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The periodical dynamics of a G0 cell cycle model of pluripotential stem cells is analyzed by DDE‐Biftool software. The cell cycle model is impressed by modeling the optional choice of Hill function, which is benefited by Fourier transformation. The cell cycle is based on DDEs with distributed time delay, in which the kernel function is denoted by Gamma‐distribution expression. Hopf bifurcation of the linear version of the cell cycle model with distribution time delay is analyzed analytically. The periodical solution continuation is simulated by the artificial handbook of DDE‐Biftool software. With the discrete time delay, the complex behavior of adding‐period bifurcation and period‐doubling bifurcation are simulated. With distribution time delay, the continuation work of the homoclinic solution is done, and the homoclinic bifurcation line crosses the generalized Hopf point nearly. JEL Classification: 34C25, 34K18, 37G15 [ABSTRACT FROM AUTHOR]

The time-dependent viscoelastic deformation of host rocks is important when considering the dynamics of fault behavior, specifically in brittle–ductile transitional regions or shallow subduction zones, because it relaxes stress heterogeneity and affects loading to the fault. For a rate-and-state fault embedded in a Maxwell viscoelastic medium, a previous study discovered a transition from repeated earthquakes to the permanent stuck of a rate-weakening patch (EQ–ST transition) with decreased viscoelastic relaxation time t c . This transition differs from the well-known seismic–aseismic transition explained by the Hopf bifurcation at the critical stiffness of an elastic system. To better understand the EQ–ST transition, quantifying the effect of heterogeneous frictional strength is important, because this effect is characteristic to the viscoelastic medium and is absent in the elastic limit. Previous experimental studies suggest a potential contrast in frictional strength Δ f ∗ in such a way that a rate-weakening patch is stronger than a rate-strengthening region containing clay minerals. Here, we conducted two-dimensional, fully dynamic earthquake sequence simulations for a fault in a Maxwell viscoelastic medium; we investigated the EQ–ST transition in the two-dimensional parameter space of t c and Δ f ∗ . With Δ f ∗ = 0.3 , the EQ–ST transition occurred at about 1 order of magnitude larger t c than in the case with Δ f ∗ = 0 . We constructed a coarse-grained model with only two degrees of freedom based on the spatial average. Consequently, the coarse-grained model behaves remarkably similar to the continuum model, and the EQ–ST transition is associated with a homoclinic bifurcation. The EQ–ST boundary in the parameter space can be quantitatively explained by considering elastic loading due to creep in the rate-strengthening region and unloading by viscoelastic relaxation of the stress heterogeneity comparable to Δ f ∗ . A larger rate-weakening patch is anticipated to become aseismic earlier as t c decreases, because the elastic loading rate is inversely correlated with the patch size. This may be qualitatively consistent with the change in the size distribution of events around the brittle–ductile transition in observations and laboratory experiments; however, further investigations on, for example, the interactions of events and changes in frictional parameters with depth are required for quantitative discussion. [ABSTRACT FROM AUTHOR]

We present a comprehensive experimental study of the dynamics of electrostatic MEMS resonators under large excitations. We identified three frequency ranges where large oscillations occur; a non-resonant region driven by fast–slow dynamic interactions and two resonant regions. In these regions, we found a plethora of dynamic phenomena including cascades of period-doubling bifurcations, a bubble structure, homoclinic and cyclic-fold bifurcations, hysteresis, intermittencies, quasiperiodicity, chaotic attractors, odd-periodic windows within those attractors, Shilnikov orbits, and Shilnikov chaos. We encountered these complex nonlinear dynamics phenomena under relatively high dissipation levels, the quality factors of the resonators examined in this study were Q = 6.2 and 2.1. In the case of MEMS with higher quality factors (Q > 100) , it is quite reasonable to expect those phenomena to appear under relatively low excitation levels (compared to the static pull-in voltage). This calls for a new paradigm in the design of electrostatic MEMS that seeks to manage dynamic phenomena rather than attempt to avoid them and, thereby, overly restricting the design space. We believe this is feasible given the repeatable and predictable nature of those phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26]. [ABSTRACT FROM AUTHOR]

