Heteroclinic Cycles Research Articles

On Lotka--Volterra Equations with Identical Minimal Intrinsic Growth Rate

We investigate the long-run behavior of time-dependent Lotka--Volterra equations $({\rm E}_{\phi}):\frac{dx_i}{dt}=x_i(1+\phi(t)+\sum_{j=1}^na_{ij}x_j)$, $i=1,\ldots,n$, on the positive orthant. It is proved that the system $({\rm E}_{\phi})$ is decomposed into $({\rm E}_0)$ and the logistic equation $({\rm L}):\frac{dg}{dt}=g(1+\phi(t)-g)$ in the sense that $({\rm D}):\Phi(t)=g(t)\Psi(\int_0^tg(s)ds)$, where $\Phi(\cdot)$, $\Psi(\cdot)$, and $g(\cdot)$ are the solutions of $({\rm E}_{\phi})$, $({\rm E}_0)$, and (L), respectively. Suppose that $\phi(t)$ is periodic with the mean value ${\cal M}\{\phi\}>-1$. Then the existence of stable equilibrium, periodic solution, and chaotic motion for $({\rm E}_0)$ implies the existence of stable periodic solution, quasiperiodic solution, and chaotic motion for $({\rm E}_{\phi})$, respectively. The complete dynamical classification for 3-dimensional competitive system $({\rm E}_0)$ is provided in terms of competitive coefficients. There are 37 topological classes, in 34 of which any trajectory converges to an equilibrium. Among the remaining three classes, the first has a heteroclinic cycle attracting all positive points except the ray joining the origin and the positive equilibrium; the second possesses a family of periodic orbits attracting all positive points; the third class, where a family of periodic orbits, an asymptotically stable equilibrium, a heteroclinic cycle, and an invariant noncoordinate plane coexist, is a new type we find for the first time. It is shown that for 3-dimensional periodic competitive system $({\rm E}_{\phi})$, any trajectory for Poincaré mapping tends to a fixed point, a periodic orbit, an invariant minimal closed curve, or a heteroclinic cycle. All their attracting domains, which are all cone-like, are exactly described via (D) and carrying simplex theory. The dynamics can be completely classified into 37 classes corresponding to the autonomous systems. All counterparts hold when $\phi(t)$ is another minimal function, such as a quasiperiodic, almost periodic, or almost automorphic function, via (D), the classification above, and skew-product flow theory.

Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics

We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in ${R}^{N}$ with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we establish the emergence of strange attractors and Sinai--Ruelle--Bowen (SRB) measures with strong statistical properties (central limit theorem, exponential decay of correlations, etc.). We identify and study the mechanism responsible for the nonuniform hyperbolicity: saddle point shear. Our results apply to concrete systems of interest in the biological and physical sciences, such as May--Leonard models of Lotka--Volterra dynamics.

A new Lorenz-type hyperchaotic system with a curve of equilibria

Little seems to be known about hyperchaotic systems with a curve of equilibria. Based on the classical Lorenz system, this paper proposes a new four-dimensional Lorenz-type hyperchaotic system which has a curve of equilibria. This new system can generate not only hyperchaotic attractors but also chaotic, quasi-periodic and periodic attractors, as well as singular degenerate heteroclinic cycles. Of particular interest is the observation that there are four types of coexisting attractors of this new hyperchaotic system: (i) chaotic attractor and quasi-periodic attractor, (ii) chaotic attractor and singular degenerate heteroclinic cycle, (iii) periodic attractor and singular degenerate heteroclinic cycle, and (iv) different periodic attractors. Furthermore, many singular degenerate heteroclinic cycles are found, which may lead to complex dynamics of hyperchaotic system with a curve of equilibria.

Dynamics in a tumor immune system with time delays

In this paper, we study the dynamical behavior of a tumor–immune system (T–IS) interaction model with two discrete delays, namely the immune activation delay for effector cells (ECs) and activation delay for helper T cells (HTCs). By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Our results exhibit that both delays do not affect the stability of tumor-free equilibrium. However, they are able to destabilize the immune-control equilibrium and cause periodic solutions. We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors. The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium. Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

Heterodimensional cycle bifurcation with two orbit flips

In this paper, we consider a heteroclinic cycle consisting of two hyperbolic equilibria with different indices, one robust heteroclinic connection and a heteroclinic connection within a codimension-2 intersection of the corresponding manifolds of the equilibria, which is called the heterodimensional cycle. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincare return map under the nongeneric conditions two orbit flips and further obtain the bifurcation equations. By the bifurcation equations, different bifurcation phenomena are discussed under small perturbations. New features produced by the degeneracy that heterodimensional cycles and periodic orbits coexist on the same bifurcation surface are shown. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional with two orbit flips.

