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In a food chain, the role of intake patterns of predators is very influential on the survival and extinction of the interacting species as well as the entire dynamics of the ecological system. In this study, we investigate the affluent and intricate dynamics of a simple three-species food chain model in a discrete-time framework by analyzing the parameter plane of the system with simultaneous changes of two crucial parameters, the predation rates of middle and top predators. From the theoretical viewpoint, we study the model by determining the fixed points' biological feasibility and local asymptotic stability criteria, and performing some analyses of local bifurcations, namely, transcritical, flip, and Neimark–Sacker bifurcations. Here, we initiate the numerical simulation by plotting the changes of the prey population density in terms of a vital parameter of the system, and we observe the switching among different dynamical behaviors of the system. We also draw some phase portraits and plot the time series solutions to show the diverse characteristics of the system dynamics. Further, we move one step ahead to explore the intricate dynamical scenarios appearing in the parameter plane by forming Lyapunov exponent and isoperiodic diagrams. In the parameter plane of the system, we see the emergence of innumerable Arnold tongues. All these Arnold tongues are organized along a particular direction, and the beautiful arrangement of these tongues forms several kinds of period-adding sequences. The study sheds more light on various types of multistability occurring in the model system. We see the coexistence of three periodic attractors in the parameter plane. In this study, the most striking observation is the coexistence of four periodic attractors, which occurs infrequently in ecological systems. We draw the basins of attraction for the tristable and tetrastable attractors, which are complex Wada basins. A system with Wada basin is very sensitive to initial conditions and more erratic in nature than a system with fractal basin. Also, we plot the density of all interacting species in terms of the predation rates of middle and top predators and observe the variation in the population densities of all species with the variability of these two key parameters. In the parameter plane created by the simultaneous changes of two parameters, the system exhibits a variety of intricate and subtle dynamics, which cannot be found by changing only a single parameter. [ABSTRACT FROM AUTHOR]

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors' dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs (α ,). These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors. [ABSTRACT FROM AUTHOR]

The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equilibrium point of the system is obtained by calculation and its local stability condition is given based on the Jury criterion. Then, two paths of the system from stability to chaos, namely flip bifurcation and Neimark-Sacker bifurcation, are analyzed by using the two-dimensional parametric bifurcation diagram. Meanwhile, the abundant Arnold tongues in the two-dimensional parametric bifurcation diagram are analyzed. Finally, the phenomenon of multistability of the system is illustrated through the basin of attraction, and the contact bifurcation occurs during the evolution of the basin of attraction with varying parameters. [ABSTRACT FROM AUTHOR]

This study is aimed at investigating the effects of anisotropic supports on the stability of slender rotors parametrically excited by external loads. An axisymmetric shaft described by scaling a spinning Timoshenko beam on anisotropic supports is studied, loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs, which represents a model including all the general features of slender rotors which are relevant for this kind of stability analysis, gyroscopic effects comprised. Stability is studied after discretization of the equations of motion into a set of coupled ordinary differential Mathieu-Hill equations. The influence on stability of angular speed combined with anisotropy in the supports (including principal stiffness, principal damping and cross-elements) is analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince-Strutt diagrams. The occurrence of different kinds of critical solutions, simple and combination, is investigated, highlighting their dependency on both the degree of anisotropy in the supports and angular speed. [ABSTRACT FROM AUTHOR]

Brain oscillations emerge during sensory and cognitive processes and have been classified into different frequency bands. Yet, even within the same frequency band and between nearby brain locations, the exact frequencies of brain oscillations can differ. These frequency differences (detuning) have been largely ignored and play little role in current functional theories of brain oscillations. This contrasts with the crucial role that detuning plays in synchronization theory, as originally derived in physical systems. Here, we propose that detuning is equally important to understand synchronization in biological systems. Detuning is a critical control parameter in synchronization, which is not only important in shaping phase-locking, but also in establishing preferred phase relations between oscillators. We review recent evidence that frequency differences between brain locations are ubiquitous and essential in shaping temporal neural coordination. With the rise of powerful experimental techniques to probe brain oscillations, the contributions of exact frequency and detuning across neural circuits will become increasingly clear and will play a key part in developing a new understanding of the role of oscillations in brain function. [ABSTRACT FROM AUTHOR]

Brain oscillations emerge during sensory and cognitive processes and have been classified into different frequency bands. Yet, even within the same frequency band and between nearby brain locations, the exact frequencies of brain oscillations can differ. These frequency differences (detuning) have been largely ignored and play little role in current functional theories of brain oscillations. This contrasts with the crucial role that detuning plays in synchronization theory, as originally derived in physical systems. Here, we propose that detuning is equally important to understand synchronization in biological systems. Detuning is a critical control parameter in synchronization, which is not only important in shaping phase-locking, but also in establishing preferred phase relations between oscillators. We review recent evidence that frequency differences between brain locations are ubiquitous and essential in shaping temporal neural coordination. With the rise of powerful experimental techniques to probe brain oscillations, the contributions of exact frequency and detuning across neural circuits will become increasingly clear and will play a key part in developing a new understanding of the role of oscillations in brain function.

