlinear feedback, thus highlighting that the non-ideal model is still affected by the first integrals influence, and justifying the importance of studying the ideal devices in order to understand the non-ideal ones. • A general systematic approach to study circuits with ideal memristors is presented • Coexistence of attractors is explained by means of the first integrals of the motion • Extreme multistability is unraveled using first integrals • Memristor's ideality is interpreted as the ability to create first integrals • A novel way to deal with non-ideal memdevices extending the classic ones is presented [ABSTRACT FROM AUTHOR]

The paper considers a wide class of circuits containing memristors, coupled and nonlinear capacitors and inductors, linear resistive multi-ports and independent voltage and current sources. A new method is proposed to analyze the invariants of motion and invariant manifolds of memristor circuits in this class. The method permits to show the existence of invariant manifolds, and analytically find their expressions, under very weak conditions, namely, when there exists the differential algebraic equations (DAE) describing the circuit and there is at least one hybrid representation of a linear resistive multi-port where memristors are connected. These conditions are satisfied by all memristor circuits in the class, except for pathological cases only. One salient feature of the method is that the dynamic problem of finding the invariant manifolds is brought back to a static problem involving the analysis of a linear resistive multi-port. In the one-memristor case, this boils down to the familiar problem of finding the Thévenin or Norton equivalent of a linear resistive one-port. The results in the paper improve previous work along several directions. The technique here developed is applicable to a broader class of memristor circuits including coupled nonlinear reactive elements and linear multi-port resistors. More importantly, it is applicable even if the state equation (SE) description of the circuit does not exist, as in the case where there are singular points as impasse points. Moreover, when the SE representation exists, additional assumptions introduced in previous work are not needed. The effectiveness of the method is illustrated via the application to selected examples. [ABSTRACT FROM AUTHOR]

• Limit Cycle Oscillations (LCOs) are investigated in a novel Pitch-Plunge model of the wing dynamics with nonlinear stiffness on the plunge displacement. • It is shown that the model can be represented as the feedback interconnection of a linear subsystem and a nonlinear one, a structure which has been thoroughly investigated in the literature for its connection with the celebrated Lur'e control problem. • The LCOs and the related bifurcation diagram as a function of the wind speed are obtained in an analytic, though approximate, way via Describing Function (DF) method. • A stable equilibrium point and two LCOs (one unstable and one stable) are predicted to coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a fold bifurcation is detected. • The comparison of the predicted bifurcation diagram with that obtained via numerical simulations makes it clear the quite good accuracy of predicted LCOs. The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur'e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small. [ABSTRACT FROM AUTHOR]

[Display omitted] • Comparing to the other research papers, this paper compares the performance of the oldest, newest, more minor and well-known algorithms on deterministic chaos generators in one massive and unique study. • Paper show that by precision tuning, the original chaotic series convert into short N periodic time series (PTS). Thus no randomness as usually understand is there. These series are then used instead of classics pseudorandom numbers with positive impact. • Paper reveal the clearly visible positive impact of PTS on evolutionary algorithms (EAs) dynamics, which is visible almost on all algorithms used in this paper. Compared with the same EAs with classic random generators. • Paper open the question of whether standard random (nonchaotic) processes are really necessary for algorithm dynamics and suggest relations between randomness in EAs and noise in dynamical system control and theory. • Paper open, sketch and suggest new ideas and strategies on how to understand algorithm dynamics as the discrete feedback dynamical systems. Random mechanisms including mutations are an internal part of evolutionary algorithms, which are based on the fundamental ideas of Darwin's theory of evolution as well as Mendel's theory of genetic heritage. In this paper, we debate whether pseudo-random processes are needed for evolutionary algorithms or whether deterministic chaos, which is not a random process, can be suitably used instead. Specifically, we compare the performance of 10 evolutionary algorithms driven by chaotic dynamics and pseudo-random number generators using chaotic processes as a comparative study. In this study, the logistic equation is employed for generating periodical sequences of different lengths, which are used in evolutionary algorithms instead of randomness. We suggest that, instead of pseudo-random number generators, a specific class of deterministic processes (based on deterministic chaos) can be used to improve the performance of evolutionary algorithms. Finally, based on our findings, we propose new research questions. [ABSTRACT FROM AUTHOR]