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Matsuda and Abrams (Theor Popul Biol 45(1):76–91, 1994) initiated the exploration of self-extinction in species through evolution, focusing on the advantageous position of mutants near the extinction boundary in a prey-predator system with evolving foraging traits. Previous models lacked theoretical investigation into the long-term effects of harvesting. In our model, we introduce constant-effort prey and predator harvesting, along with individual logistic growth of predators. The model reveals two distinct evolutionary outcomes: (i) Evolutionary suicide, marked by a saddle-node bifurcation, where prey extinction results from the invasion of a lower forager mutant; and (ii) Evolutionary reversal, characterized by a subcritical Hopf bifurcation, leading to cyclic prey evolution. Employing an innovative approach based on Gröbner basis computation, we identify various bifurcation manifolds, including fold, transcritical, cusp, Hopf, and Bogdanov-Takens bifurcations. These contrasting scenarios emerge from variations in harvesting parameters while keeping other factors constant, rendering the model an intriguing subject of study. [ABSTRACT FROM AUTHOR]

This study models the subcritical Hopf bifurcation in thermoacoustic Stirling engines using the Stuart–Landau model, highlighting the role of nonlinear dynamics. By inducing self-sustained oscillations and measuring pressure fluctuations across different temperature gradients imposed on the regenerator, we reveal the engine's transition to a nonlinear domain, characterized by heightened oscillation amplitudes and unique periodic patterns. Interpreted Landau constants and growth rates illuminate the stabilizing effects of nonlinear dynamics, demonstrating the Stuart–Landau model's applicability in thermoacoustic engine analysis. Our research confirms that this empirically refined model reliably describes oscillation amplitudes and transient phenomena, contributing valuable perspectives for advancing thermoacoustic engine design and operational understanding. [ABSTRACT FROM AUTHOR]

This study models the subcritical Hopf bifurcation in thermoacoustic Stirling engines using the Stuart–Landau model, highlighting the role of nonlinear dynamics. By inducing self-sustained oscillations and measuring pressure fluctuations across different temperature gradients imposed on the regenerator, we reveal the engine’s transition to a nonlinear domain, characterized by heightened oscillation amplitudes and unique periodic patterns. Interpreted Landau constants and growth rates illuminate the stabilizing effects of nonlinear dynamics, demonstrating the Stuart–Landau model’s applicability in thermoacoustic engine analysis. Our research confirms that this empirically refined model reliably describes oscillation amplitudes and transient phenomena, contributing valuable perspectives for advancing thermoacoustic engine design and operational understanding.

A nonlinear phenomenon in nature is often modeled by a system of differential equations with parameters. The bifurcation occurs when a parameter varies in such systems, causing a qualitative change in its solution. In this paper, we study one of the most exciting bifurcations, which is Hopf bifurcation. We use tools from algebraic topology to analyze and reveal supercritical and subcritical Hopf bifurcations. [ABSTRACT FROM AUTHOR]

Mixed-mode oscillations (MMOs) consisting of spikes alternating with a series of subthreshold oscillations have been observed in various neurons related to some physiological functions. In the present paper, inhibitory-autapse-induced MMOs are simulated by using the Hodgkin-Huxley neuron model, and the underlying dynamical mechanism is identified to be related to dynamics of unstable behaviors near subcritical Hopf bifurcation. For the monostable spiking, a delayed inhibitory current pulse activated by a spike can suppress the phase trajectory corresponding to depolarization phase of the next spike to the unstable focus nearby or the neighborhood outside of unstable limit cycle, respectively. Then the trajectory rotates multiple cycles away and converges to the stable limit cycle, resulting in an evolution process of membrane potential from small-amplitude subthreshold oscillations to a large-amplitude spike, i.e., MMOs. For the spiking coexisting with the resting state, inhibitory autapse induces MMOs and resting state from the spiking. The difference in the MMOs from those induced by the excitatory autapse is identified. The result presents the underlying nonlinear mechanisms of inhibitory autapse to suppress the neuronal firing and reveals the potential role to control the neuronal firing patterns near subcritical Hopf bifurcation.

