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EN
This study investigates the complex dynamics of a five-dimensional artificial neural network (ANN) system with a hyperbolic tangent activation function. The objective is to analyze the chaotic behavior and validate multi-platform implementation of the proposed system. The methodology involves numerical simulations in MATLAB-Simulink and LabVIEW, followed by circuit design in Multisim for hardware feasibility verification. Through systematic parameter variation, the system exhibits rich dynamical regimes, including periodic oscillations and chaos. Lyapunov exponent analysis reveals a positive value (LE1 = 0.035) and Kaplan-Yorke dimension DKY = 2.488, confirming chaotic dynamics with fractional attractor geometry. Bifurcation diagrams demonstrate transitions between periodic and chaotic states as parameter b varies. The Multisim circuit simulation results demonstrate excellent agreement with MATLAB-Simulink and LabVIEW outputs, confirming the system's robustness and practical implementability across different platforms for potential applications in secure communications and cryptography.
PL
W niniejszym badaniu analizowana jest złożona dynamika pięciowymiarowego systemu sztucznej sieci neuronowej (ANN) z hiperboliczną funkcją aktywacji tangens. Celem jest analiza chaotycznego zachowania i walidacja wieloplatformowej implementacji proponowanego systemu. Metodologia obejmuje symulacje numeryczne w MATLAB-Simulink i LabVIEW, a następnie projektowanie obwodów w Multisim w celu weryfikacji wykonalności sprzętowej. Dzięki systematycznej zmianie parametrów system wykazuje bogate reżimy dynamiczne, w tym oscylacje o kresowe i chaos. Analiza wykładnika Lyapunowa ujawnia wartość dodatnią (LE1 = 0,035) i wymiar Kaplana-Yorke'a DKY = 2,488, potwierdzając chaotyczną dynamikę z frakcyjną geometrią atraktorową. Diagramy bifurkacji pokazują przejścia między stanami okresowymi i chaotycznymi w miarę zmiany parametru b. Wyniki symulacji obwodu Multisim wykazują doskonałą zgodność z wynikami MATLAB-Simulink i LabVIEW, potwierdzając niezawodność systemu i praktyczną możliwość wdrożenia na różnych platformach do potencjalnych zastosowań w bezpiecznej komunikacji i kryptografii.
EN
In this research work, we first obtain a new 3-D chaotic Lü system by modifying the dynamics of the classical Lü chaotic system (2002). Next, by introducing a state feedback to the new 3-D modified Lü chaotic system, we obtain a new 4-D hyperchaotic Lü system with a curve equilibrium. We carry out a detailed bifurcation analysis of the new 4-D hyperchaotic system with a curve equilibrium and describe the bifurcation transition diagrams and Lyapunov exponents diagrams. We also derive new multistability results of the new 4-D hyperchaotic Lü system with a curve equilibrium. For engineering applications, we provide an electronic circuit simulation of the proposed hyperchaotic Lü system using MultiSim 14.0. As a control application, we derive new results for the complete synchronization for a pair of new hyperchaotic Lü systems taken as the master and slave systems. We have used integral sliding mode control for the derivation of the synchronizing control law for the complete synchronization design for the new hyperchaotic Lü system. MATLAB simulations are provided to illustrate the main results of this research work.
EN
This study investigates the chaotic behavior in the rotational speed of internal combustion engines. High-precision measurements of engine rotational speed were taken at discrete intervals of 0.36 degrees with time measured to a precision of 41 nanoseconds. The data was analyzed using various techniques from chaos theory and nonlinear dynamics, including Lyapunov exponent calculations, phase space reconstruction, and power spectral density analysis. Results reveal that engine rotational speed exhibits complex, chaotic behavior across different operating conditions. Lyapunov exponents ranged from -0.004 to 0.024, indicating varying degrees of chaos from near-stability to strong chaotic behavior. The strongest chaos was observed at certain idle speeds, while full gas conditions showed milder but persistent chaotic characteristics. The study demonstrates that rotational speed fluctuations in internal combustion engines go beyond simple periodic or random variations, suggesting that traditional linear models may be insufficient for accurately predicting and controlling engine behavior. These findings have significant implications for engine design, control strategies, and diagnostics. The authors provide access to the original datasets and analysis code, encouraging further research and collaboration in this field. This work contributes to a deeper understanding of engine dynamics and may lead to the development of more sophisticated, nonlinear approaches to engine analysis and optimization.
