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This paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation. [ABSTRACT FROM AUTHOR]

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under m$$ m $$‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits γ1$$ {\gamma}_1 $$ and γ2$$ {\gamma}_2 $$ connected by three hyperbolic saddle points q1,q2,q3$$ {q}_1,{q}_2,{q}_3 $$. Assume that the degeneracy of the unperturbed heteroclinic orbit γi$$ {\gamma}_i $$ is ni$$ {n}_i $$, the splitting index is δi$$ {\delta}_i $$. By applying Lin's method, we construct heteroclinic orbits connected q1$$ {q}_1 $$ and q3$$ {q}_3 $$ near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from ℝn1+n2+m$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+{\delta}_1+{\delta}_2} $$. Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices δ1$$ {\delta}_1 $$, δ2$$ {\delta}_2 $$ and then obtain the existence of the heteroclinic orbits which backward asymptotic to q1$$ {q}_1 $$ and forward asymptotic to q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter ϵ$$ \epsilon $$, which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of ϵ$$ \epsilon $$ because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of ϵ$$ \epsilon $$ cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period‐doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the instability of a constant equilibrium solution in a general activator–inhibitor (depletion) model with passive diffusion, cross‐diffusion, and nonlocal terms. It is shown that nonlocal terms produce linear stability or instability, and the system may generate spatial patterns under the effect of passive diffusion and cross‐diffusion. Moreover, we analyze the existence of bifurcating solutions to the general model using the bifurcation theory. At last, the theoretical results are applied to the spatial water–biomass system combined with cross‐diffusion and nonlocal grazing and Holling–Tanner predator–prey model with nonlocal prey competition. [ABSTRACT FROM AUTHOR]

This paper deals with a model of chemostat with two cross‐feeding species and involving two inhibitions. The excess of hydrogen inhibits the growth of acetogenic bacteria which, in turn, inhibit the growth of methanogenic hydrogenotrophic bacteria. The model is described by a system of four ordinary differential equations. We established the conditions of existence and stability of equilibria with respect to the operating parameters which are the dilution rate and the input substrate concentrations of the species of the model, then we illustrate the operating and bifurcation diagrams. This technique makes it easy to interpret different regions of operating diagrams. Inhibitions let some regions of stability rise and others vanish. [ABSTRACT FROM AUTHOR]

Subsynchronous oscillation events of grid‐connected voltage source converter (VSC) pose a potential risk to the security and stability of power systems. It is particularly challenging to reveal the subsynchronous oscillation mechanism precisely because the grid‐connected VSC consists of non‐linear components such as saturation. The describing function and the generalized Nyquist criterion are used in this paper to investigate the bifurcation types and dominant parameters of grid‐connected VSC. First, the condition of a supercritical Hopf bifurcation is investigated, which generates subsynchronous oscillations. Then, the dominant parameters of supercritical Hopf bifurcation are deduced, including the grid strength, the active power reference, or the proportional coefficient of phase‐locked loop. Finally, it is examined how the saturation's limit value impacts the limit cycle's amplitude and stability. The findings demonstrate that a system's stability depends on whether its closed‐loop frequency characteristic curve intersects the describing function's negative reciprocal curve. When the two curves intersect, the supercritical Hopf bifurcation appears, and changing the limit value can affect the limit cycle's amplitude. However, if the two curves do not intersect, altering the limit value cannot cause the supercritical Hopf bifurcation, and the subsynchronous oscillation will not happen. Variations in grid strength, active power reference, or proportional coefficient of phase‐locked loop will significantly affect the Nyquist curve, determining if supercritical Hopf bifurcation appears. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence, nonexistence, and multiplicity of positive solutions for a impulsive periodic boundary value problem with positive parameter λ$$ \lambda $$. Using the critical point theorem, we obtain a bifurcation type result about positive solutions; that is, there exists 0<Λ<∞$$ 0<\Lambda <\infty $$ such that our problem has at least two positive solutions for all λ∈(0,Λ)$$ \lambda \in \left(0,\Lambda \right) $$, one positive solution for λ=Λ$$ \lambda =\Lambda $$, and no positive solution for λ>Λ$$ \lambda >\Lambda $$ under the appropriate conditions. [ABSTRACT FROM AUTHOR]

