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In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension- 1 Hopf bifurcation and a codimension- 2 cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence. [ABSTRACT FROM AUTHOR]

In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov's direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point O (0 , 0 , 0 , 0 , 0 , 0) of this system is globally exponentially stable. [ABSTRACT FROM AUTHOR]

Phantom attractors in nonlinear systems under additive stochastic excitation have been recently discovered. This paper uncovers the existence of phantom attractors in a single-degree-of-freedom smooth nonlinear equation, which characterizes the vibration of an inextensible beam subjected to lateral stochastic excitation. It also elucidates that the stochastic averaging method, in this context, may lead to qualitatively erroneous probability density functions, identified as one of the reasons why these attractors were previously overlooked. The study then proceeds to analyze the formation process of the phantom attractor and the critical noise intensity associated with it. Subsequently, the key nonlinear term related to the emergence of phantom attractors is identified by observing whether the system still exhibits phantom attractors after the corresponding nonlinear terms are removed. It is revealed that in this system, the presence of phantom attractors is closely linked to the inertia nonlinearity of the hardening type. The system investigated in this paper is simpler compared to previously identified systems capable of generating phantom attractors. This simplicity aids in facilitating research focused on unraveling the general principles behind the formation of phantom attractors. [ABSTRACT FROM AUTHOR]

In prior work [Katsanikas & Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas & Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom. [ABSTRACT FROM AUTHOR]

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system. [ABSTRACT FROM AUTHOR]

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in ℝ 3 . In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in ℝ 3 , where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles. [ABSTRACT FROM AUTHOR]

The bifurcation theory for homoclinic networks with singular and nonsingular equilibriums is a key to understand the global dynamics of nonlinear dynamical systems, which will help one determine the dynamical behaviors of physical and engineering nonlinear systems. In this paper, the appearing and switching bifurcations for homoclinic networks through equilibriums in planar polynomial dynamical systems are studied. The appearing and switching bifurcations are discussed for the homoclinic networks of nonsingular and singular sources, sinks, saddles with singular saddle-sources, saddle-sinks, and double-saddles in self-univariate polynomial systems. The first integral manifolds for nonsingular and singular equilibrium networks are determined. The illustrations of singular equilibriums to networks of nonsingular sources, sinks and saddles are given. The appearing and switching bifurcations are studied for homoclinic networks of singular and nonsingular saddles and centers with singular parabola-saddles and double-inflection saddles in crossing-univariate polynomial systems, and the first integral manifolds of such homoclinic networks are determined through polynomial functions. The illustrations of singular equilibriums to networks of nonsingular saddles and centers are given. This paper may help one understand higher-order bifurcation theory in nonlinear dynamical systems, which is completely different from the classic bifurcation theories. [ABSTRACT FROM AUTHOR]

In our earlier research (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b, 2023c]), we developed two methods for creating dividing surfaces, either based on periodic orbits or two-dimensional generating surfaces. These methods were specifically designed for Hamiltonian systems with three or more degrees of freedom. Our prior work extended these dividing surfaces to more complex structures such as tori or cylinders, all within the energy surface of the Hamiltonian system. In this paper, we introduce a new method for constructing dividing surfaces. This method differs from our previous work in that it is based on 3D surfaces or geometrical objects, rather than periodic orbits or 2D generating surfaces (see [Katsanikas & Wiggins, 2023a]). To explain and showcase the new method and to present the structure of these 3D surfaces, the paper provides examples involving Hamiltonian systems with three degrees of freedom. These examples cover both uncoupled and coupled cases of a quadratic normal form Hamiltonian system. Our current paper is the first in a series of two papers on this subject. This research is likely to be of interest to scholars and researchers in the field of Hamiltonian systems and dynamical systems, as it introduces innovative approaches to constructing dividing surfaces and exploring their applications. [ABSTRACT FROM AUTHOR]

Our paper is a continuation of a previous work referenced as [Katsanikas & Wiggins, 2024b]. In this new paper, we present a second method for computing three-dimensional generating surfaces in Hamiltonian systems with three degrees of freedom. These 3D generating surfaces are distinct from the Normally Hyperbolic Invariant Manifold (NHIM) and have the unique property of producing dividing surfaces with no-recrossing characteristics, as explained in our previous work [Katsanikas & Wiggins, 2024b]. This second method for computing 3D generating surfaces is valuable, especially in cases where the first method is unable to achieve the desired results. This research aims to provide alternative techniques and solutions for addressing specific challenges in Hamiltonian systems with three degrees of freedom and improving the accuracy and reliability of generating surfaces. This research may find applications in the broader field of dynamical systems and attract the attention of researchers and scholars interested in these areas. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]