In this work, we investigate the effects of background noise characteristics — specifically noise color (or correlation time) and intensity — arising from velocity-coupled and additive noise sources, on the early warning indicators (EWIs) of thermoacoustic instability. We employ a nonlinear reduced-order combustion dynamics model for our investigation. In the absence of noise, the nonlinear model undergoes a subcritical Hopf bifurcation, and our focus lies within the linearly stable region of the system (subthreshold region). The studied class of EWIs encompasses those derived from time series (variance), spectral analysis (coherence factor), Hurst exponent, and nonlinear methods (permutation entropy). We find that when the background noise is purely multiplicative, the trends in EWIs are primarily influenced by noise characteristics rather than the control parameter. Further, the EWIs cannot be estimated at low noise levels for large correlation times. In case of purely additive noise driven system, the coherence factor and variance are reliable EWIs across all range of investigated noise characteristics. The Hurst exponent can serve as effective EWI when the system features large noise correlation times, while permutation entropy is effective only when the system features small noise correlation time, i.e., where white noise assumption is acceptable. When the background noise includes contributions from both multiplicative and additive sources, coherence factor and variance emerges as the most reliable EWIs. These results provide insights for selecting appropriate EWIs to be employed in practical systems, considering potential variations in noise characteristics with simultaneous changes in combustor operating conditions. Novelty and significance This study conclusively shows that background noise (multiplicative and additive) characteristics – noise correlation time (color) and intensity – can significantly change the trends in early warning indicators (EWIs) for predicting the impending thermoacoustic oscillations. In gas turbine combustors, where thermoacoustic instability poses a critical challenge, inherent noise dynamics undergo variations with changing operating conditions and combustor designs. The development of effective EWIs to foresee the onset of thermoacoustic instability is crucial for preventing potential damage and ensuring the reliable operation of combustion systems. Previous works on stochastic dynamics of combustors simplify noise with the additive white noise assumption. Thus, our results demonstrated on a nonlinear combustion dynamics model advance the state-of-the-art with respect to the early prediction and control of thermoacoustic instability. The results provide valuable insights for selecting appropriate EWIs for engine monitoring in the absence of information on noise properties and their variation with operating conditions. This practical information is of direct relevance and interest to any gas turbine manufacturer/user. [ABSTRACT FROM AUTHOR]

Based on the post-inhibitory rebound (PIR) spike induced by inhibitory current pulse, in the present paper, a novel counterintuitive phenomenon that the inhibitory autapse with time delay can induce the resting state changed to stable spiking pattern is identified near subcritical Hopf bifurcation of Hodgkin–Huxley model. The delayed inhibitory autaptic current pulse induced by the preceded action potential can induce the preceding PIR spike via the hyperpolarization, rebound, and depolarization processes, which is compared with spiking induced by excitatory autapse via only a depolarization process. The threshold of inhibitory or excitatory autaptic conductance to induce spiking with increasing time delay, and the threshold curve of inhibitory or excitatory pulse current to evoke a spike exhibit damping oscillations can be well interpreted with the damping dynamics of focus near subcritical Hopf bifurcation. However, due to PIR mechanism, the threshold conductance of inhibitory autapse is stronger than that of excitatory autapse, and the spiking period for inhibitory autapse, which is composed of time delay and durations of the other three processes, is longer than the one for excitatory autapse, which is composed of time delay and duration of only a depolarization process. Therefore, a linear correlation between spiking period and time delay is identified, which shows that autapse can modulate the spike timing related to temporal coding. The results present a novel viewpoint and a potential function that inhibitory autapse can facilitate spiking like the excitatory autapse and provide effective measures to modulate neuronal spiking pattern, which is related to subcritical Hopf bifurcation. [ABSTRACT FROM AUTHOR]

Under the threat of predation, a species of prey can evolve to its own extinction. Matsuda and Abrams (Theor Popul Biol 45:76–91, 1994a) found the earliest example of evolutionary suicide by demonstrating that the foraging effort of prey can evolve until its population dynamics cross a fold bifurcation, whereupon the prey crashes to extinction. We extend this model in three directions. First, we use critical function analysis to show that extinction cannot happen via increasing foraging effort. Second, we extend the model to non-equilibrium systems and demonstrate evolutionary suicide at a fold bifurcation of limit cycles. Third, we relax a crucial assumption of the original model. To find evolutionary suicide, Matsuda and Abrams assumed a generalist predator, whose population size is fixed independently of the focal prey. We embed the original model into a three-species community of the focal prey, the predator and an alternative prey that can support the predator also alone, and investigate the effect of increasingly strong coupling between the focal prey and the predator's population dynamics. Our three-species model exhibits (1) evolutionary suicide via a subcritical Hopf bifurcation and (2) indirect evolutionary suicide, where the evolution of the focal prey first makes the community open to the invasion of the alternative prey, which in turn makes evolutionary suicide of the focal prey possible. These new phenomena highlight the importance of studying evolution in a broader community context. [ABSTRACT FROM AUTHOR]