Multiple-timescale systems often display intricate dynamics, yet of great mathematical interest and well suited to model real-world phenomena such as bursting oscillations. In the present work, we construct a piecewise-linear version of the Morris–Lecar neuron model, denoted PWL-ML, and we thoroughly analyse its bifurcation structure with respect to three main parameters. Then, focusing on the homoclinic connection present in our PWL-ML, we study the slow passage through this connection when augmenting the original system with a slow dynamics for one of the parameters, thereby establishing a simplified framework for this slow-passage phenomenon. Our results show that our model exhibits equivalent behaviours to its smooth counterpart. In particular, we identify canard solutions that are part of spike-adding transitions. Focusing on the one-spike and on the two-spike scenarios, we prove their existence in a more straightforward manner than in the smooth context. In doing so, we present several techniques that are specific to the piecewise-linear framework and with the potential to offer new tools for proving the existence of dynamical objects in a wider context. [ABSTRACT FROM AUTHOR]

Homoclinic bifurcation in a class of SD oscillators under external quasi-periodic excitation with two frequencies is investigated. By using the Melnikov method, the Melnikov function of the system under two-frequency excitation is derived, and the threshold condition of chaos in the system is obtained. Based on the threshold condition, then a complete description of the bifurcation sets and the chaotic regions in the parameter space are presented. The parameters in the chaotic regions are selected for numerical simulation, and the chaotic motion is verified by calculating the largest Lyapunov exponent of the system. In addition, due to the geometric strong nonlinearity of the SD oscillator, there are infinitely many extreme points in the frequency-dependent function of the Melnikov function. The previous research on the frequency-dependent function of the Melnikov function is mainly focused on one or two extremes. In this work, the dynamical system with infinitely many extreme points in the frequency-dependent function is considered, and a conjecture of chaotic region under this condition is given. [ABSTRACT FROM AUTHOR]

‎We consider a planar symmetric vector field that undergoes a homoclinic bifurcation‎. ‎In order to study the existence of exterior periodic solutions of the vector field around broken symmetric homoclinic orbits‎, ‎we investigate the existence of fixed points of the exterior Poincare map around these orbits‎. This Poincare map is the result of the combination of flows inside and outside the homoclinic orbits. It shows how ‎a big periodic orbit converts to two small periodic orbits by passing through a double homoclinic structure‎. Finally‎, ‎we use the results to investigate the existence of periodic solutions of the perturbed Duffing equation.

We consider a planar symmetric vector field that undergoes a homoclinic bifurcation‎. ‎In order to study the existence of exterior periodic solutions of the vector field around broken symmetric homoclinic orbits‎, ‎we investigate the existence of fixed points of the exterior Poincaré map around these orbits‎. This Poincaré map is the result of the combination of flows inside and outside the homoclinic orbits. It shows how ‎a big periodic orbit converts to two small periodic orbits by passing through a double homoclinic structure‎. Finally‎, ‎we use the results to investigate the existence of periodic solutions of the perturbed Duffing equation. [ABSTRACT FROM AUTHOR]

The effects of time delay on dynamics of a predator–prey system with multiple coexisting equilibria are explored. The system exhibits focus-node and cycle-node bistability in the absence of delay. The delay-induced stability and bifurcations of the coexisting equilibria, and the evolution of the bistability are analyzed. Criteria for different delay-driven stability scenarios including stability and instability switches are derived. Our investigation indicates that the delay controls the bistability through different bistable modes. For focus-node bistability, the system evolves through the bistable modes: focus-node → focus-focus → focus-cycle as the delay grows. On the other hand, we obtain two different scenarios, viz., cycle-node → cycle-focus and cycle-node → cycle-focus → cycle-cycle for the effect of delay on the cycle-node bistability. We report the existence of a homoclinic bifurcation for transition from bistable to monostable dynamics. Furthermore, it is also revealed by computing mean density that due to bistability, time delay can be beneficial or harmful for biological conservation of the populations. [ABSTRACT FROM AUTHOR]