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period- p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence , where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold , where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on R n to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle. • Piecewise-linear maps restricted to attracting invariant circles are often degree-one • On such circles periodic solutions have rotational symbolic encodings • Such periodic solutions rarely undergo persistence type border-collision bifurcations • The results help explain the sausage-string structure of mode-locking regions [ABSTRACT FROM AUTHOR]

We explore a system of two coupled nonlinear pendula, with the lowest-order coupling term being quadratic, rather than the more typical linear coupling term. We show that synchronization can occur in this system, but with the synchronization frequency lying below either of the original independent frequencies, in contrast to the usual case where the synchronization frequency is between the two frequencies of the independent pendula. Both phase synchronization and amplitude synchronization are found, and one can have situations, for example, where phase synchronization exists, but not amplitude synchronization. We develop analytic formulas to understand the frequency synchronization. [ABSTRACT FROM AUTHOR]

For quasiperiodically forced (qpf) continuous circle maps with degree one, the set of fibred rotation numbers is a closed interval or a singleton. We show in this paper that if a number ρ is in the interior of this interval, then there exist ρ -bounded orbits. We also discuss extensions of the structure of Arnold tongues for qpf circle homeomorphisms to qpf continuous circle maps with degree one. [ABSTRACT FROM AUTHOR]

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits an attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the network and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov–Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory. [ABSTRACT FROM AUTHOR]

In ecology, the predator's impact goes beyond just killing the prey. In the present work, we explore the role of fear in the dynamics of a discrete-time predator-prey model where the predator-prey interaction obeys Holling type-II functional response. Owing to the increasing strength of fear, the system becomes stable from chaotic oscillations via inverse Neimark–Sacker bifurcation. Extensive numerical simulations are carried out to investigate the intricate dynamics for the organization of periodic structures in the bi-parameter space of the system. We observe fear induced multistability between different pairs of coexisting heterogeneous attractors due to the overlapping of multiple periodic domains in the bi-parameter space. The basin sets of the coexisting attractors are obtained and discussed at length. Multistability in the predator-prey system is important because the dynamics of the predator and prey populations in the critical parameter zone becomes uncertain. [ABSTRACT FROM AUTHOR]

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of (2) ⊕ ℤ 2 -symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to 3 of a polynomial vector field in ℝ 4 . We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue. [ABSTRACT FROM AUTHOR]

Climate change causing large seasonal fluctuations is likely to lead to an increase in the average threshold of the Allee effect as well as an increase in its seasonal variability. In this paper, we show that a seasonally synchronized predator-prey system can be strongly destabilized by these changes in the threshold of the Allee effect. The typical result first leads to two-year and multiyear cycles by the principle of period doubling on a folded Möbius strip, up to the emergence of a chaotic and hyper-chaotic attractor living close to the trivial equilibrium corresponding to the extinction of populations. Moreover, this instability of the ecosystem can be hidden for a long time and the transition to its basin of attraction can occur by external perturbation in a random and irreversible manner. We also reveal a double folding of the 1:1 synchronized cycle manifold inside the corresponding Arnold tongue and hysteresis similar to well-known Duffing oscillator. • Increase in level and fluctuation of Allee effect destabilize seasonally synchronized system. • Period doubling route to chaos on folded Möbius strip inside 1:1 Arnold tongue. • Local dynamics of a subsystem and global chaotic dynamics of a seasonally driven system connected. • Instability can stay long-time hidden and lead to extinction of populations. • 1:1 cycle manifold double folding inside Arnold tongue and hysteresis similar to Duffing oscillator. [ABSTRACT FROM AUTHOR]

