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A dynamic model of biogeochemical carbon cycles in swamp ecosystems is proposed. There are fast and slow biogeochemical cycles. Fast cycles operate in the biosphere and include photosynthesis, vegetative growth, and decomposition. Swamp ecosystems are one of the significant reservoirs of biogeochemical cycles. It is known that huge reserves of carbon in the form of slightly decomposed organic matter are preserved in the swamps. They are active sources of methane and carbon dioxide runoff from the atmosphere. Mathematical models of dynamic processes in ecology can be divided into two categories: quantitative and qualitative. Quantitative models, as a rule, are aimed at solving problems of predicting numerical indicators of the dynamics of real systems. They must be modified to consider specific climatic conditions, special types of swamp vegetation, and hydrological regime for their successful application. Qualitative models written as systems of differential equations assume the finding of singular points, their classification and study for stability, the construction of phase portraits, etc. Such models rarely lend themselves to quantitative verification, but provide important knowledge and understanding of processes in nature. A qualitative study of the system of ordinary differential equations describing carbon cycles is carried out, the types of singular points are investigated, integral curves and phase portraits are constructed., Предложена динамическая модель биогеохимических циклов углерода в болотных экосистемах. Существуют быстрые и медленные биогеохимические циклы. Быстрые циклы действуют в биосфере и включают фотосинтез, вегетативный рост и разложение. Болотные экосистемы являются одним из значимых резервуаров биогеохимических циклов. Известно, что в болотах законсервированы огромные запасы углерода в виде слаборазложившегося органического вещества и они являются активным источникам метана и стоком углекислого газа из атмосферы. Математические модели динамических процессов в экологии можно разделить на две категории: количественные и качественные. Количественные модели, как правило, направлены на решение задач прогнозирования числовых показателей динамики реальных систем. Чтобы успешно применять количественные модели, они должны быть модифицированы с учетом специфических климатических условий, особых видов болотных растений, растительности и гидрологического режима. Качественные модели, записанные как системы дифференциальных уравнений, предполагают нахождение особых точек, их классификацию и исследование на устойчивость, построение фазовых портретов и т.п. Такие модели редко поддаются количественной проверке, но дают важные знания и понимание процессов в природе. Проведено качественное исследование системы обыкновенных дифференциальных уравнений, описывающих циклы углерода, исследованы типы особых точек, построены интегральные кривые и фазовые портреты.

The question of targeted control over trajectories of systems of differential equations encountered in the theory of genetic and neural networks is considered. Examples are given of transferring trajectories corresponding to network states from the basin of attraction of one attractor to the basin of attraction of the target attractor. This article considers a system of ordinary differential equations that arises in the theory of gene networks. Each trajectory describes the current and future states of the network. The question of the possibility of reorienting a given trajectory from the initial state to the assigned attractor is considered. This implies an only partial control of the network. The difficulty lies in the selection of parameters, the change of which leads to the goal. Similar problems arise when modeling the response of the body’s gene networks to serious diseases (e.g., leukemia). Solving such problems is the first step in the process of applying mathematical methods in medicine and pharmacology.

One of the classical and difficult problems in the theory of planar differential systems is to classify their centers. Here we classify the global phase portraits in the Poincaré disk of the class continuous piecewise differential systems separated by one straight line and formed by two cubic Hamiltonian systems with nilpotent bi-center at (± 1, 0). The main tools for proving our results are the Poincaré compactification, the index theory, and the theory of sign lists for determining the exact number of real roots or negative real roots of a real polynomial in one variable.

The current work is devoted to studying the dynamical behavior of the Sakovich equation with beta derivatives. We announce the conditions of problem parameters leading to the existence of periodic, solitary, and kink solutions by applying the qualitative theory of planar dynamical systems. Based on these conditions, we construct some new solutions by integrating the conserved quantity along the possible interval of real wave propagation in order to obtain real solutions that are significant and desirable in real-world applications. We illustrate the dependence of the solutions on the initial conditions by examining the phase plane orbit. We graphically show the fractional order beta effects on the width of the solutions and keep their amplitude approximately unchanged. The graphical representations of some 3D and 2D solutions are introduced.