A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets. [ABSTRACT FROM AUTHOR]

Birhythmicity oscillators have been extensively applied in fields such as biology, physics, and engineering. Studying their global dynamics is crucial for gaining a comprehensive understanding of the intrinsic mechanisms that govern oscillator behavior. This paper focuses on investigating the influence of memory effects on the global dynamics of the fractional birhythmic van der Pol (BVDP) oscillator. To determine the system's global properties, we employ an improved cell mapping method. Specifically, the system's evolution is computed by introducing additional auxiliary variables, creating a space for storing historical information. This method allows us to examine the system's global properties without memory loss. Through a comparison of the global dynamic behaviors of BVDP oscillators with memory to those without memory, we observe that the presence of memory effects results in the emergence of chaotic attractors in the system. This, in turns, results in system instability and heightened sensitivity to initial conditions. Furthermore, our findings suggest that changes in the fractional-order can induce various crises in the oscillator and may have the potential to suppress chaotic oscillations. [ABSTRACT FROM AUTHOR]

This study comprehensively investigates synchronization control of the memristive- and resistive-coupled neural network with noise effect. In this context, a sub-network is constructed with three chaotic memristive Hindmarsh–Rose neurons coupled by the memristive synapses. Then, two equivalent sub-networks are linked by the diffusive resistive synapse under a distinctive global network structure including two sub-networks. Thus, a double (memristive/resistive) control mechanism is used to achieve the synchronization state. The first mechanism is the memristive synaptic coupling in each sub-network, and the second one is the diffusive resistive coupling between two sub-networks constructing the global network. The synchronization between the neurons is proven theoretically through the Lyapunov stability and Lipschitz theorems and this proof process is verified with numerical experiments. According to the theoretical and numerical results, by adjusting memristive/resistive synaptic coupling weights, a synchrony behavior is generated successfully in these sub-networks and global networks. Additionally, this paper also handles the Gaussian white noise effect on the global network to examine continuity capability and stability of synchronization. Two different frameworks are built by applying the noise with multiple noise power levels and periods to (i) one of the resistive coupling neurons in each sub-network directly and (ii) one neuron in a sub-network without direct contact with the other sub-network. With these two characteristic designs, it is proven that the second design provides more robust characteristics since it utilizes the double control mechanism, which enhances the stability throughout external interferences enabling the network to sustain synchronous states. [ABSTRACT FROM AUTHOR]

Cascaded H-bridge inverters, which belong to strongly nonlinear time-varying system with complex dynamics, have been widely used in high-voltage and high-power engineering fields. In this paper, a carrier phase-shifted sinusoidal pulse-width modulation cascaded H-bridge inverter is taken as an example with piecewise analysis to establish the discrete iterative mathematical model. Furthermore, the nonlinear dynamic behavior of this inverter system is investigated using folding diagrams, spectrum diagrams, bifurcation diagrams, Lyapunov exponents, etc. We find that this inverter can exhibit multiperiod bifurcation, tangential bifurcation and paroxysmal chaos within certain ranges of control coefficients and other circuit parameters, which will influence the system's stability seriously. On the basis of this discovery, the sinusoidal time-delay feedback control method is proposed. The difference between the load current of the controlled object and its own delay is used to form a feedback term, which is then fed into the sine function and proportional link to obtain the control term. Moreover, the Jacobian matrix of this feedback inverter system is derived, and the constraints on the feedback control coefficient are obtained based on the Jury criterion. Simulation results have verified the effectiveness of such a sinusoidal time-delay feedback control method, which can improve the stability and anti-interference of the system. The research can be used to provide some reference and guidance for the circuit design, debugging and control of cascaded H-bridge inverters. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems. [ABSTRACT FROM AUTHOR]

In this paper, we detect Unstable Periodic Orbits (UPOs) of both self-excited and hidden attractors using the Damped Newton–Raphson–Mees algorithm. This study answers one of the long-standing questions about the size of a hidden attractor. We observe that the UPOs of the self-excited attractor are spread in a thick region within the phase space at varying distances from the fixed point, whereas the UPOs of hidden attractors are confined to a thin layer. This reveals details concerning the size of a hidden attractor. We demonstrate this for a hidden attractor without a fixed point and for a hidden attractor with a single stable fixed point. To compare the results with self-excited attractors, we have taken Lorenz and Rössler attractor as examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well. [ABSTRACT FROM AUTHOR]