• rTMS treatment outcome is quadratically related to the individual alpha peak frequency (IAF) in depression. • Absolute distance of IAF to 10 Hz is linearly related to clinical outcome. • These results strengthen the theory that 10 Hz rTMS entrains brain oscillations. The aim of the current study was to attempt to replicate the finding that the individual alpha frequency (IAF) as well as the absolute difference between IAF and 10 Hz stimulation frequency (IAF-prox) is related to treatment outcome. Correlations were performed to investigate the relationship between IAF-prox and percentage symptom improvement in a sample of 153 patients with major depressive disorder treated with 10 Hz (N = 59) to the left dorsolateral prefrontal cortex (DLPFC) or 1 Hz (N = 94) to the right DLPFC repetitive Transcranial Magnetic Stimulation (rTMS). There was a significant negative correlation between IAF-prox and the percentage of symptom improvement only for the 10 Hz group. Curve fitting models revealed that there was a quadratic association between IAF and treatment response in the 10 Hz group, with a peak at 10 Hz IAF. The main result of Corlier and colleagues was replicated, and the findings suggest that the distance between 10 Hz stimulation frequency and the IAF may influence clinical outcome in a non-linear manner. rTMS is often administered at a frequency of 10 Hz, which is the center of the EEG alpha frequency band. The results can make a significant contribution to optimizing the clinical application of rTMS. [ABSTRACT FROM AUTHOR]

The dynamical phase localization of a single qubit in a dissipative bath is investigated. We show that phase preference persists in the long time limit only for a non-Markovian evolution with a finite detuning implying that non-Markovianity is required for phase localization. The maximum of the shifted phase distribution exhibits an Arnold tongue like feature where the phase localized and phase delocalized regions are distinctly separated. We show the formation of Arnold tongue using the relative entropy of coherence as well. From our work we propose that Arnold tongue can be used to detect dynamical phase localization even when there is no synchronization present in the system. • Dynamical phase localization of a single qubit in a dissipative bath is investigated. • Both Markovian and non-Markovian dynamics of a two-level system is investigated. • The Arnold tongue demarcates the phase localized and delocalized regions. • Relative entropy of coherence reveals an interplay between quantum phase and coherence. • Arnold tongue is a feature of quantum synchronization as well as phase localization. [ABSTRACT FROM AUTHOR]

Studies in the literature have demonstrated the significance of the synchronization of neuronal electrical activity for signal transmission and information encoding. In light of this importance, we investigate the synchronization of the Chay neuron model using both theoretical analysis and numerical simulations. The Chay model is chosen for its comprehensive understanding of neuronal behavior and computational efficiency. Additionally, we explore the impact of electromagnetic induction, leading to the magnetic flux Chay neuron model. The single neuron model exhibits rich and complex dynamics for various parameter choices. We explore the bifurcation structure of the model through bifurcation diagrams and Lyapunov exponents. Subsequently, we extend our study to two coupled magnetic flux Chay neurons, identifying mode locking and structures reminiscent of Arnold's tongue. We evaluate the stability of the synchronized manifold using Lyapunov theory and confirm our findings through simulations. Expanding our study to networks of diffusively coupled flux Chay neurons, we observe coherent, incoherent, and imperfect chimera patterns. Our investigation of three network types highlights the impact of network topology on the emergent dynamics of the Chay neuron network. Regular networks exhibit diverse patterns, small-world networks demonstrate a critical transition to coherence, and random networks showcase synchronization at specific coupling strengths. These findings significantly contribute to our understanding of the synchronization patterns exhibited by the magnetic flux Chay neuron. To assess the synchronization stability of the Chay neuron network, we employ master stability function analysis. [ABSTRACT FROM AUTHOR]

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold's tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold's tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses. [ABSTRACT FROM AUTHOR]

Through the past few million years large ice sheets have repeatedly grown and disappeared on the Northern hemisphere. These are the Pleistocene glaciations. They are related to the changing solar heating of the Earth due to changes in Earth's orbit and axis of rotation. The climate response to these changes is highly non-trivial and non-linear, expressing the complex nature of the climate system. Many aspects of glacial cycles still need a convincing explanation, one particular mystery being the change from approximately 40 kyr (kilo year) glacial cycles to approximately 100 kyr cycles around 1 million years ago. This transition is called the middle Pleistocene transition (MPT). Here we review some conceptual models to explain the dynamics of glacial cycles and possible dynamical causes of the MPT. We especially focus on the well studied van del Pol oscillator as a conceptual model for the glacial cycles and propose that the MPT is a result of changes in frequency locking of the climate system to the astronomical forcing. This is compared to a recently presented model that relates the MPT to a transcritical bifurcation in the structure of a generic critical/slow manifold for a fast-slow dynamical system.

