Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

Dynamics of a discrete-time pioneer–climax model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

Dynamics of a Discrete-time Pioneer–climax Model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population density growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and exclusion of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect

This paper deals with a discrete-time predator–prey system which is subject to an Allee effect on prey. We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system. Moreover, using bifurcation theory, it is shown that the system undergoes Neimark–Sacker bifurcation in a small neighborhood of the unique positive fixed point and an invariant circle will appear. Then the direction of bifurcation is given. Furthermore, numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages. Thus, the main theoretical results are supported with numerical simulations.

Invariant circles and depinning transition

We associate the existence or non-existence of rotational invariant circles of an area-preserving twist map on the cylinder with a physically motivated quantity, the depinning force, which is a critical value in the depinning transition. Assume that $H:\mathbb{R}^{2}\mapsto \mathbb{R}$ is a $C^{2}$ generating function of an exact area-preserving twist map $\bar{\unicode[STIX]{x1D711}}$ and consider the tilted Frenkel–Kontorova (FK) model: $$\begin{eqnarray}{\dot{x}}_{n}=-D_{1}H(x_{n},x_{n+1})-D_{2}H(x_{n-1},x_{n})+F,\quad n\in \mathbb{Z},\end{eqnarray}$$ where $F\geq 0$ is the driving force. The depinning force is the critical value $F_{d}(\unicode[STIX]{x1D714})$ depending on the mean spacing $\unicode[STIX]{x1D714}$ of particles, above which the tilted FK model is sliding, and below which the particles are pinned. We prove that there exists an invariant circle with irrational rotation number $\unicode[STIX]{x1D714}$ for $\bar{\unicode[STIX]{x1D711}}$ if and only if $F_{d}(\unicode[STIX]{x1D714})=0$. For rational $\unicode[STIX]{x1D714}$, $F_{d}(\unicode[STIX]{x1D714})=0$ is equivalent to the existence of an invariant circle on which $\bar{\unicode[STIX]{x1D711}}$ is topologically conjugate to the rational rotation with rotation number $\unicode[STIX]{x1D714}$. Such conclusions were claimed much earlier by Aubry et al. We also show that the depinning force $F_{d}(\unicode[STIX]{x1D714})$ is continuous at irrational $\unicode[STIX]{x1D714}$.

Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others

Phase resetting curves (PRCs) are phenomenological and quantitative tools that tabulate the transient changes in the firing period of endogenous neural oscillators as a result of external stimuli, for example, presynaptic inputs. A brief current perturbation can produce either a delay (positive phase resetting) or an advance (negative phase resetting) of the subsequent spike, depending on the timing of the stimulus. We showed that any planar neural oscillator has two remarkable points, which we called neutral points, where brief current perturbations produce no phase resetting and where the PRC flips its sign. Since there are only two neutral points, all PRCs of planar neural oscillators are bimodal. The degree of bimodality of a PRC, that is, the ratio between the amplitudes of the delay and advance lobes of a PRC, can be smoothly adjusted when the bifurcation scenario leading to stable oscillatory behavior combines a saddle node of invariant circle (SNIC) and an Andronov-Hopf bifurcation (HB).

Superstable manifolds of invariant circles and codimension-one Böttcher functions

Let $f: X~\dashrightarrow ~X$ be a dominant meromorphic self-map, where $X$ is a compact, connected complex manifold of dimension $n\gt 1$. Suppose that there is an embedded copy of ${ \mathbb{P} }^{1} $ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose that $f$ restricted to this line is given by $z\mapsto {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a\geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

The Shape of Phase-Resetting Curves in Oscillators with a Saddle Node on an Invariant Circle Bifurcation

We introduce a simple two-dimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.