PL
W prezentowanym artykule przedstawiono symulacyjne wyniki badań zachowania chaotycznego w wyodrębnionym silnie nieliniowym fragmencie systemu elektroenergetycznego. Do badania zachowań chaotycznych wykorzystano dwa graficzne wskaźniki: trajektorie fazowe oraz mapy Poincarego. Analizie poddano zamodelowany układ ferrorezonasowy stanowiący fragment sieci z pracującym w stanie jałowym transformatorem z uwzględnieniem doziemnych i międzyfazowych sprzężeń pojemnościowych. W badaniach uwzględniono odkształcenia zasilania w sieci zarówno składowymi harmonicznymi, jak i sub- i interharmonicznymi występującymi na przykład w miękkich elektroenergetycznych sieciach okrętowych.
EN
The article presents simulation results of research on chaotic behaviour in a separate, highly nonlinear fragment of the power system. Two graphical indicators were used to study chaotic behaviour: phase trajectories and Poincaré maps. The modeled ferroresonant system, which is a fragment of the network with a transformer operating in an idle state, was analyzed, taking into account ground and phase-to-phase capacitive couplings. The study considers the distortion of the power supply in the network by both harmonics and interharmonics occurring for example in soft ship power electricity.
EN
This paper analyzes the chaos and coexistence of attractors of a heavy gyroscope with parametric nonlinear damping. After having found the mathematical model of the dynamics, we used the multiple scale technique to look for secondary resonances. Subsequently, the amplitude of the harmonic oscillations is determined based on the harmonic balance technique. Impact of each of the system parameters is analyzed on the amplitudes and frequency of resonances. By applying order four Runge-Kutta algorithm, the different dynamics of the gyroscope are determined and analyzed. Subharmonic resonances and harmonic oscillations are derived from the limited development the basic equation solved numerically to investigate the dynamics and coexistence of the gyroscope attractors. The analysis of impact of each parameter on the existence and disappearance of coexisting attractors is done numerically.
EN
This article presents circuit realization of the pulse chaotic generator that can be used in digital modern telecommunication systems for masking and decrypt of the information. This generator based a classical Chua’s circuit. The results of computer simulation of a nonlinear element that realizes the chaotic behavior of the classical Chua's circuit are presented. For modelling was used a modern software MultiSim. Also, such basic results as chaotic attractor and time distributions of signals were obtained.
PL
W artykule przedstawiono realizację obwodu impulsowego generatora chaotycznego, który może być stosowany w nowoczesnych cyfrowych systemach telekomunikacyjnych do maskowania i deszyfrowania informacji. Generator ten bazuje na klasycznym obwodzie Chuy. Przedstawiono wyniki symulacji komputerowej nieliniowego elementu realizującego chaotyczne zachowanie klasycznego obwodu Chuy. Do modelowania wykorzystano nowoczesne oprogramowanie MultiSim. Uzyskano również takie podstawowe wyniki jak atraktor chaotyczny i rozkłady czasowe sygnałów.
7
Content available remote Noise sensitivity for stochastic heat and Schrödinger equations
EN
In this note, we consider the stochastic heat and Schrödinger equations, and show that, at time t, the onset of the chaos occurs on the scale of 1/t, and the Fourier spectrum of the solution is asymptotically Gaussian after centering and rescaling.
EN
In this work, we present a new four-dimensional chaotic hyperjerk system with a half-line of equilibrium points. In the chaos literature, it is well-known that chaotic systems with an infinite number of equilibrium points exhibit hidden attractors. Thus, we deduce in this research work that the new chaotic hyperjerk system has hidden attractors. We next study the new chaotic hyperjerk system for a dynamic analysis using bifurcation plots and Lyapunov Exponents (LE) diagrams.We exhibit that the new hyperjerk system has a special property of multistability with coexisting attractors. Using Multisim version 14.2, we carry out an electronic circuit simulation for the proposed 4-D chaotic hyperjerk system with a half-line of equilibrium points. Finally, as an application in control engineering, we apply backstepping control for achieving antisynchronization of a pair of new chaotic hyperjerk systems taken as master-slave systems, which has important applications in communication systems.