Nonfeedback methods of chaos and bifurcation control are suited for practical applications because of their speed, flexibility, no online monitoring, and processing requirements. In this paper, we analyze the control performance of various nonfeedback methods such as (i) adding a weak periodic force, (ii) adding a second periodic force, and (iii) adding a quasiperiodic force. We apply these methods to control chaos and bifurcation in an autocatalytic chemical system. By choosing the amplitudes (f,g)$(f,g)$ of the external excitation as control parameters, we investigate what effect the amplitudes have on the dynamics of the chemical system with suitable system's parameters value. Controlling of chaotic and bifurcation behaviors has been investigated through the bifurcation diagram, phase portrait, and time series. [ABSTRACT FROM AUTHOR]

In this paper, an improved railway wheelset system is presented. The dynamical behaviors of the system are investigated, including dissipativity and invariance of the system, stability of zero‐equilibrium point, and the bifurcation characteristics of railway wheelset system at zero equilibrium point. Furthermore, the chaotic behaviors of the motion of railway wheelset and the dynamical behaviors of the railway wheelset system under geometric constraint are studied. It shows that the railway wheelset system has complex dynamical phenomena owing to nonlinear factor and the railway wheelset system may have different complex dynamical behaviors with different nonlinear parameters. In addition, the motion of railway wheelset also has chaotic behaviors under geometric constraint. Finally, the chaos control of the railway wheelset system is achieved by using linear feedback control method. The numerical simulations are carried out in order to analyze the complex phenomena of the railway wheelset system. [ABSTRACT FROM AUTHOR]

In this paper, a discrete‐time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, resonance and bifurcation of a nonlinear damped fractional‐order Duffing system are studied. The amplitude and phase of the steady‐state response of system are obtained by means of the average method, the stability is analyzed, and then the amplitude‐frequency characteristic curves of the system with different parameters are drawn based on the implicit function equation of amplitude. Grunwald–Letnikov fractional derivative is used to discretize the system numerically, the response curve and phase trajectory of the system under different parameters are obtained; meanwhile, the dynamic behavior is analyzed. The bifurcation and saddle bifurcation behavior of the system is studied through numerical simulation with different parameters. [ABSTRACT FROM AUTHOR]

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ℝ+4$$ {\mathbb{R}}_{+}^4 $$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has an interior equilibrium solution under definite parametric condition. We have explored the local dynamics with topological classifications about boundary and interior equilibrium solutions of the discrete‐time COVID‐19 epidemic model by linear stability theory. Further, for the discrete‐time COVID‐19 epidemic model, existence of periodic points and convergence rate are also investigated. It is also studied the existence of possible bifurcations about boundary and interior equilibrium solutions and proved that there exists no flip bifurcation about boundary equilibrium solution. Moreover, it is proved that about interior equilibrium solution, there exist Hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Moreover, by feedback control strategy, chaos in the discrete COVID‐19 epidemic model is also explored. Finally, theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

In this paper, we detect a magnetic null inside the MMS space‐craft (SC) tetrahedron using a linear interpolation that might be nearly impossible for very small inter‐SC separation like MMS (previously implemented on Cluster mission which is larger separation SC). We also report, successive or identical magnetic nulls that enter/reenter and cross the MMS spacecraft tetrahedron, and the "bifurcations" (change of the null type) of magnetic nulls are captured when it crosses one of four MMS tetrahedral surfaces during the 19 September 2015 event. During this event, we observe the eigenvalues of the radial null that enters and crosses the MMS tetrahedron changes from all real to one conjugate complex pair and one real eigenvalues, when the null exits through one of the tetrahedral boundary cells. This bifurcation is also evidenced by measuring the topological distance between two nulls before and after "bifurcation" using the Earth Mover's Distance. Key Points: First, we find a magnetic null and track it using a linear interpolation method in MMSWe identify successive magnetic nulls that enter/reenter, inside, and cross the MMS tetrahedron during a reconnection event on 19 September 2015 at 07:43:30We measure the topological distance between each successive magnetic null and try to capture the "bifurcations" or the change of magnetic topology when null crosses one of the boundary cells of the MMS tetrahedron [ABSTRACT FROM AUTHOR]

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses. [ABSTRACT FROM AUTHOR]