We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields. This work has been realized thanks to the Ministerio de Ciencia e Innovación (PID2019-104658GB-I00 and PID2021-122954-I00 grants), the grant Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M) and also the Agència de Gestió d’Ajuts Universitaris i de Recerca (2021 SGR 00113 and 2021 SGR 01039 grants).

The present work is attentive to studying the qualitative analysis for a nonlinear strain wave equation describing the finite deformation elastic rod taking into account transverse inertia, and shearing strain. The strain wave equation is rewritten as a dynamic system by applying a particular transformation. The bifurcation of the solutions is examined, and the phase portrait is depicted. Based on the bifurcation constraints, the integration of the first integral of the dynamic system along specified intervals leads to real wave solutions. We prove the strain wave equation has periodic, solitary wave solutions and does not possess kink (or anti-kink) solutions. In addition, the set of discovered solutions contains Jacobi-elliptic, trigonometric, and hyperbolic functions. This model contains many kinds of solutions, which are always symmetric or anti-symmetric in space. We study how the change in the physical parameters impacts the solutions that are found. Numerically, the behavior of the strain wave for the elastic rod is examined when particular periodic forces act on it, and moreover, we clarify the existence of quasi-periodic motion. To clarify these solutions, we present a 3D representation of them and the corresponding phase orbit.

In the present work, nonlinear dynamics of a simply-supported fluid-conveying composite pipe subjected to axial tension in sub- and super-critical regimes are investigated. The tension acting on the pipe is assumed to comprise an average component and a harmonically changing disturbance component. More specifically, the nonlinear equation of motion is derived based on the Euler-Bernoulli beam theory considering the von Karman nonlinearity for large deformation assumption, which is then truncated via the Galerkin's approach. In addition, the Kelvin-Voigt model is adopted on the pipe materials with visco-elastic structural damping. Afterwards, parametric studies are performed to evaluate the effects of the axial tension and fiber orientation angle on pre- and post-buckling frequencies, critical fluid velocities and static buckling displacements. The global dynamic behaviors of the pipe are studied through the forms of bifurcation diagrams, phase portraits, Poincare maps, power spectral density (PSD) diagrams and linear instability maps, in both sub- and super-critical regimes. The results show that, based on symmetric layers, pipes with smaller fiber orientation angles have larger critical divergence fluid velocities but smaller static buckling displacements. Some interesting dynamical phenomena, including periodic, multi-periodic, quasi-periodic and chaotic motions, can be observed with different fiber orientation angles. Moreover, the system shows ‘soft spring’ characteristic in super-critical regime, compared with the ‘hard spring’ characteristic in sub-critical case.

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles.

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ for the case $$d>0$$ , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case $$d

In this study, a new variable-order fractional chaotic neural network with frequency effect is proposed based on Hopfield Neural Network Under Electromagnetic Radiation, which is used to model brain functions in the literature. The numerical solution of this proposed system is carried out by the Grunwald–Letnikov (G–L) method, and time series and phase portraits are presented. In addition, the chaotic behavior is analyzed by Lyapunov exponent analysis according to the frequency parameter used for the variable-order function. A Pseudo-Random Number Generator (PRNG) is designed for data security applications using the obtained state variables and the usability of the PRNG is demonstrated with the NIST-800-22 test. Afterward, a data-hiding algorithm for personal data is presented for multi-channel biomedical signals based on the designed PRNG. With this algorithm, implementations are made for EEG, ECG, and EMG signals. It is shown in time and frequency domains that the method is usable for data-hiding applications. In addition, the effectiveness of the newly developed algorithm is proven with statistical methods.