EN
Fine particulate matter (PM2.5) is a global environmental issue and a serious threat to human health. Reducing PM2.5 emissions is particularly crucial for China and India, which have the highest mortality rates associated with PM2.5 pollution. Prediction and modeling as a vital tool for making accurate PM2.5 concentration reduction policies require understanding the time series behavior of these pollutant. As the PM2.5 concentration is influenced by multiple pollutant and meteorological parameters, it is needed to investigate that if the PM2.5 has chaotic, periodic, or stochastic behavior. Hence, this paper investigates the dynamics of the daily time series of the PM2.5 in Beijing and Delhi during 2014–2021 implementing the most common nonlinear methods of chaotic data analysis: false nearest neighbors, average mutual information, local divergence rates, correlation exponent, and the recurrence plot (RP). Based on the results, the correlation exponent does not saturate with increasing m, and the value for maximal Lyapunov exponent is negative. Hence, the PM2.5 daily time series in Beijing and Delhi is non-chaotic. According to the RP analysis, the PM2.5 daily time series in Beijing and Delhi has white noise and auto-regressive pattern, respectively. Understanding these data characteristics will be useful for designing accurate environmental policy and pollution reduction programs, as it helps to improve the estimation power and the robustness of the model.
EN
In this research work, we investigate a new three-dimensional jerk system with three parameters in which one of the nonlinear terms is a sinusoidal nonlinearity. We show that the new jerk system has two unstable equilibrium points on the 𝑥-axis. Numerical integrations show the existence of periodic and chaotic states, as well as unbounded solutions. Consideration of the Poincaré sphere at infinity found no periodic states. We show that the new jerk system exhibits multistability with coexisting attractors. We also present results for the offset boosting of the proposed chaotic jerk system. Using MultiSim version 14.1, we design an electronic circuit for the new jerk system with a sinusoidal nonlinearity. As a control application, we design complete synchronization for the master-slave jerk systems using backstepping control technique. Simulations are presented to illustrate the main results of this research work.
EN
In this research work, we propose a new four-dimensional chaotic hyperjerk system with four quadratic nonlinearities. We carry out a detailed bifurcation analysis and derive conditions for the existence of a Hopf bifurcation for the new hyperjerk system. A linear analysis shows that there is only a unique trivial equilibrium state, whose stability depends solely on the parameter p. The only bifurcation possible is a Hopf bifurcation when p = 2. This is verified from bifurcation transition diagrams. We derive new results showing multistability and the existence of coexisting attractors for the new chaotic hyperjerk system. Using MultiSim, a new electronic circuit is designed for the new chaotic hyperjerk system with four quadratic nonlinearities. Finally, we present a control application for the proposed chaotic hyperjerk system with four quadratic nonlinearities. Using active backstepping control, we design a new controller that achieves complete synchronization for the master-slave chaotic hyperjerk systems with four quadratic nonlinearities.
EN
The main goal of this work is summed up in a univariate chaotic analysis of the runoff series using a topological approach. As such, nineteen series of daily runoff from eight large watersheds in northern Algeria were analyzed. Firstly, preliminary analyses of two traditional dynamic and metric approaches have been tested. In the dynamic approach, three algorithms have shown that the largest Lyapunov exponent for the series is positive, which supports the hypothesis of the existence of chaos. In the metric approach, the Grassberger and Procaccia algorithm clearly shows the saturation of the correlation dimension, which indicates the existence of deterministic dynamics for all studied stations. Secondly, the application of the topological approach in this study constitutes in itself a new contribution to the demonstration of chaos in hydrology by using the recurrence plot (RP) and the recurrence quantification analysis (RQA). As such, the RP structures of the runoff series seem to be more comparable to chaotic systems. In addition, RQA parameters give high values of determinism and laminarity, which supports the hypothesis of the existence of deterministic chaos. The presence of chaos in the runoff series can be identified by the existence of a strong probability of recurrence, indicating a fairly low level of complexity and fairly high predictability. Ultimately, the comparison of these approaches together made it possible to confirm the hypothesis according to which the process generating the runoff series is deterministic and suggests low-dimensional chaotic dynamics.