This paper proposes an L p optimal control method to shift the homoclinic bifurcation of single-degree-of-freedom nonlinear oscillators. The homoclinic intersection between stable and unstable manifolds is detected analytically using the Melnikov method. An optimization strategy is formulated to obtain the control gain that minimizes the absolute value of the largest magnitude characteristic multiplier of the periodic orbit that emerges due to the shift of the homoclinic bifurcation. The optimal control gain is obtained by considering the stability of this periodic orbit. The cross-entropy method is used to obtain a numerical optimal solution to the optimization problem and, consequently, to obtain an optimal controller for the feedback method presented here. As an example of the L p optimal control strategy, we consider the periodically forced Duffing oscillator with a twin-well potential. The numerical results demonstrate the capability of the L p optimal control procedure to shift the homoclinic bifurcation. Moreover, the L p optimal control can help regularize the fractal basin boundaries of the two confined attractors in the two potential wells. [ABSTRACT FROM AUTHOR]

Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant D to study the homoclinic bifurcations to limit cycles for the case | λ 1 | = λ 3 . Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In a previous publication, we established a new paradigmatic model of laser with feedback including the minimal nonlinearity leading to chaos. In this paper, the jerk dynamics of this minimal universal model of laser is presented. It is proved that two equivalent forms of the model in jerk dynamics can be derived. The electronic circuit of the simpler dynamics is designed and implemented. The link between the minimal universal model of laser and the search for simple jerk circuits is established. [ABSTRACT FROM AUTHOR]

A spike induced by inhibitory stimulation instead of excitatory stimulation, called post-inhibitory rebound (PIR) spike, has been found in multiple neurons with important physiological functions, which presents counterintuitive behavior mainly related to focus near Hopf bifurcation. In the present paper, the condition for the PIR spike is extended to small homoclinic orbit (SHom) and saddle-node (SN) bifurcations, and the underlying mechanism is acquired in a neuron model. Firstly, PIR spike is evoked from a stable node near the SHom or SN bifurcation by a strong inhibitory stimulation. Then, the dynamics of threshold curve for a spike, vector fields, and nullcline of recovery variable are used to well explain the cause for the PIR spike. The shape of threshold curve for the node resembles that of focus. The nullcline plays an important role in forming PIR spike, which is analytically identified at last. Besides, a sufficient condition is acquired from the integration to a differential equation, and the range of parameters for the PIR spike is presented. The extended bifurcation types and the underlying mechanisms for the PIR spike such as the nullcline present comprehensive and deep understandings for the PIR spike, which also provides potential strategy to modulate the PIR phenomenon and even related physiological functions of neurons.

An islanded microgrid consisting of grid-forming converters, being a high-order nonlinear system, exhibits rich nonlinear dynamical phenomena. The use of appropriate reduced-order models offers useful physical insights into the behavior of the system without the need for excessive computational resources. In this article, we derive a number of reduced-order models capable of describing the slow-scale dynamics of an islanded microgrid comprising a number of grid-forming converters. It is shown that slow-scale Hopf and homoclinic bifurcation behaviors arise from the stability of the voltage loops of grid-forming converters and are unrelated to the transmission network dynamics. Therefore, omitting the network dynamics does not affect the accuracy of reduced-order models in representing the slow-scale dynamics of the system. This is especially beneficial for modeling the microgrid with a complex transmission network. Furthermore, on this basis, all inner loops can be omitted when studying saddle-node bifurcation, leading to the development of power-flow-based reduced-order models. Finally, the stability of an islanded microgrid with a complex transmission network is evaluated. [ABSTRACT FROM AUTHOR]