In this paper, the nonlinear dynamics of the biological system modeled by the fractional incommensurate order Van der Pol equations are investigated. The stability of the proposed fractional non-autonomous system is analyzed by varying both the fractional order derivative and system parameters. Moreover, very interesting phenomena such as symmetry, multi-stability and coexistence of attractors are discovered in the considered biological system. Numerical simulations are performed by considering the Caputo fractional derivative and results are reported by means of bifurcation diagrams, computation of the largest Lyapunov exponent, phase portraits in 2D and 3D projections.

In this chapter we will study global bifurcations corresponding to the appearance of homoclinic or heteroclinic orbits connecting hyperbolic equilibria in continuous-time dynamical systems. First we consider in detail two- and three-dimensional cases where geometrical intuition can be fully exploited. Then we show how to reduce generic n-dimensional cases to the considered ones plus a four-dimensional case treated in Appendix A.

In this chapter we introduce and discuss the following fundamental notions that will be used throughout the book: topological equivalence of dynamical systems and their classification, bifurcations and bifurcation diagrams, and topological normal forms for bifurcations. The last section is devoted to the more abstract notion of structural stability. In this chapter we will be dealing only with dynamical systems in the state space X = ℝ n .

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied.

Пояснювальна записка до магістерської дисертації «Програмно-алгоритмічне забезпечення інформаційно-вимірювальних систем аналізу часових рядів», складає 86 сторінок тексту, 27 ілюстрацій, 28 таблиць, 19 бібліографічних посилань. Інформаційно-вимірювальні системи випадкових сигналів Предметом дослідження є методи фазових портретів Кластеризація часового ряду по певним критеріям Для виконання поставленої цілі використовувалися: середа розробки Python, експериментальні дані. В результаті було досліджено: кластеризацію фазових портретів сигналів, отриманих експериментальним шляхом фазові портрети сигналів, отриманих експериментальним шляхом, а також їх класифікація. The explanatory note to the master's thesis "Software-algorithmic support of information and measurement systems of time series analysis" consists of 86 pages of text, 27 illustrations, 28 tables, 19 bibliographic references. Information and measurement systems of random signals The subject of research is the methods of phase portraits Clustering of time series according to certain criteria To fulfill the set goal, the Python development environment and experimental data were used. As a result, the following was investigated: clustering of phase portraits of signalsobtained experimentally phase portraits of signals obtained experimentally, as well as their classification.

Resonant elements can generate small amounts of energy that make them pertinent for feeding miniaturized accelerometers with the energy needed. Suitable oscillator candidates are Helmholtz resonators, which have been, for a long time, analyzed and designed within the context of linear vibration. This study focuses on extracting nonlinear characteristics of a dual Helmholtz resonator (HR), with a neck-cavity–neck-cavity configuration, mounted on an acoustic waveguide with harmonically oscillating pressure. The mathematical model used for describing the resonator embraces inherent nonlinear air stiffness and the damping nonlinearity of hydrodynamic origin. Numerical solutions for the resonator’s nonlinear oscillations are obtained. Bifurcation diagrams are produced, indicating that the dual HR behaves in a deterministic fashion within the engineering practical limits. Phase portraits are drawn for the system, showing a quasi-periodic motion. Frequency response curves (FRC) are found to shift to the left at the lower resonant frequency indicating a softening behavior. FRC keep generally symmetric curves at the higher resonant frequency indicating a mostly linear behavior.

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

The aim of this work is to analytically investigate the nonlinear dynamic behaviors of a proposed reduced Lorenz system based on q-deformations. The effects of varying the new q-deformation parameter on the dynamical behaviors of the system along with the induced bifurcations of fixed points are explored. In particular, the codimension-one bifurcation analysis is carried out at interior fixed point of the q-deformed system. Explicit conditions for the existence of pitchfork and Neimark–Sacker bifurcations are obtained. Numerical simulations are performed to confirm stability and bifurcation analysis in addition to investigate the effects of variations in system parameters. The changes in system dynamics are explored via the bifurcation diagrams, phase portraits and time series diagrams. Moreover, the quantification of system complex behaviors is depicted through the maximal Lyapunov exponent plots. A cascaded version of the model is proposed to boost its complex dynamics. Then, a chaos-based image encryption scheme, relying on a proposed key-distribution algorithm, is introduced as an application. Finally, several aspects of security analysis are examined for the encryption system to prove its efficiency and reliability against possible attacks.