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate λ, which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider "flat stationary solution" in this paper. We first show that there exists a unique flat stationary solution for any λ>0. Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points λ=λk(ρ*) (k≥1). [ABSTRACT FROM AUTHOR]

Both the International Financial Reporting Standards (IFRS) and the codified accounting standards (ASC) for the US GAAP categorize hedging relationships as falling into several buckets. The two buckets of relevance in this paper are (i) hedging the volatility of fair values, and (ii) hedging the volatility of future cash flow. In this paper, I argue that at least three accounting treatments of derivatives and hedging lead to creating serious distortion of reporting actual transactions, to combining hard and plastic valuations, and to violating adherence to the principle of 'faithful representation'. The three accounting treatments are as follows: (1) creating the fictional Hypothetical Derivatives Method; (2) allowing for the establishment of purely discretionary valuation adjustments for all over‐the‐counter derivative assets (Credit Valuation Adjustment) and liabilities (Debt Valuation Adjustment) without any guides or constraints; (3) requiring subjective metaphysical separation of embedded derivatives with the main guide being the management's own perception of the instrument's embodiment of unrelated value and risk generators. To remedy the resulting distortion in financial reporting, significant revisions of certain accounting standards are sorely needed. [ABSTRACT FROM AUTHOR]

This paper reviews laboratory observations of earthquake initiation and describes new experiments on a 3‐m rock sample where the nucleation process is imaged in detail. Many of the laboratory observations are consistent with previous work that showed a slow and smoothly accelerating earthquake nucleation process that expands to a critical nucleation length scale Lc, before it rapidly accelerates to dynamic fault rupture. The experiments also highlight complexities not currently considered by most theoretical and numerical models. This includes a loading rate dependency where a "kick" above steady state produces smaller and more abrupt initiation. Heterogeneity of fault strength also causes abrupt initiation when creep fronts coalesce on a stuck patch that is somewhat stronger than the surrounding fault. Taken together, these two mechanisms suggest a rate‐dependent "cascade up" model for earthquake initiation. This model simultaneously accounts for foreshocks that are a by‐product of a larger nucleation process and similarities between initial P wave signatures of small and large earthquakes. A diversity of nucleation conditions are expected in the Earth's crust, ranging from slip limited environments with Lc < 1 m, to ignition‐limited environments with Lc > 10 km. In the latter case, Lc fails to fully characterize the initiation process since earthquakes nucleate not because a slipping patch reaches a critical length but because fault slip rate exceeds a critical power density needed to ignite dynamic rupture. Plain Language Summary: In uniquely large‐scale laboratory experiments, a 3‐m rock sample is squeezed until earthquake‐like slip events spontaneously develop on a planar fault cut through the sample. This paper describes the initiation of those slip events—where one part of the fault begins to slip a fraction of a second before the rest of it ruptures (i.e., preslip). The laboratory observations are compared to theoretical models, computer simulations, and field studies of foreshock sequences and other earthquake precursors. Many observations are consistent with previous work that showed slow and smoothly accelerating earthquake initiation—a process termed earthquake nucleation. When the preslip region grows larger than a critical length scale Lc (~1 m), it accelerates unstably and radiates seismic waves like an earthquake. However, some observations show an order of magnitude variation in apparent Lc. The initiation process is sensitive to details such as naturally occurring variation in the strength of the rock/rock fault and perturbations in the rate at which the rock is loaded. Put together, the laboratory work suggests that smoothly accelerating earthquake nucleation is a property of unnaturally smooth and homogenous faults and that Lc is an incomplete metric for characterizing the initiation of earthquakes on realistically rough natural faults. Key Points: Experiments on a 3‐m rock show nucleation at many locations, including at the edges of creeping regions, consistent with numerical modelsLoading perturbations and mild strength heterogeneity produce order of magnitude variations in estimates of nucleation length scaleEarthquake initiation on heterogeneous faults is better characterized by critical power density than a time‐invariant critical length scale [ABSTRACT FROM AUTHOR]

Summary: This paper deals with the stabilization of direct current/direct current (DC/DC) boost converter and the nonlinear phenomena elimination using a constrained backstepping technique. Based on the converter averaged model, the proposed control approach is designed and the input to state stability concept is used to prove the system global stability. Furthermore, the structural stability is proven to show the efficiency of the proposed approach to suppress the nonlinear phenomena exhibited by the converter. The simulation results illustrate the different stability regions of the system as functions of the controller and the system parameters. The bifurcation diagrams are used to show the effectiveness of the proposed approach in terms of nonlinear phenomena suppression in wide regions of the system operating domains. [ABSTRACT FROM AUTHOR]