In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We study a family of one-dimensional quasi-periodically forced maps F a , b (x ,) = (f a , b (x ,) , + ω) , where x is real, is an angle, and ω is an irrational frequency, such that f a , b (x ,) is a real piecewise-linear map with respect to x of certain kind. The family depends on two real parameters, a > 0 and b > 0. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For a < 1 and any b , there is only one continuous invariant curve. For a > 1 , there exists a smooth map b = b 0 (a) such that: (a) For b < b 0 (a) , f a , b has two continuous attracting invariant curves and one continuous repelling curve; (b) For b = b 0 (a) , it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For b > b 0 (a) , it has one continuous attracting invariant curve. The case a = 1 is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family G a , b (x ,) = (arctan (a x) + b sin () , + ω) for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when a → ∞. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

The impact of power-law memory on the dynamics of the Caputo standard fractional map is investigated in this paper. The definition of a complexity measure for the Caputo standard fractional map is introduced. This measure evaluates both the average algebraic complexity of a trajectory and the distribution of the trajectories of different types in the phase space of the system. The interplay between the small-scale spatial chaos and the large-scale spatial behavior is observed and measured during the transition of the Caputo standard fractional map from the circle map to the classical standard map. It is demonstrated that the impact of the fractional derivative on the complexity of the fractional system is not straightforward and is predetermined by the physical properties governing the dynamics of that system. [ABSTRACT FROM AUTHOR]

Robust synchrony in network dynamics is governed by balanced colorings and the corresponding quotient network, also formalized in terms of graph fibrations. Dynamics and bifurcations are constrained — often in surprising ways — by the associated synchrony subspaces, which are invariant under all admissible ordinary differential equations (ODEs). The class of admissible ODEs is determined by a groupoid, whose objects are the input sets of nodes and whose morphisms are input isomorphisms between those sets. We define the coloring subgroupoid corresponding to a coloring, leading to groupoid interpretations of colorings and quotient networks. The first half of the paper is mainly tutorial. The second half, which is new, characterizes the structure of the network groupoid and proves that the groupoid of the quotient network is the quotient of the network groupoid by a normal subgroupoid of transition elements. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model of fluid flows in a convective thermal system is developed, and a five-dimensional dynamical system is developed for the investigation of the convective fluid dynamics. The analytical solutions of periodic motions to chaos of the convective fluid flows are developed for steady-state vortex flows, and the corresponding stability and bifurcations of periodic motions in the five-dimensional dynamical system are studied. The harmonic frequency-amplitude characteristics for periodic flows are obtained, which provide energy distribution in the parameter space. Analytical homoclinic orbits for the convective fluid flow systems are developed for the asymptotic convection through the infinite-many homoclinic orbits in the five-dimensional dynamical system. The dynamics of fluid flows in the convective thermal systems are revealed, and one can use such methodology to predict atmospheric and oceanic phenomena through thermal convections. [ABSTRACT FROM AUTHOR]

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures. [ABSTRACT FROM AUTHOR]

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points. [ABSTRACT FROM AUTHOR]

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a host–parasitoid model with a Holling type-II functional response and incorporate harvest effort. The Holling type-II response leads to saturation in parasitized hosts, creating a potential economic harvesting opportunity. To address overexploitation risks, we integrate a harvest effort function, determining an optimal threshold to prevent depletion. We explore model dynamics and bifurcations, including co-dimension one behaviors such as flip and Neimark–Sacker bifurcations, we provide numerical examples for validation. Our suggested difference-algebraic model, compared to continuous-time models, exhibits rich dynamics within the Nicholson–Bailey host–parasitoid framework. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities. [ABSTRACT FROM AUTHOR]

In this work, a new glucose–insulin model incorporating time delay and obesity is developed to gain insights of its dynamical mechanisms. Through the method of multiple scales, we theoretically demonstrate that time delay can drive the system to yield Hopf bifurcation, thereby producing oscillating solutions that are consistent with the simulation results. Moreover, obesity changes the level of glucose, but cannot induce oscillations. In particular, it is found that under the combined effect of obesity and time delay, obesity delays the appearance of Hopf bifurcation which is induced by time delay. Results show that a low calorie diet can achieve therapeutic effects including reducing blood glucose fluctuations and insulin resistance, which can be used as an adjuvant for the treatment of diabetes. In addition, our results indicate that the delay, together with an optimal rate of model parameters can cause a variety of dynamics and induce glucose oscillations. The result obtained in this paper may help to better understand the obesity, diabetes, and the interaction between glucose and insulin, so that control strategies can be designed to better regulate blood glucose levels and fluctuations and mitigate the occurrence of type-2 diabetes. [ABSTRACT FROM AUTHOR]