A quasi-linear Mathieu-type equation is investigated by means of the averaging method in the neighborhood of the main resonance. All possible types of phase portraits are found, and steady-state solutions are identified. The periodic solutions of the primary equation that correspond to the steady-state solutions of the averaged equation are determined. Analytical expressions for probabilities of the dissipation-induced capture into feasible steady-state solutions are obtained.

Since laminar separation bubbles are neutrally shaped on the suction side of full-span wings in low Reynolds number flows, a roughness element can be used to improve the performance of micro aerial vehicles. The purpose of this article was to investigate the leading-edge roughness element’s effect and its location on upstream of the laminar separation bubble from phase portrait point of view. Therefore, passive control might have an acoustic side effect, especially when the bubble might burst and increase noise. Consequently, the effect of the leading-edge roughness element features on the bubble’s behavior is considered on the acoustic pressure field and the vortices behind the NASA-LS0417 cross-section. The consequences express that the distribution of roughness in the appropriate dimensions and location could contribute to increasing the performance of the airfoil and the interaction of vortices produced by roughness elements with shear layers on the suction side has increased the sound frequency in the relevant sound pressure level (SPL). The results have demonstrated that vortex shedding frequency was increased in the presence of roughness compared to the smooth airfoil. Also, more complexity of the phase portrait circuits was found, retrieved from velocity gradient limitation. Likewise, the highest SPL is related to the state where the separation bubble phenomenon is on the surface versus placing roughness elements on the leading edge leads to a negative amount of SPL.

In (Euzebio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincare disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

A new 3D chaotic system with only one nonlinear term is constructed in this paper, which has conservative and dissipative features respectively by choosing different parameters. The complex dynamic behaviors of the system are analyzed theoretically and numerically by using nonlinear analytical tools, such as Lyapunov exponents, phase portraits, and bifurcation diagrams. In a certain range of parameters, the new system is a non-Hamiltonian conservative chaotic system, which can produce infinitely many coexisting conservative flows. With the variation of parameters, the divergence of the system is less than zero, and the system can also generate hidden and self-excited attractors with multi-scroll. To verify the feasibility of the new system, it has been implemented by field-programmable gate array; phase diagrams obtained in the experiment are consistent with the simulation of the data.

Studies on the nonlinear dynamics of a modified Van der Pol oscillator based electronic circuit has been reported. It has been found that inclusion of an active RC section (or filter) in the Van der Pol oscillator is able to generate rich nonlinear behavior. Effect of a low pass filter as well as an all pass filter has been reported. The behavior of the proposed system has been explained analytically and numerically with respect to its parameters by exploiting standard nonlinear analysis tools such as bifurcation diagrams, Lyapunov exponent, phase portraits, Poincare sections, and basins of attraction. Occurrence of some fascinating features like antimonotonicity, hysteresis and coexisting attractors have been found in the modified system. The analytical and numerical studies has been verified experimentally with a prototype experiment.

In gear transmissions, vibration causes noise and malfunction. In actual applications, misalignments contribute to intensifying the destructive effect of vibrations. In this paper, the nonlinear dynamics of a spiral bevel gear pair, with small helix angle, considering different misalignments, are deeply investigated. Axial misalignment, radial misalignment, and the combination of these two types are considered in this study. The governing equation is numerically solved through an implicit Runge–Kutta scheme. Since the main goal of this study is the analysis of the dynamic scenario, the mesh stiffness of the gear pair is obtained from the literature. The dynamical system is nonlinear and time-varying, it is analyzed through time responses, phase portraits, Poincaré maps, and bifurcation diagrams. Results show that, among the considered three cases with different types of misalignments, the spiral bevel gear with axial misalignment is the worst destructive case, aperiodic, subharmonic, and multiperiod responses are observable for this case. It is interesting that the chaotic responses for the case, having both types of misalignments, are less likely for the case with axial misalignment, only.