Summary: With the rapid development of renewable energy, inverters have been widely used in the distributed power system. For the safety of the distributed power system, the stable operation of the inverter is vital. Hence, this paper detects slow‐scale instability fast‐scale instability and in the double‐loop proportional‐integral (PI) control single‐phase full H‐bridge inverter with inductive impedance load by the discrete‐mapping model firstly. Next, the mechanism of the mechanism slow‐scale and fast‐scale behaviors are explained by state‐space average model and Filippov method, respectively. Then, passive damping method and sine wave compensation techniques are extended to the system to suppress the nonlinear behaviors. Finally, the PLECS simulation and experiments are utilized to verify the correctness of the investigations. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical transition and chaos for a five‐dimensional Lorenz system. Based on the eigenvalue analysis, the principle of exchange of stabilities conditions is obtained. By using the dynamical transition theory, three different types of dynamical transition for the five‐dimensional Lorenz system are derived. More precisely, when the control parameter r=1, the system has a continuous transition and bifurcates to two stable steady states. As r further increases, the system undergoes two successive transitions. That is, under some condition, the transition is continuous and a stable limit cycle is bifurcated; if not, the system undergoes a jump transition and an unstable periodic orbit occurs. Especially, the chaotic orbits occur when r=36.91. Finally, numerical results are given to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the discrete Sturm–Liouville problem −Δ(p(k)Δu(k−1))+q(k)u(k)=λm(k)u(k)+f1(k,u(k),λ)+f2(k,u(k),λ),k∈[1,T]Z,a0u(0)−b0Δu(0)=0,a1u(T+1)+b1Δu(T)=0,where λ∈ℝ is a parameter, [1,T]Z:={1,2,...,T},f1,f2∈C([1,T]Z×ℝ2,ℝ),f1 is not differentiable at the origin and infinity. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcate from intervals of the line of trivial solutions or from infinity, respectively. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the evolution of the transmembrane potential on the cardiac cell under different rates and amplitudes of stimulation. For modeling this potential, the modification of the Fenton‐Karma model was applied. It is a phenomenological model with 3 degrees of freedom that corresponds to nondimensional transmembrane potential and gating variables for regulation of inward and outward ion currents that can better reproduce the shape of the transmembrane potential than the original Fenton‐Karma model. The model was newly forced by stimulus with the shape of the half‐sine period. As the main goal of the paper is to show that this model is showing regular as well as irregular motion; periodic and chaotic patterns are detected using bifurcation diagrams, the Fourier spectra, Poincaré sections, and 0‐1 test for chaos. [ABSTRACT FROM AUTHOR]

This paper considers the Alpazur oscillator, which is composed of a Rayleigh oscillator unit and a dc power supply controlled by a switch, and examines the bifurcation set by numerical calculation. In such a case, a bifurcation diagram is constructed in order to see the phase-space behavior of the system. In the conventional method, however, the calculation is possible only when the trajectory is continuous and differentiable. The method cannot directly be applied to the case where the trajectory is discontinuous or not smooth at a point (not differentiable) as in the case of on–off circuits. From such a viewpoint, this paper proposes a bifurcation trajectory tracking algorithm for the that nonlinear on–off circuit, containing a switch that is made on and off periodically by an external force. As the method of analysis, the composite Poincaré mapping is defined based on the periodic operation of the switch. The method of calculating the bifurcation parameter is described precisely. Using the proposed method, the Alpazur oscillator is analyzed. The phase-synchronized solutions that exist along the Neimark–Sacker bifurcation as well as the chaos are investigated. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(1): 75–83, 2001 [ABSTRACT FROM AUTHOR]