This paper presents a novel discrete memristor model that incorporates exponential and absolute value functions. A discrete coupled memristor neural network model is constructed based on this memristor design. The periodic and chaotic regions of the discrete neural network model are determined using bifurcation and Lyapunov exponent spectrum. Furthermore, by varying the initial values of the discrete memristive neural network, we observe the coexistence of chaos and periodic attractors, as well as periodic attractors. Additionally, an application to color image encryption based on the discrete system model is given. Security analysis is conducted in the aspects of key space, histogram analysis, correlation analysis, sensitivity analysis, Peak Signal-to-Noise Ratio (PSNR), and information entropy analysis. The analysis results show that the algorithm has a key space size of 2 2 5 8 , and the information entropy of baboon graph is 7.9993, which is very close to the ideal value of 8. It shows that the image encryption algorithm is feasible and effective. Finally, the implementation of the discrete memristive neural network model is realized using Field Programmable Gate Array (FPGA). The experimental implementation is conducted using the Verilog language on the Vivado 2018.3 platform, and the obtained results align with the numerical simulation results obtained through MATLAB. [ABSTRACT FROM AUTHOR]

In recent years, many image encryption schemes have adopted Hilbert curves for encryption. In this approach, the Hilbert curve is used to encrypt grayscale images by traversal scrambling. However, the correlation between pixels has not been fully considered and those algorithms are not safe enough. To solve this problem, a new image encryption algorithm based on a new chaotic system of 2D-LICM (Two-Dimensional Linear-Infinite-Collapse Chaotic Map) and an improved Hilbert curve is proposed in this paper. First, we propose a new 2D-chaotic system to address the shortcoming that the commonly used chaotic systems are too simple in scope and complexity. Then, a new image encryption algorithm is proposed using the newly designed 2D-LICM and the improved Hilbert curve. The proposed algorithm uses Hilbert curve to reduce the correlation between adjacent pixels of the image at the pixel and bit levels and increase the scrambling and diffusion effects. Simulation and security analysis results show that the proposed scheme has high security and is superior to several advanced image encryption algorithms. [ABSTRACT FROM AUTHOR]

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua's circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds. [ABSTRACT FROM AUTHOR]

In this paper, a 5D Lorenz system is discussed. The discrete mappings are developed to solve the periodic motions in the 5D Lorenz system. Then the stability and bifurcations are determined by eigenvalue analysis. A bifurcation tree is presented to demonstrate that the discrete mapping method can provide not only stable orbits but also unstable motions. Finally, trajectory illustrations are given to show bifurcation influences on periodic orbits and homoclinic orbits in the 5D Lorenz system. [ABSTRACT FROM AUTHOR]

This paper investigates a type of iterated implicit function systems composed of equations F n (x , y) = c , where F n (x , y) is a continuous function, and c is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation F n (x , y) = c containing the implicit function or being α n -contractive about y. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated. [ABSTRACT FROM AUTHOR]

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one. [ABSTRACT FROM AUTHOR]

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]