• We derive an ecological predator-prey model from individual-level processes with cannibalistic predators and timid prey. • Bifurcation analysis reveals ecological bistability between equilibrium and periodic population attractors. • We investigate the evolution of timidity of the prey along each ecological attractor, obtaining ten qualitatively different evolutionary scenarios. • Ecological bistability plays a central role in the evolutionary scenarios, which involve abrupt switching between alternative ecological attractors through catastrophic bifurcations. • We find that the end result can be an evolutionary cycle of the level of timidity, driven by ecological attractor switching. We investigate the evolution of timidity in a prey species whose predator has cannibalistic tendencies. The ecological model is derived from individual-level processes, in which the prey seeks refuge after detecting a predator, and the predator cannibalises on the conspecific juveniles. Bifurcation analysis of the model reveals ecological bistability between equilibrium and periodic attractors. Using the framework of adaptive dynamics, we classify ten qualitatively different evolutionary scenarios induced by the ecological bistability. These scenarios include ecological attractor switching through catastrophic bifurcations, which can reverse the direction of evolution. We show that such reversals often result in evolutionary cycling of the level of timidity. In the absence of cannibalism, the model never exhibits ecological bistability nor evolutionary cycling. We conclude that cannibalistic predator behaviour can completely change both the ecological dynamics and the evolution of prey. [ABSTRACT FROM AUTHOR]

Railway vehicles suffer from hunting motion, even when traveling below the critical speed obtained by linear analysis, due to the nonlinear characteristics of the wheel system. Nonlinear characteristics in Hopf bifurcations can be characterized as subcritical or supercritical, depending on whether the cubic nonlinearity is softening or hardening, respectively. In a system with softening cubic nonlinearity, third-order nonlinear analysis cannot detect nontrivial stable steady-state oscillations because they are affected by quintic nonlinearity. Therefore, in such a system, it is necessary to apply fifth-order nonlinear analysis to a system model in which quintic nonlinearity is taken into account. In this study, we investigated the cubic and quintic nonlinear phenomena in hunting motion with a roller rig that is widely used for hunting motion research. Previous experimental studies using a roller rig were restricted to the linear stability and the cubic nonlinear stability. We clarified that roller rig experiments can observe the hysteresis phenomenon and the existence of subcritical Hopf and saddle-node bifurcations, indicating that not only the cubic but also the quintic nonlinearity of the wheel system plays an important role. In addition, we obtained the normal form governing the nonlinear dynamics. We developed an experimental identification method to obtain the coefficients of the normal form. The validity of our method was confirmed by comparing the bifurcation diagrams obtained from the experimental time history and the normal form whose coefficients were experimentally identified using the proposed method. [ABSTRACT FROM AUTHOR]

The FitzHugh–Nagumo model is improved to consider the effect of the electromagnetic induction on single neuron. On the basis of investigating the Hopf bifurcation behavior of the improved model, stochastic resonance in the stochastic version is captured near the bifurcation point. It is revealed that a weak harmonic oscillation in the electromagnetic disturbance can be amplified through stochastic resonance, and it is the cooperative effect of random transition between the resting state and the large amplitude oscillating state that results in the resonant phenomenon. Using the noise dependence of the mean of interburst intervals, we essentially suggest a biologically feasible clue for detecting weak signal by means of neuron model with subcritical Hopf bifurcation. These observations should be helpful in understanding the influence of the magnetic field to neural electrical activity.

Biofilm communities of Bacillus subtilis bacteria have recently been shown to exhibit collective growth-rate oscillations mediated by electrochemical signaling to cope with nutrient starvation. These oscillations emerge once the colony reaches a large enough number of cells. However, it remains unclear whether the amplitude of the oscillations, and thus their effectiveness, builds up over time gradually or if they can emerge instantly with a nonzero amplitude. Here we address this question by combining microfluidics-based time-lapse microscopy experiments with a minimal theoretical description of the system in the form of a delay-differential equation model. Analytical and numerical methods reveal that oscillations arise through a subcritical Hopf bifurcation, which enables instant high-amplitude oscillations. Consequently, the model predicts a bistable regime where an oscillating and a nonoscillating attractor coexist in phase space. We experimentally validate this prediction by showing that oscillations can be triggered by perturbing the media conditions, provided the biofilm size lies within an appropriate range. The model also predicts that the minimum size at which oscillations start decreases with stress, a fact that we also verify experimentally. Taken together, our results show that collective oscillations in cell populations can emerge suddenly with nonzero amplitude via a discontinuous transition. [ABSTRACT FROM AUTHOR]

This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor. [ABSTRACT FROM AUTHOR]