Onset of intermittent octahedral patterns in spherical Bénard convection

The onset of convection for spherically invariant Rayleigh–Bénard fluid flow is driven by marginal modes associated with spherical harmonics of a certain degree ℓ, which depends upon the aspect ratio of the spherical shell. At certain critical values of the aspect ratio, marginal modes of degrees ℓ and ℓ+1 coexist. Initially motivated by an experiment of electrophoretic convection between two concentric spheres carried in the International Space Station (GeoFlow project), we analyze the occurrence of intermittent dynamics near bifurcation in the case when marginal modes with ℓ=3,4 interact. The situation is by far more complex than in the well studied ℓ=1,2 mode interaction, however we show that heteroclinic cycles connecting equilibria with octahedral as well as axial symmetry can exist near bifurcation under certain conditions. Numerical simulations and continuation (using the software AUTO) on the center manifold help understanding these scenarios and show that the dynamics in these cases exhibit intermittent behavior, even though the heteroclinic cycles may not be asymptotically stable in the usual sense.

Arnold diffusion in a restricted planar four-body problem

In this paper, we construct a certain planar four-body problem which exhibits fast energy growth under certain assumption. The system considered is a quasi-periodic perturbation of the restricted planar circular three-body problem (RPC3BP). Gelfreich–Turaev's and de la Llave's mechanism is employed to obtain the fast energy growth. The diffusion is created by a heteroclinic cycle formed by two Lyapunov periodic orbits surrounding L1 and L2 Lagrangian points and their heteroclinic intersections. Our model is the first known example in celestial mechanics about the a priori chaotic case of Arnold diffusion (Delshams et al 2006 Memoirs of the AMS vol 179).

Stability of heteroclinic cycle connecting hyperbolic saddles in higher dimensional space

In this paper, the stability of nontwisted heteroclinic loop to 2 hyperbolic saddles is considered in arbitrarily finite dimensional spaces. Under the condition that the Poincar′e map is well-defined, the criterion is given for the asymptotic stability of the heteroclinic loop confined in its partial neighborhood. The stability results for 3-dimensional system are extened to m + n + 2 dimensional space, where m 0, n 0. By taking a suitable linear transformation, we get the first normal form, by a coordinate change to straighten the local stable manifold and the local unstable manifold, we establish the second normal form. Then, in the small neighborhood of the saddle P1, P2, we select two cross sections transversal to the heteroclinic orbit Γ respectively, and construct the Poincar′e map by two steps: in the small neighborhood of the saddle, we build the main part of the singular flow map by the linearly approximate system, in the regular tubular neighborhood of Γ, we use a differential homeomorphism to express the regular flow map. The Poincar′e map is achieved by composing the singular flow map and the regular flow map. At last, by estimating the modules of some vectors rather skilled, we obtain the ratio of the distance between the first recurrent point and the heteroclinic point to the distance between the initial point and the heteroclinic point. As a direct consequence, we derive two quite concise stability criteria for the non-resonant heteroclinic cycle to hyperbolic saddles.

More Dynamical Properties Revealed from a 3D Lorenz-like System

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

Hierarchical Heteroclinics in Dynamical Model of Cognitive Processes: Chunking

Combining the results of brain imaging and nonlinear dynamics provides a new hierarchical vision of brain network functionality that is helpful in understanding the relationship of the network to different mental tasks. Using these ideas it is possible to build adequate models for the description and prediction of different cognitive activities in which the number of variables is usually small enough for analysis. The dynamical images of different mental processes depend on their temporal organization and, as a rule, cannot be just simple attractors since cognition is characterized by transient dynamics. The mathematical image for a robust transient is a stable heteroclinic channel consisting of a chain of saddles connected by unstable separatrices. We focus here on hierarchical chunking dynamics that can represent several cognitive activities. Chunking is the dynamical phenomenon that means dividing a long information chain into shorter items. Chunking is known to be important in many processes of perception, learning, memory and cognition. We prove that in the phase space of the model that describes chunking there exists a new mathematical object — heteroclinic sequence of heteroclinic cycles — using the technique of slow–fast approximations. This new object serves as a skeleton of motions reflecting sequential features of hierarchical chunking dynamics and is an adequate image of the chunking processing.