Shells of revolution under axisymmetric conditions exhibit a circumferentially uniform pre‐buckling stress state and are important fundamental systems which often serve as reference systems for those under more complex conditions. Given this status, work is continuing on a careful and complete characterization of their buckling response with the aid of the Reference Resistance Design (RRD) framework for the ultimate benefit of the EN 1993‐1‐6 Eurocode on the strength of stability of metal shells. The situation is greatly complicated by the fact that while modern finite element software packages offer axisymmetric shell elements in an efficient 2D modelling plane, these are not capable of detecting bifurcation buckling into non‐axisymmetric modes which are often critical for slender systems. Reverting to a full 3D plane is possible, but grossly inefficient and the explicitly modelled circumferential direction is parasitic and detrimental to the overall solution quality. AQUINAS is an accessible and intuitive toolbox developed by the Authors in MATLAB for the efficient analysis of axisymmetric shell structures, aiming to reintroduce a modelling capability that was once standard in the field. Data input is entirely object‐oriented and matrix assembly is parallelized with pre‐compiled C++ routines, with users being able to take direct advantage of MATLAB's visualization properties. The software natively supports the LA, LBA, MNA, GMNIA etc. Eurocode analysis taxonomy. This paper demonstrates the current capabilities of the toolbox, describes the extensive programme of verification against existing established solutions that has been performed, and illustrates its ability to efficiently compute very detailed capacity curves using the EN 1993‐1‐6 capacity curve framework. [ABSTRACT FROM AUTHOR]

This paper focuses on the different bifurcations of fixed points of a delayed discrete neural network model analytically and numerically. The conditions and critical values of different bifurcations including the pitchfork, flip, Neimark–Sacker, and flip–Neimark–Sacker are analyzed. By using the critical coefficients, the structure for each bifurcation are determined. By taking one and two parameters, the critical coefficients are calculated and the curves associated with each bifurcation are plotted. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional‐order delay differential model for tuberculosis (TB) transmission with the effects of endogenous reactivation and exogenous reinfections. We investigate the qualitative behaviors of the model throughout the local stability of the steady states and bifurcation analysis. A discrete time delay is introduced in the model to justify the time taken for diagnosis of the disease. Existence and positivity of the solutions are investigated. Some interesting sufficient conditions that ensure the local asymptotic stability of infection‐free and endemic steady states are studied. The fractional‐order TB model undergoes Hopf bifurcation with respect to time delay and disease transmission rate. The presence of fractional order and time delay in the model improves the model behaviors and develops the stability results. A numerical example is provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the bifurcation problem for the fractional Laplace equation [ABSTRACT FROM AUTHOR]

Power converter circuits, such as current-controlled or voltage-controlled converters and inverters often have multiple inputs in the controller. The multiple inputs cause high-frequency and low-frequency oscillations. In earlier studies, the characteristics of circuits in fast-scale and slow-scale dynamics have been investigated. However, in many cases, circuits with multiple inputs have three or more dimensional topology which makes detailed analysis difficult. In this paper, we analyze a simple interrupted electric circuit in order to understand essential characteristics of fast-scale and slow-scale dynamics. The advantage of this simple interrupted circuit is that it is possible to derive a 1-dimensional map, which facilitates rigorous studies. Based on the structure of the return map and the characteristic multiplier, we explain the characteristics of the system. We report the occurrence of pitchfork, period doubling, and border collision bifurcations in slow scale, and period doubling bifurcation in fast scale. We found that local bifurcation, which appears in fast-scale dynamics, does not significantly affect the global behavior of the system while instabilities in the slow-scale dynamics strongly affect the system behavior. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper derives a renormalization formula defined on the parameter space where mapping behavior is preserved, together with the equivalent potential function. In contrast to the universal function given by Feigenbaum, the behavior near the critical point is governed by the potential function. There are several interesting features which are shared by Feigenbaum's universal function, such as the representation of the critical point in terms of the unstable fixed point of the potential function, but the mapping differs from the scaling. The one-dimensional mapping is considered as an example, and the critical point and the scaling, which are major constants characterizing chaos from the potential function, are calculated precisely. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(8): 41–51, 1998 [ABSTRACT FROM AUTHOR]

This paper considers the case where hysteresis is given by the Preisach model and discusses the stability analysis for the periodic solution in the nonautonomous system containing hysteretic elements. The behavior of the system containing hysteretic elements is affected by the history of the system until it arrives at the steady state. In this paper, it is shown first that even periodic solutions with the same hysteretic loop may have different variational systems for the steady-state solution if the histories until arriving at the steady state are different, Next, two kinds of stability criteria are considered based on the difference of the history up to the steady state and the stability against the external disturbance is investigated. The dependence of the stability of the solution on the bifurcation of the periodic solution also is discussed. Finally, examples of numerical analysis are presented to verify the result of discussions. [ABSTRACT FROM AUTHOR]

