Nikov Bifurcation Research Articles

Stabilization and complex dynamics in a predator–prey model with predator suffering from an infectious disease

We study the effects of a non-specified infectious disease of the predator on the dynamics a predator–prey system, by evaluating the dynamics of a three-dimensional model. The predator population in this (PSI) model is split into a susceptible and an unrecoverable infected population, while all newborn are susceptible. The incidence rate at which susceptible become infectious is described by a Holling type II functional response giving saturation when the number of susceptibles increases. From a modeling context this three-dimensional model is in the limit case similar to the well-known 3D Rosenzweig–MacArthur (RM) model, with the infected population replacing the top-predator. The RM model is known for the Shil’nikov bifurcation, which is associated to the chaotic behaviour. The effects of the disease are considered to be changes in the parameters that represent relative predation efficiency and mortality rates. A combination of analysis, numerical integration and numerical continuation techniques are used to perform a bifurcation analysis of the model. The positive stationary solution of the disease free, two-dimensional predator–prey system is either a stable equilibrium or a stable limit cycle where the transition occurs at the Hopf bifurcation. For a biologically applicable parameter set, it is found that when the infected individuals feed less fast or less effective than the susceptibles there is bi-stability where the two-dimensional disease free state co-exists with a stable equilibrium for the three-dimensional PSI system. The introduction of a disease can also cause chaos when the infected predator individuals are ecologically not functioning (not feeding and no offspring). However, under small parameter changes first the Shil’nikov bifurcation, and hence the chaotic behaviour, disappears followed by the Hopf bifurcation that marks the existence of limit cycles of the three-dimensional PSI system. As such, an infectious disease has a strongly stabilizing effect on the predator–prey system, similar to the existence of weak links in food webs.

Bifurcations of self-excitation regimes in a Van der Pol oscillator with a nonlinear energy sink

The paper investigates regimes of self-excitation in a Van der Pol oscillator with an attached nonlinear energy sink (NES). Initial equations are reduced by averaging to a 3D system. The small relative mass of the NES justifies analysis of this averaged system as singularly perturbed with two “slow” and one “super–slow” variable. Such an approach, in turn, provides a complete analytic description of possible response regimes. In addition to almost unperturbed limit cycle oscillations (LCOs), the system can exhibit complete elimination of self-excitation, small-amplitude LCOs as well as excitation of a quasiperiodic strongly modulated response (SMR). In the space of parameters, the latter can be approached by three distinct bifurcation mechanisms: canard explosion, Shil’nikov bifurcation and heteroclinic bifurcation. Some of the above oscillatory regimes can co-exist for the same values of the system parameters. In this case, it is possible to establish the basins of attraction for the co-existing regimes. Direct numeric simulations demonstrate good coincidence with the analytic predictions.

Dynamical systems modeling of low-frequency variability in low-order atmospheric models

The understanding of atmospheric and oceanic low-frequency variability is an old problem having both theoretical interest and practical importance, e.g., for the assessment of climate change. In this paper possible relations with dynamical systems theory are given, in particular through bifurcation theory. Firstly, a specific type of oceanic low frequency variability is described, the so-called Atlantic Multidecadal Oscillation (AMO). Then recent work is reviewed, that investigates bifurcations as these occur in a few low-order models of the atmospheric circulation. It is shown that the Shil′nikov bifurcation in the Hopf-saddle-node bifurcation scenario takes place in each of the above atmospheric models. Related strange attractors and intermittency behavior are also found, both in agreement with the theoretical expectations and with qualitative aspects of the climate variability, like the AMO. It is discussed how the latter connection may be consolidated in higher dimensional and in PDE models.

Chaos and strange attractors in coupled oscillators with energy-preserving nonlinearity

We present an analysis for a particular singularly perturbed conservative system. This system comes from the normal form of two coupled oscillator systems with widely-separated frequencies and energy-preserving nonlinearity. The analysis is done in this paper for a degenerate case of such a system, while the generic one has been treated in the literature. To understand the relation with the strong resonance case, we have computed the normal form of the 2:1 resonance, and found that the latter is contained in our system. We present a theorem that gives the existence of a nontrivial equilibrium for a general singularly perturbed conservative system. We detect that the nontrivial equilibrium undergoes two Hopf bifurcations. Furthermore, the periodic solutions created through these Hopf bifurcations undergo a sequence of period doubling bifurcations. This leads to the presence of chaotic dynamics through Shil'nikov bifurcation of a homoclinic orbit. Also, we measure the size of the chaotic attractor which is created in our system.

Three dimensional nilpotent singularity and Sil’nikov bifurcation

Abstract In this paper, by using the normal form, blow-up theory and the technique of global bifurcations, we study the singularity at the origin with threefold zero eigenvalue for nonsymmetric vector fields with nilpotent linear part and 4-jet C ∞ -equivalent to y ∂ ∂ x + z ∂ ∂ y + ax 3 y ∂ ∂ z , with a ≠ 0, and analytically prove the existence of Sil’nikov bifurcation, and then of the strange attractor for certain subfamilies of the nonsymmetric versal unfoldings of this singularity under some conditions.