In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated. Interestingly, the system shows novel multi-bifurcation cascaded bursting oscillations (MBCBOs) when the frequency of the two excitations is much lower than the mean frequency of the original system. For instance, periodic, quasi-periodic and chaotic bursting oscillations induced by a variety of cascaded bifurcations are first observed, and the phenomenon of spiking transfer is also revealed. Besides, stability and local bifurcations of the system are comprehensively investigated to analyze the mechanism of the observed MBCBOs, in which bifurcation diagram, Lyapunov exponents, time series, phase portraits, and transformed phase diagrams are used. Finally, through a circuit simulation and hardware experiment, these complex dynamics phenomena are verified physically.

In almost all works done up to now on proton dynamics in hydrogen-bonded systems, dipole–dipole interactions have been overlooked. Here, we examine the role of these interactions on the dynamics and statistical properties of ions transfer and bonding defects in those systems. A one-dimensional model for hydrogen-bonded chains is investigated by including dipole–dipole interactions created by protons and heavy ion charge movement, generalizing thus the original Antonchenko–Davydov–Zolotaryuk model. The inclusion of dipole–dipole interactions is not only interesting from a physical point of view; it also provides an extraordinary mathematical model, a new class of differential equations possessing several key parameters and many singular straight lines. The dynamics around these singularities gives new information of great interest, such as to better explain analytically the formation of cracks resulting from dislocations observed in semiconductor heterostructures. Despite major individual differences in initial models, our results can generalize the results previously established in DNA, in the wave diffusion by static localized inhomogeneities or by generic time-dependent potentials, in nonlinear discrete electrical transmission lines and many others. Therefore, this work comes, among other things, to present a method of analytical resolution of these new equations. We use dynamic system methods to first determine the crucial parameters of the system (among them is the dipole–dipole interaction coefficient), and then we discuss on bifurcations of phase portraits and vector fields defined by the singular system. These dipole–dipole interactions have crucial effects on the response of nonlinear excitations that can propagate along these hydrogen-bonded systems. For each orbit of phase portraits with corresponding conditions, highlighting the importance of each component in the hydrogen-bonded system with dipole–dipole interactions on the dynamics, we compute all possible exact parametric representations of solutions. Our findings show that, under given parameter conditions, there are smooth solitary wave solutions, periodic wave solutions, periodic peaks, pseudo-peaks, compacton families and obviously kink and antikink solitons, specially interesting in systems with hydrogen bonds.

Cough signal analysis for understanding the pathological condition has become important from the outset of the exigency posed by the epidemic COVID-19. The present work suggests a surrogate approach for the classification of cough signals - croup cough (CC) and pertussis (PT) – based on spectral, fractal, and nonlinear time-series techniques. The spectral analysis of CC reveals the presence of more frequency components in the short duration cough sound compared to PT. The musical nature of CC is unveiled not only through the spectral analysis but also through the phase portrait features – sample entropy (S), maximal Lyapunov exponent (L), and Hurst exponent (Hb). The modifications in the internal morphology of the respiratory tract, giving rise to more frequency components associated with the complex airflow dynamics, get staged through the higher fractal dimension of CC. Among the two supervised classification tools, cubic KNN (CKNN) and neural net pattern recognition (NNPR), used for classifying the CC and PT signals based on nonlinear time series parameters, NNPR is found better. Thus, the study opens the possibility of identification of pulmonary pathological conditions through cough sound signal analysis.

A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).

This poster was made as partial credits towards LIFE731 module at the University of Liverpool. The aim is to communicate a mathematical research model to a scientific audience. My chosen template is the Better Science Poster created by Mike Morrison: https://osf.io/ef53g/. Welcome comments and suggestions on how the poster may be improved.