Fractional calculus as a new approach for modeling has been used widely to study the non-linear behavior of physical and biological systems with some degrees of fractionality or fractality using differential and integral operators with non integer-orders. In this paper, to explore different dynamical classes of the Morris–Lecar neuronal model with chloride channel, we extend its integer-order domain into a new fractional-order space using fractional calculus. The nonstandard finite difference (NSFD) method following the Grünwald–Letnikov discretization may be applied to discretize the model and obtain the fractional-order form. Fractional derivative order has been used as a new control parameter to extract variety of neuronal firing patterns that happen in real world application but the integer-order operator may not be able to reveal them. To find the impact of chloride channel on dynamical behaviors of this neuronal model, the phase portrait and time series analysis have been performed for different fractional-orders and input currents. Depending on different values for γ, the fractional-order Morris–Lecar model with a chloride channel reproduces quiescent, spiking and bursting activities the same as the fractional-order Morris–Lecar model without a chloride channel. We numerically discover the occurrence of hopf bifurcation, and homoclinic bifurcation for these two models. These results show that adding a chloride channel to this neuron does not affect the overall spiking patterns of the model, however, when we add this new ionic channel, the neuron needs higher input current as stimulus to fire action potential and spike.

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A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modified numerical response. This numerical response accounts for the reduction in the herbivore's growth and reproduction due to chemical defenses from plants. It is shown that the system exhibits very rich dynamics including saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical simulations are presented to illustrate the occurrence of multitype bistability, limit cycles, homoclinic orbits and heteroclinic orbits. We also discuss the ecological implications of the resulting dynamics.

• A bistable energy harvester subjected to galloping and base excitations is proposed. • The threshold for homoclinic bifurcation is obtained by Melnikov method. • The accuracy of Melnikov analysis is analyzed by numerical simulation. • The effect of wind speed on system's nonlinear dynamics is verified experimentally. In this paper, we studied the homoclinic bifurcation and nonlinear characteristics of a bistable piezoelectric energy harvester while it is concurrently excited by galloping and base excitation. Firstly, the electromechanical model of the energy harvester is established analytically by the energy approach, the Kirchhoff's law and quasi-steady hypothesis. Then, by the Melnikov method, the threshold for underlying snap-through in the system is derived, and the necessary conditions for homoclinic bifurcation and chaos are presented. The threshold is a determinant for the occurrence of high-energy oscillation. The analysis results reveal that the wind speed and the distance between magnets could affect the threshold for inter-well chaos and high energy oscillation. Finally, numerical simulation and experiments are carried out. Both results from numerical simulation and experiment support the theoretical prediction from Melnikov theory. The study could provide a guideline for the optimum design of the bi-stable piezoelectric energy harvester for wind and vibration in practice. [ABSTRACT FROM AUTHOR]

A spike induced by inhibitory stimulation instead of excitatory stimulation, called post-inhibitory rebound (PIR) spike, has been found in multiple neurons with important physiological functions, which presents counterintuitive behavior mainly related to focus near Hopf bifurcation. In the present paper, the condition for the PIR spike is extended to small homoclinic orbit (SHom) and saddle-node (SN) bifurcations, and the underlying mechanism is acquired in a neuron model. Firstly, PIR spike is evoked from a stable node near the SHom or SN bifurcation by a strong inhibitory stimulation. Then, the dynamics of threshold curve for a spike, vector fields, and nullcline of recovery variable are used to well explain the cause for the PIR spike. The shape of threshold curve for the node resembles that of focus. The nullcline plays an important role in forming PIR spike, which is analytically identified at last. Besides, a sufficient condition is acquired from the integration to a differential equation, and the range of parameters for the PIR spike is presented. The extended bifurcation types and the underlying mechanisms for the PIR spike such as the nullcline present comprehensive and deep understandings for the PIR spike, which also provides potential strategy to modulate the PIR phenomenon and even related physiological functions of neurons. [ABSTRACT FROM AUTHOR]