EN
This paper investigates the influence of the preamplifier stage on the acquisition of low-amplitude nonlinear dynamics signals. Five different amplifiers were compared with a gain of about 60 dB and a bandwidth range from 10 Hz to 1 MHz. The reconstruction of the attractor correlation dimension was investigated using three chaotic signal generators built according to different principles. The results of the conducted research indicate a significant influence of the amplifier's self-noise and the bandwidth of the frequency response on the estimated dimension. In addition, a different nature of the impact on the correlation dimension caused by preamplifiers based on operational amplifiers and transistor circuits was also noticed.
PL
W niniejszej pracy zaprezentowano wpływy stopnia przedwzmacniacza na akwizycję słabych sygnałów dynamiki nieliniowej. Porównano pięć różnych wzmacniaczy o wzmocnieniu 60 dB oraz paśmie przenoszenia z przedziału od 10 Hz do 1 MHz. Za pomocą trzech generatorów sygnału chaotycznego zbudowanych według różnych zasad, zbadano odtwarzanie się wymiaru korelacyjnego atraktora. Przeprowadzone badania wskazują na istotny wpływ szumów własnych wzmacniacza oraz na wpływ szerokości pasma przenoszenia na estymowany wymiar. Ponadto zauważono również odmienny charakter wpływu na wymiar korelacyjny spowodowany przez przedwzmacniacze bazujące na scalonych wzmacniaczach operacyjnych oraz na układach tranzystorowych.
EN
In this paper, we propose an efficient non-linear post-processing placed downstream of an image encryption scheme. It consists firstly to encrypt the plaintext image by the confusion-diffusion technique using chaotic functions. Then, the resulting image is added to a chaotically generated image having the same dimensions. Obtained result passed through the arctangent function to give the encrypted image. Computer simulations have proven the support that a nonlinear function can give an image encryption scheme. In addition, the performance measurements carried out prove the superiority of the proposed method towards existing algorithms in the literature from the point of view of histogram analysis, correlation test and key space.
PL
W tym artykule proponujemy wydajne nieliniowe przetwarzanie końcowe umieszczone poniżej schematu szyfrowania obrazu. Polega ona po pierwsze na zaszyfrowaniu obrazu tekstu jawnego techniką zamieszania-dyfuzji z wykorzystaniem funkcji chaotycznych. Następnie powstały obraz jest dodawany do chaotycznie generowanego obrazu o tych samych wymiarach. Otrzymany wynik przeszedł przez funkcję arcus tangens dając zaszyfrowany obraz. Symulacje komputerowe dowiodły, że funkcja nieliniowa może zapewnić schemat szyfrowania obrazu. Ponadto przeprowadzone pomiary wydajności dowodzą wyższości proponowanej metody w stosunku do algorytmów istniejących w literaturze z punktu widzenia analizy histogramu, testu korelacji oraz przestrzeni klucza.
15
Content available Spike patterns and chaos in a map-based neuron model
EN
The work studies the well-known map-based model of neuronal dynamics introduced in 2007 by Courbage, Nekorkin and Vdovin, important due to various medical applications. We also review and extend some of the existing results concerning β-transformations and (expanding) Lorenz mappings. Then we apply them for deducing important properties of spike-trains generated by the CNV model and explain their implications for neuron behaviour. In particular, using recent theorems of rotation theory for Lorenz-like maps, we provide a classification of periodic spiking patterns in this model.