This paper presents a new way of designing a multi-wing chaotic system. The proposed design is based on 3D continuous chaotic system of Lorenz, improved by introducing a saw-tooth and sine functions. The basic proprieties of the proposed system are analyzed using of equilibrium points, phase portrait, Lyapunov exponent, and bifurcation diagram. Furthermore, the modeling of the design is based on Euler method using hardware description language (VHDL) and validated on Xilinx Virtex-II-Pro FPGA platform. Fixed-point arithmetic coding is employed to represented data on 32 bits (16Q16). Finally, the proposed system used to design a new chaos-based TRNG True Random Number Generators by analyzing its chaotic dynamical behavior and FPGA implementation performances. The proposed hardware architecture is based on two stages of pipeline and parallel structure (only 2 clock cycles). Experimental implementation results demonstrate that the design can achieve a maximum operating frequency of 12.649 MHz and a throughput of 202 Mbit/s. Besides, the random bit sequences produced by TRNG have been successfully passed the NIST-800-22 statistical standards tests. The proposed multi-wing attractor presents also complex dynamics and it can be applied in many engineering applications, especially in embedded cryptographic applications.

The electronic implementation, synchronization, and control of hyperchaos in a five-dimensional (5D) autonomous homopolar disc dynamo are investigated in this paper. The hyperchaotic behavior is found numerically using phase portraits and time series in 5D autonomous homopolar disc dynamo is ascertained on Orcad-PSpice software. The synchronization of the unidirectional coupled 5D hyperchaotic system is also studied by using the feedback control method. Finally, hyperchaos found in 5D autonomous homopolar disc dynamo is suppressed thanks to the designed single feedback. Numerical simulations and electronic implementation reveal the effectiveness of the single proposed control.

The microcontroller implementation, chaos control, synchronization, and antisynchronization of the nonlinear resistive-capacitive-inductive shunted Josephson junction (NRCISJJ) model are reported in this paper. The dynamical behavior of the NRCISJJ model is performed using phase portraits, and time series. The numerical simulation results reveal that the NRCISJJ model exhibits different shapes of hidden chaotic attractors by varying the parameters. The existence of different shapes of hidden chaotic attractors is confirmed by microcontroller results obtained from the microcontroller implementation of the NRCISJJ model. It is theoretically demonstrated that the two designed single controllers can suppress the hidden chaotic attractors found in the NRCISJJ model. Finally, the synchronization and antisynchronization of unidirectional coupled NRCISJJ models are studied by using the feedback control method. Thanks to the Routh Hurwitz stability criterion, the controllers are designed in order to control chaos in JJ models and achieved synchronization and antisynchronization between coupled NRCISJJ models. Numerical simulations are shown to clarify and confirm the control, synchronization, and antisynchronization.

It has been shown that account of elastic nonlinearity during the propagation of an acoustic wave in a solid lead to the appearance of a quadratic nonlinearity, which in its turn leads to the possibility of generating a wave of double frequency, the interaction of harmonics is asymmetric. The conditions under which nonlinear stationary waves are formed are considered. A phase portrait is constructed, and the dependence of the wavenumber of a nonlinear wave on its amplitude is estimated.

We deal with Hamiltonian bifurcations associated with the reversible umbilic in two degrees of freedom systems defined by 0:1 resonance, i.e. the unperturbed equilibrium has two purely imaginary eigenvalues and a semisimple double-zero one. The Hamiltonian is written as the sum of integrable Hamiltonian $$N\big (\frac{1}{2}(x^{2}+y^{2}),q,p;\lambda \big )$$ and a small perturbation $$P(x,y,q,p;\lambda )$$ by the normalization procedure. The phase portrait of N on each level of the integral $$I_{1}=\frac{1}{2}(x^{2}+y^{2})$$ is then studied in detail, obtaining an unfolding related to the reversible hyperbolic umbilic catastrophe. The persistence of 2-tori (i.e. two-dimensional tori) for the full system is analysed via KAM theory pointed out just as in Broer et al. (Z Angew Math Phys 44: 389–432, 1993), (in: Langford, Nagata (eds) Normal forms and homoclinic chaos, Waterloo, (1992), Fields Institute Communications 4 (1995)). In a sense, our results can be seen as a four-dimensional extension of a planar problem studied by Hansmann (Phys D 112: 81–94, 1998).