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold's tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons. [ABSTRACT FROM AUTHOR]

In this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view. [ABSTRACT FROM AUTHOR]

Additionally to being endogenous, a circadian system must be able to communicate with the outside world and align its rhythmicity to the environment. As a result of such alignment, external Zeitgebers can entrain the circadian system. Entrainment expresses itself in coinciding periods of the circadian oscillator and the Zeitgeber and a stationary phase difference between them. The range of period mismatches between the circadian system and the Zeitgeber that Zeitgeber can overcome to entrain the oscillator is called an entrainment range. The width of the entrainment range usually increases with increasing Zeitgeber strength, resulting in a wedge-like Arnold tongue.This classical view of entrainment does not account for the effects of photoperiod on entrainment. Zeitgebers with extremely small or large photoperiods are intuitively closer to constant environments than equinoctial Zeitgebers and hence are expected to produce a narrower entrainment range. In this paper, we present theoretical results on entrainment under different photoperiods. We find that in the photoperiod-detuning parameter plane, the entrainment zone is shaped in the form of a skewed onion. The bottom and upper points of the onion are given by the free running periods in DD and LL, respectively. The widest entrainment range is found near photoperiods of 50%. Within the onion, we calculated the entrainment phase that varies over a range of 12h. The results of our theoretical study explain the experimentally observed behaviour of the entrainment phase in dependence on the photoperiod.

The functional relevance of brain oscillations in the alpha frequency range (8-13 Hz) has been repeatedly investigated through the use of rhythmic visual stimulation. The underlying mechanism of the steady-state visual evoked potential (SSVEP) measured in EEG during rhythmic stimulation, however, is not known. There are two hypotheses on the origin of SSVEPs: entrainment of brain oscillations and superposition of event-related responses (ERPs). The entrainment but not the superposition hypothesis justifies rhythmic visual stimulation as a means to manipulate brain oscillations, because superposition assumes a linear summation of single responses, independent from ongoing brain oscillations. Here, we stimulated participants with a rhythmic flickering light of different frequencies and intensities. We measured entrainment by comparing the phase coupling of brain oscillations stimulated by rhythmic visual flicker with the oscillations induced by arrhythmic jittered stimulation, varying the time, stimulation frequency, and intensity conditions. In line with a theoretical concept of entrainment (the so called Arnold tongue), we found the phase coupling to be more pronounced with increasing stimulation intensity as well as at stimulation frequencies closer to each participant's intrinsic frequency. Only inside the Arnold tongue did the conditions significantly differ from the jittered stimulation. Furthermore, even in a single sequence of an SSVEP, we found non-linear features (intermittency of phase locking) that contradict the linear summation of single responses, as assumed by the superposition hypothesis. Our findings provide unequivocal evidence that visual rhythmic stimulation entrains brain oscillations, thus validating the approach of rhythmic stimulation as a manipulation of brain oscillations. [ABSTRACT FROM AUTHOR]

We study a two-parameter family of nonautonomous ordinary differential equations on the 2-torus. This family models the Josephson effect in superconductivity. We study its rotation number as a function of the parameters and the Arnold tongues (also known as the phase locking domains) defined as the level sets of the rotation number that have nonempty interior. The Arnold tongues of this family of equations have a number of nontypical properties: they exist only for integer values of the rotation number, and the boundaries of the tongues are given by analytic curves. (These results were obtained by Buchstaber-Karpov-Tertychnyi and Ilyashenko-Ryzhov-Filimonov.) The tongue width is zero at the points of intersection of the boundary curves, which results in adjacency points. Numerical experiments and theoretical studies carried out by Buchstaber-Karpov-Tertychnyi and Klimenko-Romaskevich show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in an asymptotically vertical direction. Recent numerical experiments have also shown that for each Arnold tongue all of its adjacency points lie on one and the same vertical line with integer abscissa equal to the corresponding rotation number. In the present paper, we prove this fact for an open set of two-parameter families of equations in question. In the general case, we prove a weaker claim: the abscissa of each adjacency point is an integer, has the same sign as the rotation number, and does not exceed the latter in absolute value. The proof is based on the representation of the differential equations in question as projectivizations of linear differential equations on the Riemann sphere and the classical theory of linear equations with complex time. [ABSTRACT FROM AUTHOR]

A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range versus forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Hölder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For 1:1 entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in L p -space with p > 1 , and (ii) a construction of the global optimal pulse-like forcings in L 1 -space, and for m : n entrainment ( m ≠ n ), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for 1 < p ≤ ∞ and these pulse-like forcings for p = 1 . These theoretical findings are verified by systematic, extensive numerical calculations and simulations. [ABSTRACT FROM AUTHOR]