EN
This study investigates Thomas’ cyclically symmetric attractor dynamics with mathematical and electronic simulations using a proportional fractional derivative to comprehend the dynamics of a given chaotic system. The three-dimensional chaotic flow was examined in detail with Riemann-Liouville derivative for different values of the fractional index to highlight the sensitivity of chaotic systems with initial conditions. Thus, the dynamics of the fractional index system were investigated with Eigenvalues, Kaplan-Yorke dimension, Lyapunov exponent, and NIST testing, and their corresponding trajectories were visualized with phase portraits, 2D density plot, and Poincaré maps. After obtaining the results, we found that the integer index dynamics are more complex than the fractional index dynamics. Furthermore, the chaotic system circuit is simulated with operational amplifiers for different fractional indices to generate analog signals of the symmetric attractor, making it an important aspect of engineering. The qualitative application of our nonlinear chaotic system is then applied to encrypt different data types such as voice, image, and video, to ensure that the developed nonlinear chaotic system can widely applied in the field of cyber security.
EN
The purpose of this paper is to introduce a new chaotic oscillator. Although different chaotic systems have been formulated by earlier researchers, only a few chaotic systems exhibit chaotic behaviour. In this work, a new chaotic system with chaotic attractor is introduced for triangular wave non-linearity. It is worth noting that this striking phenomenon rarely occurs in respect of chaotic systems. The system proposed in this paper has been realized with numerical simulation. The results emanating from the numerical simulation indicate the feasibility of the proposed chaotic system. More over, chaos control, stability, diffusion and synchronization of such a system have been dealt with.
EN
In this work, we report a new chaotic population biology system with one prey and two predators. Our new chaotic population model is derived by introducing two nonlinear interaction terms between the prey and predator-2 to the Samardzija-Greller population biology system (1988). We show that the new chaotic population biology system has a greater value of Maximal Lyapunov Exponent (MLE) than the Maximal Lyapunov Exponent (MLE) of the Samardzija-Greller population biology system (1988). We carry out a detailed bifurcation analysis of the new chaotic population biology system with one prey and two predators. We also show that the new chaotic population biology model exhibits multistability with coexisting chaotic attractors. Next, we use the integral sliding mode control (ISMC) for the complete synchronization of the new chaotic population biology system with itself, taken as the master and slave chaotic population biology systems. Finally, for practical use of the new chaotic population biology system, we design an electronic circuit design using Multisim (Version 14.0).
EN
In this work, a new 3-D modified WINDMI chaotic jerk system with exponential and sinusoidal nonlinearities is presented and its dynamical behaviours and properties are investigated. Firstly, some properties of the system are studied such as equilibrium points and their stability, Lyapunov exponents and Kaplan-Yorke dimension. Also, we study the new jerk system dynamics using numerical simulations and analyses, including phase portraits, Lyapunouv exponent spectrum, bifurcation diagram and Poincaré map, 0-1 test. Next, we exhibit that the new 3-D chaotic modified WINDMI jerk system has multistability with coexisting chaotic attractors. Moreover, we design an electronic circuit using MultiSim 14.1 for real implementation of the modified WINDMI chaotic jerk system. Finally, we design an active synchronization scheme for the complete synchronization of the modified WINDMI chaotic jerk systems via backstepping control.
PL
Stały wzrost mocy odbiorników nieliniowych w tym przekształtników powoduje pojawienie się w sieci zasilającej, zwłaszcza typu miękkiego (np. okrętowy system elektroenergetyczny), zaburzeń charakteryzujących się pojawieniem subharmonicznych i interharmonicznych. Artykuł prezentuje wyniki badań symulacyjnych wskazujących stopień wpływu zawartości tych zaburzeń w napięciu zasilania silnika indukcyjnego na prądy silnika. Wpływ powyższych zaburzeń badano dodatkowo z użyciem wskaźników zachowań chaotycznych. Do badań wykorzystano autorski model silnika indukcyjnego zrealizowany w środowisku ANSYS Electronics Desktop.
EN
The continuous increase in the power of non-linear receivers, including converters, causes the appearance in the supply network, especially of the soft type (e.g. the marine power system), disturbances characterized by the appearance of subharmonics and interharmonics. This article presents the results of simulation studies showing the degree of influence of the content of these disturbances in the supply voltage of an induction motor on its currents. The impact of the above mentioned disorders was additionally investigated with the use of indicators of chaotic behaviour. The original model of the induction motor, realized in the ANSYS Electronics Desktop environment, was used for the tests.
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