The present paper investigates the efficacy of controlling friction induced vibration by normal load modulation. Friction-induced self-excited vibration, attributed to the low-velocity drooping characteristics of friction (Stribeck effect), is modelled by a mass-on-belt model where the normal force of the mass is being modulated based on the acceleration feedback followed by a second order filtering. Polynomial model is employed to study the friction phenomenon between the mass and the belt. The pole crossover design (to ensure faster transient and greater relative stability) is implemented to optimize the filter parameters with an independent choice of the belt velocity and control gain. These sets of optimized parameter values are then used to construct local stability boundaries in the plane of control parameters. Numerical simulations in a MATLAB SIMULINK model and bifurcation diagrams obtained in AUTO (while using belt velocity as the bifurcation parameter) indicate that a significantly small-amplitude limit cycle resulting from a supercritical Hopf bifurcation stabilizes the extreme low velocity region at higher values of the control gain. With the increase of the control gain the subcritical nature of Hopf bifurcation changes to a supercritical one. The efficacy of this optimization (based on numerical results) in the delicate low velocity region is also enclosed. [ABSTRACT FROM AUTHOR]

The FitzHugh–Nagumo model is improved to consider the effect of the electromagnetic induction on single neuron. On the basis of investigating the Hopf bifurcation behavior of the improved model, stochastic resonance in the stochastic version is captured near the bifurcation point. It is revealed that a weak harmonic oscillation in the electromagnetic disturbance can be amplified through stochastic resonance, and it is the cooperative effect of random transition between the resting state and the large amplitude oscillating state that results in the resonant phenomenon. Using the noise dependence of the mean of interburst intervals, we essentially suggest a biologically feasible clue for detecting weak signal by means of neuron model with subcritical Hopf bifurcation. These observations should be helpful in understanding the influence of the magnetic field to neural electrical activity. [ABSTRACT FROM AUTHOR]

Recently it was shown that in the dynamical model of Chua circuit both the classical self-excited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of self-excited and hidden attractors is studied. A pitchfork bifurcation is shown in which a pair of symmetric attractors coexist and merge into one symmetric attractor through an attractor-merging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh-Nagumo model, the Hodgkin-Huxley model, and the Morris-Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit connected across a current source. [ABSTRACT FROM AUTHOR]

Nonlinear stability study of the neutron coupled thermal hydraulics instability has been carried out by several researchers for boiling water reactors (BWRs). The focus of these studies has been to identify subcritical and supercritical Hopf bifurcations. Supercritical Hopf bifurcation are soft or safe due to the fact that stable limit cycles arise in linearly unstable region; linear and global stability boundaries are same for this bifurcation. It is well known that the subcritical bifurcations can be considered as hard or dangerous due to the fact that unstable limit cycles (nonlinear phenomena) exist in the (linearly) stable region. The linear stability leads to a stable equilibrium in such regions, only for infinitesimally small perturbations. However, finite perturbations lead to instability due to the presence of unstable limit cycles. Therefore, it is evident that the linear stability analysis is not sufficient to understand the exact stability characteristics of BWRs. However, the effect of these bifurcations on the stability boundaries has been rarely discussed. In the present work, the identification of global stability boundary is demonstrated using simplified models. Here, five different models with different thermal hydraulics feedback have been investigated. In comparison to the earlier works, current models also include the impact of adding the rate of change in temperature on void reactivity as well as effect of void reactivity on rate of change of temperature. Using the bifurcation analysis of these models the globally stable region in the parameter space has been identified. The globally stable region has only stable solutions and does not have even unstable limit cycles. Hence, the system is stable irrespective of the size of the perturbation in these regions. [ABSTRACT FROM AUTHOR]

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris-Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris-Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris-Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris-Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris-Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon. [ABSTRACT FROM AUTHOR]

Tangential control action is studied on a phenomenological mass-on-belt model exhibiting friction-induced self-excited vibration attributed to the low-velocity drooping characteristics of friction which is also known as Stribeck effect. The friction phenomenon is modelled by the exponential model. Linear stability analysis is carried out near the equilibrium point and local stability boundary is delineated in the plane of control parameters. The system is observed to undergo a Hopf bifurcation as the eigenvalues determined from the linear stability analysis are found to cross the imaginary axis transversally from RHS s-plane to LHS s-plane or vice-versa as one varies the control parameters, namely non-dimensional belt velocity and the control gain. A nonlinear stability analysis by the method of Averaging reveals the subcritical nature of the Hopf bifurcation. Thus, a global stability boundary is constructed so that any choice of control parameters from the globally stable region leads to a stable equilibrium. Numerical simulations in a MATLAB SIMULINK model and bifurcation diagrams obtained in AUTO validate these analytically obtained results. Pole crossover design is implemented to optimize the filter parameters with an independent choice of belt velocity and control gain. The efficacy of this optimization (based on numerical results) in the delicate low velocity region is also enclosed. [ABSTRACT FROM AUTHOR]