We continue to study the nonsmooth van der Pol-Duffing oscillator x ˙ = y x ˙ = y , y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y , where a i , b i a i , b i are real and a 2 b 2 ≠ 0 a 2 b 2 ≠ 0 , i = 1 , 2 i = 1 , 2. Notice that the sum of indices of equilibria is − 1 − 1 for a 2 > 0 a 2 > 0 and 1 1 for a 2 < 0 a 2 < 0. When a 2 > 0 a 2 > 0 , the nonsmooth van der Pol-Duffing oscillator has been studied completely in the companion paper. Attention goes to the bifurcation diagram and all global phase portraits in the Poincaré disc of the nonsmooth van der Pol-Duffing oscillator for a 2 < 0 a 2 < 0 in this paper. The bifurcation diagram is more complex, which includes two Hopf bifurcation surfaces, one pitchfork bifurcation surface, one homoclinic bifurcation surface, one double limit cycle bifurcation surface and one bifurcation surface for equilibria at infinity. When b 2 > 0 b 2 > 0 is fixed, this nonsmooth van der Pol-Duffing oscillator cannot be changed into a near-Hamiltonian system for small a 1 , b 1 a 1 , b 1. Moreover, the global dynamics of the nonsmooth van der Pol-Duffing oscillator and the van der Pol-Duffing oscillator are different. [ABSTRACT FROM AUTHOR]

We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps Tc,ɛ(x) = (−1 + c|x|1−ɛ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

We have applied the Melnikov criterion to examine a homoclinic bifurcation and transition to chaos in the Duffing oscillator driven by different forms of periodic piecewise linear forces. The periodic piecewise linear forces considered are triangular, hat, trapezium, quadratic and rectangular types of forces. For all the forces, an analytical threshold condition for the homoclinic transition to chaos is derived using Melnikov method and Melnikov threshold curves are drawn in a parameter space. Using the Melnikov threshold curves, we have found a critical forcing amplitude fc above which the system may behave chaotically. We have analyzed both analytically and numerically the homoclinic transition to chaos in the Duffing system with ε-parametric force also. The predictions from Melnikov method have been further verified numerically by integrating the governing equation and finding areas of chaotic behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation x ¨ + a x ˙ + b x ˙ | x ˙ | + c x + d x 3 = 0 for the case d > 0 , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case d < 0 , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Under transient disturbance, the grid-forming voltage-source converter may lose its synchronization with the grid, inducing sustained low-frequency oscillation in instantaneous power, current, and phase angle. The physical origin of such oscillations is found to be a homoclinic bifurcation in this article. Before the system runs into a homoclinic bifurcation, a stable equilibrium point (SEP) and a stable periodic orbit coexist. When a large transient disturbance is applied, the system exhibits a periodic orbit, which manifests itself as low-frequency oscillation. Moreover, after the homoclinic bifurcation, the periodic orbit subsides, and only a single attractor, the SEP, exists in the phase space. In this case, the grid-forming converter is able to resynchronize with the grid even under transient disturbances. Bifurcation diagrams are derived as the boundaries of stable operation in the parameter space, which serve as practical design guidelines to avoid sustained oscillations. Cycle-by-cycle simulations and laboratory experiments are performed to verify the analytical findings. [ABSTRACT FROM AUTHOR]

The occurrence of homoclinic bifurcation in a nonlinearly damped Duffing-vander Pol (DVP) oscillator under the excitation of two-forcing terms with different frequencies (fl, co) is analyzed with both analytical and numerical techniques. For our study, we consider two periodic cosine waves and two modulus of sine waves with different frequencies. We assume that the nonlinear damping term is proportional to the power of velocity @) in the form lilp-1. Applying the Melnikov analytical technique, the threshold condition for the occurrence of horseshoe chaos is obtained for each excitations. Melnikov threshold curves, separating the chaotic and nonchaotic regions are obtained. The chaotic features on the system parameters are discussed in detail. Numerical results are given, which verify the analytical ones. [ABSTRACT FROM AUTHOR]

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model's parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system's coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes. [ABSTRACT FROM AUTHOR]

Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions. [ABSTRACT FROM AUTHOR]

A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor's theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system. [ABSTRACT FROM AUTHOR]

The bifurcation of an ordinary differential equation model describing interaction of the wild and the released sterile mosquitoes is analyzed. It is shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, supercritical Hopf bifurcation, subcritical Hopf bifurcation, homoclinic bifurcation and Bogdanov-Takens bifurcation. We also find that the model displays monostable, bistable or tristable dynamics. This analysis suggests that the densities of the initial wild mosquitoes and the released sterile ones determine the asymptotic states of both populations. This study may give an insight into the estimation number of the released sterile mosquitoes.

We revisit the model of the laser with feedback and the minimal nonlinearity leading to chaos. Although the model has its origin in laser physics, with peculiarities related to the CO 2 laser, it belongs to the class of the three-dimensional paradigmatic nonlinear oscillator models giving chaos. The proposed model contains three key nonlinearities, two of which are of the type x y , where x and y are the fast and slow variables. The third one is of the type x z 2 , where z is an intermediate feedback variable. We analytically demonstrate that it is essential for producing chaos via local or global homoclinic bifurcations. Its electronic implementation in the range of kilo Hertz region confirms its potential in describing phenomena evolving on different time scales. [ABSTRACT FROM AUTHOR]

Nonlinear dynamic performances such as homoclinic bifurcation and chaos were investigated for tristable vibration energy harvesting systems. The analytical expression of the symmetric and asymmetric homoclinic solution was obtained through the Padé approximation, which was consistent with the numerical solution. According to the Melnikov theory, the qualitative method of studying the homoclinic bifurcation of the energy harvesting system with a triple well was developed, and the necessary condition for the occurrence of homoclinic bifurcation was obtained. Numerical simulations yielded bifurcation diagrams and maximum Lyapunov exponents that demonstrated the inter-well responses predicted with the Melnikov method. Compared with the system with symmetric potential energy, the system with asymmetric potential energy has a lower threshold of homoclinic bifurcation. For a low excitation level, the system with asymmetric potential energy witnesses inter-well chaos, while the response of the system with symmetric potential energy still keeps trapped in a single well. The change of symmetry of the system potential energy function improves the output voltage due to the increase in the probability of generating a large periodical inter-well oscillation response. The research on the homoclinic bifurcation of nonlinear energy harvesting systems with symmetric and asymmetric triple potential wells provides an effective tool for the parametric design of high-performance energy harvesters. [ABSTRACT FROM AUTHOR]

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the X , Y , and Z state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The X and Z solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function (sech) and a hyperbolic secant squared function ( sech 2 ), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the Z component and the KdV equation were obtained. Additionally, the ODE for the X component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

In this work, we consider a family of Lotka–Volterra maps (x ′ , y ′) = (x (a − x − y) , b x y) for a > 1 and b > 0 which unfold a map originally proposed by Sharkosky for a = 4 and b = 1. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region y < 0. Some properties and bifurcations are described. The x -axis is invariant, on which the map reduces to the logistic. For any a > 1 an interval of values for b exists for which all the cycles on the x -axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the x -axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent. [ABSTRACT FROM AUTHOR]

Recently, it has been shown that DC–AC and AC–DC power converters whose dynamics is governed by two vastly different frequencies lead to a special class of piecewise smooth models characterized by a practically unpredictable number of switching manifolds between partitions in the state space associated with different dynamics. In the previous publications, we have shown that transformations of chaotic attractors in such models can be caused by their interactions with unstable cycles occurring in the regions of the attractors associated with an ultra-low invariant density, which is beyond a practical observability in physical or numerical experiments. The appearance of such low-density regions has been explained by the so-called spiking phenomenon. In the present paper, we make the next step and ask the natural question what are the reasons behind the appearance of spikes. Applying a novel technique (cobweb diagrams for non-autonomous maps), we explain a connection between spiking and the presence of two vastly different frequencies. [ABSTRACT FROM AUTHOR]