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

In order to construct the high-dimensional discrete hyperchaotic systems systemically, this paper proposes a new coupled chaotic model. It has a wide chaotic parameter range and can relax the election of the coupling coefficient. Sufficient conditions are derived to prove the existence of Li–Yorke chaos in the proposed model. Meanwhile, the existence of hyperchaos is also demonstrated. To discern the effects of different coupling types on the chaotic dynamics more comprehensively, we further explore the dynamical behaviors with various coupling structures by using Lyapunov spectrum, bifurcation analysis, and phase portraits. We investigate the interaction relationship between coupled units and give suggestions for selecting coupling types. The results indicate that the coupled model has more complex and more stable chaotic performance when there exists a loop in its topological structure. Further, synchronization is also discussed in this work; analysis results illustrate that the proposed model cannot be suppressed to periodic points at sufficiently high coupling strengths. This paper suggests an effective method that may contribute to studying hyperchaos design and coupled chaotic systems.

Phase variables are continuous real-valued functions which parameterize the gait cycle. As such, they can be used to estimate a powered prosthesis user's walking state. It is common to represent the phase variable as a function of the thigh angle. This approach assumes the thigh angle profile and its integral to be sinusoids. However, this assumption does not hold at lower walking speeds which leads to inaccurate heel-strike detection. In this study, we show that the thigh angle and its integral are phase-shifted from the ideal sinusoids by analyzing both able-bodied and amputee subject walking data. We then propose a novel phase variable incorporating phase-shift that is accurate even at lower walking speeds. We tested two variants of the proposed phase-shifted phase variable on a transfemoral prosthesis in an emulator study. Results show that phase-shifting improves heel-strike detection. Additionally, phase-shifting improves the linearity of the phase variable over the gait cycle. Analysis of the knee and ankle phase portraits showed that phase-shifting results in less deviations from the limit cycle.

In this article, we propose a model for plankton dynamics in presence of dormancy and time delay. Dormancy is a better adaptive strategy to survive in a harsh environmental condition than active ones. The dormancy in zooplankton receives considerable interest as it reduces the environmental stress and helps to regulate the predator’s life cycle. The density of phytoplankton is affected by gestation delay and the density of zooplankton is benefited by the adaptation of dormant phase (such as resting eggs). The interaction between the phytoplankton and zooplankton is taken as Beddington–DeAngelis (B-D)-type functional response. The feasibility and boundedness criterion of the considered model system are proved by the comparison principle. We have discussed the stability and direction of bifurcating periodic solution by applying the normal form theory and center manifold arguments. Computational illustrations are executed to support analytical findings by plotting the bifurcation diagram, time evolution plot and phase portraits for biologically feasible parameter values. Our numerical results revealed that the system depicts chaotic attractor as the magnitude of feedback time delay beyond the threshold value but the system behavior may become stable for some parametric values.

This paper is presented on the theory and applications of the fractional-order chaotic system described by the Caputo fractional derivative. Considering the new fractional model, it is important to establish the presence or absence of chaotic behaviors. The Lyapunov exponents in the fractional context will be our fundamental tool to arrive at our conclusions. The variations of the model’s parameters will generate chaotic behavior, in general, which will be established using the Lyapunov exponents and bifurcation diagrams. For the system’s phase portrait, we will present and apply an interesting fractional numerical discretization. For confirmation of the results provided in this paper, the circuit schematic is drawn and simulated. As it will be observed, the results obtained after the simulation of the numerical scheme and with the Multisim are in good agreement.