ABSTRACT This paper studies the static and dynamic behavior of a biological wastewater treatment process consisting of an aerobic bioreactor and a settler. The process is described by a three-variable model that takes explicitly into account the mass balance of oxygen. The analysis is carried out for the general case where the biomass specific growth rate is assumed to depend arbitrarily on substrate, biomass, and oxygen. The biomass yield coefficient is also assumed to depend arbitrarily on the substrate. General conditions describing the behavior of the model are derived. The general analysis is applied to an experimentally validated model where the specific growth rate depends on substrate and oxygen following a multiplicative Monod form. Practical diagrams are constructed that delineate the effect of various operating parameters on the performance of the biodegradation. The model is also studied for the case of a variable yield coefficient. The analysis shows the existence of a periodic behavior for some range of model parameters. The results of the paper allowed a useful understanding of the dynamics of the aerobic biodegradation including the effect of operating parameters, the specific growth rate, and the biomass yield coefficient on the existence of instabilities in the process. © 2013 Curtin University of Technology and John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a novel four‐dimensional fractional‐order financial system (FFS) with time delay is presented. Unlike traditional bifurcation analysis of financial systems, the selection rules of two bifurcation points within the system are discussed. In addition, the motion state of the system in the vicinity of two bifurcation points are analyzed separately, such that the dynamic analysis of this novel nonlinear fourth‐dimensional FFS is more comprehensive. The detailed dynamical behaviors of this financial system, such as oscillation, stability, and bifurcation points, are deduced via rigorous mathematical analysis. Finally, some simulations are performed to verify the dynamic characteristics of the FFS around the two bifurcation points which satisfy the selection conditions of the bifurcation point. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge-Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0-1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The paper presents an analytical and numerical investigation carried out on part of an electrical power network exhibiting ferroresonant behaviour as well as experimental investigation on its physical model. The investigations were carried out in order to estimate the extent of predicting the range of voltage source amplitude values at which the initiation of ferroresonance depends on values of initial conditions and phase shift. For this range of voltage source amplitude values (termed as possible ferroresonant range in the paper), the impact of initial conditions and phase shift of voltage source on the initiation of ferroresonance is determined using a numerical investigation and an experimental set-up. Reasons and consequences of disagreement of parameter values of the original and physical model are determined. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper aims to study the deterministic and stochastic features of a modified Leslie–Gower predator–prey model with Holling‐type II functional response. We first investigate the dynamical properties of the deterministic model, including existence and stability of the equilibrium, and different types of bifurcations. For the stochastic model, a phenomenon of noise‐induced state transition is found. By applying the stochastic sensitivity functions technique, we construct the confidence domain of stochastic attractor and then estimate the critical value of the intensity of noise generating this transition. Numerical simulations are performed to validate the analytical findings. [ABSTRACT FROM AUTHOR]

Nonlinear dynamic behaviors such as the bifurcation and chaos of DC‐DC converters with switching constant power load are studied in this paper. Based on the consideration of constant power load switching, a discrete iterative mapping model of the system including the switching process is established, and its Lyapunov exponent spectrum algorithm is derived. The main procedure of the algorithm is as follows: Firstly, the system is divided into six smooth segments, and the zero‐time discontinuous mapping of the load and mode switching is introduced. Secondly, the local mapping of each smooth segment and the zero‐time discontinuous mapping of the switching phase are used to construct the composite Poincaré mapping of the system by the chain rule, and then the discrete iterative mapping model of the system is established. Finally, the Lyapunov exponent spectrum algorithm of the system is derived based on the model to determine the type of nonlinear dynamic behavior of the system. This algorithm overcomes the problems of the existing methods, which lead to inaccurate results due to the neglect of the switching process, and achieves the purpose of accurately describing the dynamic characteristics of power electronic circuits. The simulation and experimental results have verified the validity of the proposed model and algorithm. © 2020 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