• Symmetric and asymmetric dispersal affects the community structure of complex ecological system. • Moderate and intense predation strength provides buffer against extinction in a multiple interactions type model. • Short predator handling time support stable coexistence of species in this ecological system. • Long predator handling time disrupt the persistence of species in this multi-species system. • The complexity of distinct biotic interactions leads to an unstable limit cycle. Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system's coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management. [ABSTRACT FROM AUTHOR]

The instability mechanism of fully three-dimensional, highly separated, shear-driven confined flow inside a diagonally lid-driven cavity was investigated. The analysis was conducted on 100 and 200 stretched grids by a series of direct numerical simulations utilizing a standard second-order accuracy finite volume code, openFoam. The observed oscillatory instability was found to set in via a subcritical symmetry breaking Hopf bifurcation. Critical values of the Reynolds number Re = 2320 and the non-dimensional angular oscillating frequency $${\omega_{\rm cr}=0.249}$$ for the transition from steady to oscillatory flow were accurately determined. An oscillatory regime of the bifurcated flow was analyzed in depth, revealing and characterizing the spontaneous symmetry breaking mechanism. Characteristic spatial patterns of the base flow and the main flow harmonic were determined for the velocity, vorticity and helicity fields. Lagrangian particle tracers were utilized to visualize the mixing phenomenon of the flow from both sides of the diagonal symmetry plane. [ABSTRACT FROM AUTHOR]

We consider a Gause type predator-prey system with functional response given by ϑ(x) = arctan(ax), where a > 0, and give a counterexample to the criterion given in Attili and Mallak [Commun. Math. Anal. 1:33-40(2006)] for the nonexistence of limit cycles. When this criterion is satisfied, instead this system can have a locally asymptotically stable coexistence equilibrium surrounded by at least two limit cycles. [ABSTRACT FROM AUTHOR]

The influence of system parameters such as heater power, heater location and mass flow rate on the hysteresis characteristics of a horizontal Rijke tube is presented in this paper. It is observed that a hysteresis zone is present for all the mass flow rates considered in the present study. A power law relation is established between the non-dimensional hysteresis width and the Strouhal number, defined as the ratio between convective time scale and acoustic time scale. The transition to instability in a horizontal Rijke tube is found to be subcritical in all the experiments performed in this study. When heater location is chosen as the control parameter, period-2 oscillations are found for specific values of mass flow rate and heater power. [ABSTRACT FROM AUTHOR]

This paper shows the action potential (spikes) generated from the Hodgkin-Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an -dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction. [Copyright &y& Elsevier]

Abstract: In the present work, a mathematical model of predator–prey ecological interaction with infected prey is investigated. A saturation incidence function is used to model the behavioral change of the susceptible individuals when their number increases or due to the crowding effect of the infected individuals [V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 41–61]. Stability criteria for the infection-free and the endemic equilibria are deduced in terms of system parameters. The basic model is then modified to incorporate a time delay, describing a latency period. Stability and bifurcation analysis of the resulting delay differential equation model is carried out and ranges of the delay inducing stability and as well as instability for the system are found. Finally, a stability analysis of the bifurcating solutions is performed and the criteria for subcritical and supercritical Hopf bifurcation derived. The existence of a delay interval that preserves the stability of periodic orbits is demonstrated. The analysis emphasizes the importance of differential predation and a latency period in controlling disease dynamics. [Copyright &y& Elsevier]

The effect of delay feedback control and adaptive synchronization is studied near sub-critical Hopf bifurcation of a nonlinear dynamical system. Previously, these methods targeted the nonlinear systems near their chaotic regime but it is shown here that they are equally applicable near the branch of unstable solutions. The system is first analyzed from the view point of bifurcation, and the existence of Hopf bifurcation is established through normal form analysis. Hopf bifurcation can be either sub-critical or super-critical, and in the former case, unstable periodic orbits are formed. Our aim is to control them through a delay feedback approach so that the system stabilizes to its nearest stable periodic orbit. At the vicinity of the sub-critical Hopf point, adaptive synchronization is studied and the effect of the coupling parameter and the speed factor is analyzed in detail. Adaptive synchronization is also studied when the system is in the chaotic regime. [ABSTRACT FROM AUTHOR]