In this paper, the instability of semi-active control systems caused by the utilization of semi-active inerters is analyzed. The purpose of this study is to demonstrate the possibility of inducing instability by applying semi-active inerters in vibration control systems, a phenomenon which is essential for the applications of semi-active inerters but has not been drawn much attention in the existing results. A constructive method is proposed to prove the instable effect caused by switching-state semi-active inerters based on a general multi-degree-of-freedom (DOF) mass-chain system. The underlying idea in the proof is to construct a multi-DOF unstable semi-active control system with semi-active inerters based on its 1-DOF counterparts. First of all, it is theoretically proved that for a 1-DOF system, both undamped and damped cases, instability can be induced by semi-active inerters under certain switching laws. Then, following the idea of modal analysis, unstable n-DOF semi-active inerter-based system with proportional damping is constructed via coordinate transformation. Moreover, for general damping case, phase portraits are depicted to show the instability caused by semi-active inerters. In this way, the conclusion is reached that semi-active inerter can induce instability if the inertance improperly controlled, and such a finding is still valid when the inerter nonlinearities are considered. Hence, the stability issue should not be neglected in the practical application of semi-active inerters.

This paper mainly explores the self-reproducing dynamics in discrete-time system by constructing a two-dimension map with infinitely many fixed points. Theoretical analysis shows that the attractor of the map can not only be non-destructively reproduced by the initial values of all state variables along all axis directions, but can also be non-destructively reproduced by the parameter along all axis directions. The numerical simulations of bifurcation diagram, Lyapunov exponent, phase portrait and iterative sequence are carried out to further confirm the theoretical results. The self-reproducing behavior of different type of attractors and the corresponding iterative sequences are also confirmed by the experimental measurements performed on the DSP-based platform. This map is especially suitable for chaos-based engineering applications since the offset can be periodically switched by the initial condition and parameter on the premise of keeping robust dynamical behavior.

The paper presents a new approach to the process of regulating the basic parameters of a turbine jet engine. It presents a system for monitoring these parameters developed and put into operation and the creation of the so-called phase mapping of the engine speed increment. Its modular structure is described, which allows it to be adapted quite quickly to other types of aircraft engine units. Individual modules are based on mathematical descriptions from the theory of aircraft engines. The phase mapping of the engine speed indicates a dynamic change of this parameter. On this basis, the characteristic ranges and individual points of engine operation are presented. The following are examples of characteristics and their interpretation.

In this paper, the perturbed and unperturbed fractional regularized long wave (FRLW) equation is taken into consideration by using analytical and numerical approaches. For the unperturbed considered model diverse solitonic structures are measured using two finest approaches which are extended ( G ′ G 2 ) -expansion method and the direct algebraic method. Then, the discussed equation is transformed into the planer dynamical system by using Galilean transformation. All possible types of phase portraits are plotted in the respect of the parameters. Also, the effect of physical parameters is observed after applying an extrinsic periodic power, then the responsive analysis is applied to evaluate the periodic and quasi-periodic behavior for distinct initial value problems. On some of that, multistability analysis is applied and it is observed that for some values of physical parameters of the discussed equation is multistable.

In this paper, a predator-prey relationship in the presence of prey refuge was studied. The analysis of the dependence of locally stable equilibrium points on the parameters of the problem was carried out. Bifurcation and limit cycles for the model were analyzed to show the dynamical behaviour of the system. The results showed that the system is stable at a constant prey refuge m = 0.3 and prey harvesting rate H = 0.3. However, increasing m and decreasing H or vice versa, the predator-prey system remains stable. It was further observed that for a constant prey refuge m ≥ 0.78, the predator population undergoes extinction. Therefore, m was found to be a bifurcation parameter and m = 0.78 is a bifurcation value. Keywords: Prey refuge, bifurcation, harvesting, intraspecific competition, phase portrait