This paper studies a fractional‐order modelling chronic wasting disease (CWD). The basic results on existence, uniqueness, non‐negativity, and boundedness of the solutions are investigated for the considered model. The criterion for local as well as global stability of the equilibrium points is derived. A numerical analysis for Hopf‐type bifurcation is presented. Finally, numerical simulations are provided to justify the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic characteristics of fractional‐order nonlinear financial systems, including bifurcation and local asymptotic stability. Among them, we select the elasticity of demand of commercial (EDC) as the bifurcation point to discuss the state of the system. By calculating, the lowest order bifurcation point is obtained. Furthermore, the impulse control gains that follow a fractional‐order control law are applied to make the fractional‐order nonlinear financial system stable. In addition, some numerical simulation examples are provided to verify the effectiveness and the benefit of the proposed state form of the system near the bifurcation point and the states of the system when the impulse control is used or not. [ABSTRACT FROM AUTHOR]

The paper examines the main issues surrounding distributional effects in the domains of natural resource management and land policies, agricultural technology and research policies, agricultural market and trade policies, and consumer-oriented policies, including standards, subsidies, and labeling. Agriculture is drifting into an ever more drastic bifurcation at a global level and within many countries. Correcting that bifurcation will require large investments in rural areas and rural people, in institutions, and in information and biological technologies accessible by the poor in the world's smallholder sector. Large and growing national and international inequalities related to agriculture and rural areas threaten peace, growth, and sustainable development. [ABSTRACT FROM AUTHOR]

This paper contributes to the novel approach of a deterministic risk analysis. We introduce a risk index, which is designed such that large values of the index indicate a deterministic risk. The approach is illustrated by two examples. The new risk index allows to give the “risk area” a quantitative meaning. [ABSTRACT FROM AUTHOR]

Although the oscillatory dynamics of the p53 network have been extensively studied, the understanding of the mechanism of delay‐induced oscillations is still limited. In this paper, a comprehensive mathematical model of p53 network is studied, which contains two delayed negative feedback loops. By studying the model with and without explicit delays, the results indicate that the time delay of Mdm2 protein synthesis can well control the pulse shape but cannot induce p53 oscillation alone, while the time delay required for Wip1 protein synthesis induces a Hopf bifurcation to drive p53 oscillation. In addition, the synergy of the two delays will cause the p53 network to oscillate in advance, indicating that p53 begins the repair process earlier in the damaged cell. Furthermore, the stability and bifurcation of the model are addressed, which may highlight the role of time delay in p53 oscillations. [ABSTRACT FROM AUTHOR]

Summary: The paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is inherently nonsymmetric because of the geometrics of particle interactions and of the frictional behavior of the contacts. Internal geometric constraints are imposed by the particles' shapes, in particular, by the surface curvatures of the particles at their points of contact. Moreover, the stiffness relationship is incrementally nonlinear, and even small assemblies require the analysis of multiple stiffness branches, with each branch region being a pointed convex cone in displacement space. These aspects of the particle‐level stiffness relationship give rise to three types of microscale failure: neutral equilibrium, bifurcation and path instability, and instability of equilibrium. These three pathologies are defined in the context of four types of displacement constraints, which can be readily analyzed with certain generalized inverses. That is, instability and nonuniqueness are investigated in the presence of kinematic constraints. Bifurcation paths can be either stable or unstable, as determined with the Hill–Bažant–Petryk criterion. Examples of simple granular systems of three, 16, and 64 disks are analyzed. With each system, multiple contacts were assumed to be at the friction limit. Even with these small systems, microscale failure is expressed in many different forms, with some systems having hundreds of microscale failure modes. The examples suggest that microscale failure is pervasive within granular materials, with particle arrangements being in a nearly continual state of instability. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved method for analyzing the stability of the nonlinear hybrid dynamical systems (NHDSs) using a monodromy matrix. We define an n‐dimensional NHDS and focus on an orbit whose initial values exist near the periodic orbit. Because the system has nonlinearity, the circuit equation cannot be expressed in the form of a linear ordinary differential equation, meaning that the existing monodromy‐matrix‐based stability analysis method cannot be used because it requires a matrix exponential. Therefore, we propose a general theory for calculating orbital perturbations during Poincaré observation. Perturbation through switching events is expressed by a state‐transition matrix, which we refer to as the saltation matrix. By computing the perturbations, we obtain the monodromy matrix, whose characteristic multipliers denote the stability of the periodic orbit. We apply the proposed algorithm to an interrupted electric circuit with a nonlinear characteristic to confirm its validity. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