As digital communication and storage continue to expand, the protection of image privacy information becomes increasingly critical. To safeguard sensitive visual information from unauthorized access, this paper proposes a novel image encryption scheme that integrates multiobjective Artificial Bee Colony (ABC) optimization algorithm and DNA coding. Multiple evaluation metrics including correlation relationship, Number of Pixel Change Rate (NPCR), Unified Average Changing Intensity (UACI), and information entropy are collaboratively optimized by the ABC algorithm. The proposed method begins with the application of the SHA-256 algorithm to generate keys and random sequences using chaotic systems. These sequences are then employed for shuffling, DNA coding, decoding, and diffusion, generating initial encrypted images. Subsequently, the encrypted images serve as individuals within the ABC algorithm to determine optimal parameters of the chaotic systems and the best ciphertext image. Simulation experiments demonstrate that the ciphertext images achieved excellent results in information entropy, pixel correlation coefficient, NPCR, and UACI. The integration of the multiobjective ABC optimization algorithm with DNA coding in our proposed image encryption scheme results in heightened security, as evidenced by superior performance in various metrics. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a novel class of time-dependent controls to realize ultra-fast magnetization switching in nanomagnets driven by spin-torques produced by spin-polarized electric currents. Magnetization dynamics in such complex systems is governed by the Landau–Lifshitz–Slonczewski equation which describes the precessional motion of (dimensionless) magnetization vector on the unit-sphere. The relevant case of nanoparticles with uniaxial anisotropy having in-plane easy and intermediate axes as well as out-of-plane hard axis is considered. By exploiting the characteristic smallness of damping and spin-torque intensity, the complexity of the magnetic system's dynamic is dealt with by employing tools borrowed from Hamiltonian Perturbation Theory. More precisely, the aforementioned controls are constructed via suitable perturbative tools in a way to realize approximate latitudinal solutions (i.e. motions on a sphere in which the out-of-plane magnetization component stays constant) with the effect to fast "switch" the system from one stationary state to another. The possibility to keep a ("small") bounded value of the out-of-plane coordinate throughout this process of "transfer" turns out to be advantageous in the applications as it sensibly reduces the post-switching relaxation oscillations that may cause the failure of switching in real samples. Further relevant quantitative results on the behavior of the solutions during the pre- and post-switching stages (termed "expulsion" and "attraction", respectively) are given as a by-product. A selection of validating numerical experiments is presented alongside the corresponding theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical complexity of points with emergence behavior but without weak face behavior, especially for points without physical-like behavior in certain dynamical systems such as transitive Anosov systems. We use the tools of saturated sets to prove that these points show strong dynamical complexity in the sense of entropy, density and distributional chaos. We obtain some observations of those results related to irregular sets and level sets. These results strengthen the previous results of [Catsigeras et al., 2019; Hou et al., 2023]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point P while, simultaneously, two fixed points collide with P in a transcritical bifurcation. Moreover, all points of a line containing P become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation. [ABSTRACT FROM AUTHOR]

Benefiting from trigonometric and hyperbolic functions, a nonlinear megastable chaotic system is reported in this paper. Its nonlinear equations without linear terms make the system dynamics much more complex. Its coexisting attractors' shape is diamond-like; thus, this system is said to have diamond-shaped oscillators. State space and time series plots show the existence of coexisting chaotic attractors. The autonomous version of this system was studied previously. Inspired by the former work and applying a forcing term to this system, its dynamics are studied. All forcing term parameters' impacts are investigated alongside the initial condition-dependent behaviors to confirm the system's megastability. The dynamical analysis utilizes one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and attraction basin. Because of this system's megastability, the one-dimensional bifurcation diagrams and Kaplan–Yorke dimension are plotted with three distinct initial conditions. Its analog circuit is simulated in the OrCAD environment to confirm the numerical simulations' correctness. [ABSTRACT FROM AUTHOR]

One of the promising applications of locally-active memristors (LAMs) is to construct oscillators for oscillatory neural networks. By using two current-controlled (CC) LAMs, a fully CC LAM-based oscillator is designed in this paper. The oscillator principle originates from the small-signal inductive and capacitive impedance characteristics of two different CC LAMs, and thus extra reactance element is not required in the circuit. Based on bifurcation theory and small-signal analysis method, conditions of the equilibrium point instability are quantitatively derived. Theoretical analysis indicates that the circuit oscillation is dependent on three critical parameters. Then, according to the conditions of the equilibrium point instability, parameters design methods of the two LAMs are proposed, including the static and dynamic parameters. A simple NbOx CC LAM model is taken as an example to conduct detailed simulation analysis. The simulation results verify the feasibility of the proposed circuit and analysis methods. Finally, the effects of the LAM model parameters on the oscillator performance are investigated, which is helpful for optimal design of the oscillator. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin (0 , 0) , which are the positive x - and y -axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear bifurcation and chaos characteristics of piezoelectric composite lattice sandwich plates at 1:3 internal resonance. The nonlinear vibration partial differential equation is first discretized to become an ordinary differential equation by applying the Galerkin method. The main resonance modulation equations and the parametric resonance for the external excitation frequency that is close to the system's second-order modal frequency are then obtained by using the multiscale method. The Newton–Raphson method is subsequently applied to obtain the bifurcation diagram of the steady-state equilibrium of modulation equation with varying system parameters. The equilibrium stability is finally analyzed. We confirm the existence of static bifurcation such as saddle-node bifurcation, pitchfork bifurcation, and Hopf dynamic bifurcation. A detailed analysis is also conducted on the complex nonlinear jump phenomenon caused by the presence of multiple nonlinear steady-state solution regions. In the dynamic Hopf bifurcation interval, the fourth-order Runge–Kutta method is used to continue to track the dynamic periodic solution of the modulation equation in Cartesian coordinates. It is found that there are multiple boundary crises and attractor merging crises in the vibration system for piezoelectric composite lattice sandwich plates. [ABSTRACT FROM AUTHOR]

In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments. [ABSTRACT FROM AUTHOR]