Homoclinic and heteroclinic orbits in a modified Lorenz system

This paper presents a mathematically rigorous proof for the existence of chaos in a modified Lorenz system using the theory of Shil'nikov bifurcations of homoclinic and heteroclinic orbits. Together with its dynamical behaviors, which have been extensively studied, the chaotic dynamics of the modified Lorenz system are now much better understood, providing a rigorous theoretic foundation to support studies and applications of this important class of chaotic systems.

Baroclinic Flow and the Lorenz-84 Model

The bifurcation diagram of a truncation to six degrees of freedom of the equations for quasi-geostrophic, baroclinic flow is investigated. Period-doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The low dimension of the chaotic attractor suggests the possibility to reduce the model to three degrees of freedom. In a physically comprehensible limit of the parameters this reduction is done explicitly. The bifurcation diagram of the reduced model in this limit is compared to the diagram of the six degrees of freedom model and agrees well. A numerical implementation of the graph transform is used to approximate the three-dimensional invariant manifold away from the limit case. If the six-dimensional model is reduced to a linearisation of the invariant manifold about the Hadley state, the Lorenz-84 model is found. Its parameters can then be calculated from the physical parameters of the quasi-geostrophic model. Bifurcation diagrams at physical and traditional parameter values are compared and routes to chaos in the Lorenz-84 model are described.

Bifurcation sequences in the interaction of resonances in a model deriving from nonlinear rotordynamics: The zipper

Using numerical continuation we show a new bifurcation scenario involving resonant periodic orbits in a parametrized four-dimensional autonomous system deriving from nonlinear rotordynamics. The scenario consists of a carefully orchestrated sequence of transcritical bifurcations in which branches of periodic solutions are exchanged. Collectively, the bifurcations resemble the action of a zipper. An underlying governing mechanism clearly exists but still has to be uncovered. For a range of parameter values the sequence of bifurcations forms a global connection between a Sil'nikov bifurcation and (partial) mode-locking. The homoclinic bifurcation is introduced into the system by a Takens-Bogdanov bifurcation. The system also features an interaction between two chaotic Sil'nikov bifurcations.

The diffusionless Lorenz equations; Shil’nikov bifurcations and reduction to an explicit map

A simplified, one-parameter version of the Lorenz model is obtained in the limit of high Rayleigh- and Prandtl-numbers, physically corresponding to diffusionless convection. It is argued that the bifurcation structure of this simplified Lorenz model essentially involves only Shil’nikov bifurcations. An exact solution to this simplified dynamical system is given which serves as the limit for strong forcing and appears to be a new integrable case of the Lorenz equations. For small values of the bifurcation parameter, an approximate, analytical and multipeaked map is obtained which gives successive periods of the pulse-like motion. This map leads to self-similar behaviour in parameter-space.

A construction of three-dimensional vector fields which have a codimension two heteroclinic loop at Glendinning-Sparrow T-point

A concrete example of three-dimensional vector fields possessing a heteroclinic loop at the Glendinning-Sparrow T-point is constructed using a Melnikov-like perturbation method. A relation among the Shil'nikov bifurcation, the fishhook structure and the chaotic dynamics is also discussed.

Chaos in a low-order model of magnetoconvection

In the limit of tall, thin rolls, weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third-order set of ordinary differential equations. These equations are shown here to have three codimension-two bifurcation points: a Takens-Bogdanov bifurcation, at which a gluing bifurcation is created; a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are Lorenz-like chaotic trajectories; and a new type of bifurcation point at which the first of a cascade of period-doubling bifurcation lines originates. The last two bifurcation points are analysed in terms of a one-dimensional map. The equations also have a T-point, at which there is a heteroclinic connection between a saddle and a pair of saddle-foci; emerging from this point is a line of Šil'nikov bifurcations, involving homoclinic connections to a saddle-focus.

Šil'nikov-Hopf bifurcation

We derive a 2-parameter Poincaré map modeling a 3-dimensional vector flow near the codimension-2 point defined by a Šil'nikov homoclinic orbit to a fixed point undergoing a Hopf bifurcation. In one parameter regime the map reproduces Glendinning and Sparrow's results for the Šil'nikov bifurcation while in another regime it provides new results for a small Hopf cycle with a global reinjection mechanism. The map is analyzed finding primary homoclinic tangencies to the small Hopf cycle. Parameter dependence of periodic orbits associated with these tangencies is described. Global behavior of the tube-like unstable manifold of the Hopf cycle is studied in depth, analytically and numerically, locating its simplest subsidiary tangencies.

Chaos in models of double convection

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Homoclinic and heteroclinic orbits in the double scroll attractor

Chua, Matsumoto, and others have reported extensive simulation studies on a piecewise-linear model of a simple circuit. There is strong numerical and experimental evidence for chaotic behavior. This paper shows that the observed behavior can be understood using the theory of Shil'nikov bifurcations of homoclinic and heteroclinic orbits.

Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension

AbstractA specific singularity of a vector field on is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Šil'nikov bifurcations, which have codimension 1. Special attention is paid to the C∞-flatness of this subordinate phenomenon.