The paper investigates the complex nonlinear behavior of a fractional order four dimension power system (FOFDPS). The discrete mathematical model of the FOFDPS is derived and presented. The equilibrium points along with the Eigen values of commensurate and incommensurate FOFDPS are presented. The existence of chaotic oscillations are supported by a positive Lyapunov exponent. Bifurcation plots are derived for both parameters and fractional orders to show the impact of the same on the dynamic behavior of FOFDPS. Having shown the existence of such complex behaviors in the FOFDPS, we present an adaptive fractional order sliding mode control (FOASMC) to suppress the chaotic oscillations. Numerical results are presented to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa-Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirmthe qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV)/AIDS and cancer coexistence both in vivo and in vitro in a cancer‐immune environment leads to specific cytokines being produced by various immune cells and the cancer cells. Most of the studies have suggested that specific cytokines produced by the immune system cells and the tumor play an important role in the dynamics of non‐Hodgkin lymphomas (NHLs). In this paper, a mathematical model describing the NHL‐immune system interaction in the presence of the HIV, HIV treatment, and chemotherapy is developed. The formulated model, described by nonlinear ODEs, shows existence of multiple equilibria whose stability and bifurcation analysis are presented. From the bifurcation analysis, bistability regions are evident. We observe that with and without HIV treatment, the system results in a nonaggressive tumor size or aggressive tumor (full‐blown tumor) depending on the initial conditions. The results further suggest that at a low endemic state, patients can live for longer period with the tumor, which might explain why some patients can live with cancer for many years. However, initiation of HIV treatment in patients with NHL is observed to lower these endemic states of the tumor. Our results explain why late initiation of HIV treatment might not be helpful to NHL patients. We further investigated the effect of chemotherapy on the dynamics of the tumor. Our simulation results might explain why a few of these chemotherapeutic drugs are more effective when given at a slow continuous rate. The model provides a unique opportunity to influence policy on HIV‐related cancer treatment and management. [ABSTRACT FROM AUTHOR]

Summary: Accurate prediction of strength in rocks with distinct bedding planes requires knowledge of the bedding plane orientation relative to the load direction. Thermal softening adds complexity to the problem since it is known to have significant influence on the strength and strain localization properties of rocks. In this paper, we use a recently proposed thermoplastic constitutive model appropriate for rocks exhibiting transverse isotropy in both the elastic and plastic responses to predict their strength and strain localization properties. Recognizing that laboratory‐derived strengths can be influenced by material and geometric inhomogeneities of the rock samples, we consider both stress‐point and boundary‐value problem simulations of rock strength behavior. Both plane strain and 3D loading conditions are considered. Results of the simulations of the strength of a natural Tournemire shale and a synthetic transversely isotropic rock suggest that the mechanical model can reproduce the general U‐shaped variation of rock strength with bedding plane orientation quite well. We show that this variation could depend on many factors, including the stress loading condition (plane strain versus 3D), degree of anisotropy, temperature, shear‐induced dilation versus shear‐induced compaction, specimen imperfections, and boundary restraints. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we analyze a simple PWM-1 controlled interrupted electric circuit in order to essentially understand the circuit fundamental characteristics. First, we explain the circuit dynamics, and then we define the return map by using the exact solution. Next, we focus on the existence region of the solution (invariant interval) and bifurcation phenomena in the circuit. In particular, we find the circuit has three types of the invariant interval depending on the parameter. We also clarify that the period-doubling bifurcation and the border-collision bifurcation effect in the existence region of the periodic solution in a wide parameter plane. Finally, the mathematical results are verified by the laboratory experiment. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the dynamical analysis and chaos control in a fractional order synchronous reluctance motor (FOSyncRM). Equilibrium points, characteristic equations and Eigen values of both commensurate and incommensurate FOSyncRM are presented. finite‐time Lyapunov exponents of the FOSyncRM system for fixed and varied parameters are investigated along with the bifurcation plots. Fractional order bifurcation plots are derived to show that the system shows more complex chaotic oscillations in fractional order. Chaos control in the FOSyncRM system is achieved using adaptive sliding mode controllers and the entire control algorithm is implemented in FPGA. [ABSTRACT FROM AUTHOR]