Stability in simple heteroclinic networks in

We describe all heteroclinic networks in made of simple heteroclinic cycles of types B or C, with at least one common connecting trajectory. For networks made of cycles of type B, we study the stability of the cycles that make up the network as well as the stability of the network. We show that even when none of the cycles has strong stability properties, the network as a whole may be quite stable. We prove, and provide illustrative examples of, the fact that the stability of the network does not depend a priori uniquely on the stability of the individual cycles.

Using Lin's method to solve Bykov's problems

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points in this context. We study codimension-two T-points and their unfoldings in Rn. In our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria. In doing so we establish Lin's method as a unified approach to (re)gain and extend results of Bykov's seminal studies and related works. To a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane.

Multiple resource limitation: nonequilibrium coexistence of species in a competition model using a synthesizing unit

During the last two decades, the simple view of resource limitation by a single resource has been changed due to the realization that co-limitation by multiple resources is often an important determinant of species growth. Hence, the multiple resource limitation hypothesis needs to be taken into account, when communities of species competing for resources are considered. We present a multiple species–multiple resource competition model which is based on the concept of synthesizing unit to formulate the growth rates of species competing for interactive essential resources. Using this model, we demonstrate that a more mechanistic explanation of interactive effects of co-limitation may lead to the known complex dynamics including nonequilibrium states as oscillations and chaos. We compare our findings with earlier investigations on biological mechanisms that can predict the outcome of multispecies competition. Moreover, we show that this model yields a periodic state where more species than limiting complementary resources can coexist (supersaturation) in a homogeneous environment. We identify two novel mechanisms, how such a state can emerge: a transcritical bifurcation of a limit cycle and a transition from a heteroclinic cycle. Furthermore, we demonstrate the robustness of the phenomenon of supersaturation when the environmental conditions are varied.

On the complete classification of nullcline stable competitive three-dimensional Gompertz models

We investigate a competitive model of three species, each of which, in isolation, admits Gompertz growth. A well-known theorem by M.W. Hirsch guarantees the existence of carrying simplex. Based on this, we compare three dimensional competitive Gompertz models with three dimensional competitive Lotka–Volterra models, and we find that each Gompertz model has a corresponding Lotka–Volterra model with identical nullclines. We then present the complete classification of nullcline stable models and arrive at a total of 33 stable nullcline classes, and show that in 27 of these classes all the compact limit sets are equilibria. Despite the common results, we go on to show that the behavior on the carrying simplex of Gompertz systems is subtly different from that on Lotka–Volterra systems. The number of limit cycles is finite in 5 of the remaining 6 classes, and that only the classes 26 and 27 admit Hopf bifurcations and the other 4 do not. The class 27, which has a heteroclinic cycle, contains a system having May–Leonard phenomenon: the existence of nonperiodic oscillation, and still admitting at least two limit cycles. The numerical simulation reveals that there are some systems in class 28 with two limit cycles.

Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface

An invariant algebraic surface is calculated for a 3D autonomous quadratic system. Also, the dynamics near finite singularities and near infinite singularities on the invariant algebraic surface is analyzed. Furthermore, pitchfork bifurcation is analyzed using center manifold theorem and a first integral of this quadratic system for some special parameters is provided. Finally, the dynamics of this system at infinity using the Poincare compactification in \(R^3\) is investigated and the singularly degenerate heteroclinic cycles are presented by a first integral and verified by numerical simulations.

Complex Dynamics in the Unified Lorenz-Type System

This paper is devoted to the analysis of complex dynamics of the unified Lorenz-type system (ULTS) with six parameters, which contain common chaotic systems as its particular cases. First, some important local dynamics such as pitchfork bifurcation, Hopf bifurcation, and the stability of nondegenerate and double-zero equilibria are systematically investigated using the parameter-dependent center manifold theory combined with some bifurcation theories. Some adequate conditions for guaranteeing the occurrence of degenerate Hopf bifurcation (DHB) and the stability of the equilibria are given. Second, it is found that if DHB does not generate at the trivial equilibrium but generates at two symmetric nontrivial equilibria, then a small perturbation can lead that ULTS to exhibit a chaotic attractor. Interestingly, such a case can take place in the Chen and Lü systems (two common chaotic systems) but cannot take place in the Lorenz and Yang systems (the other two common chaotic systems), essentially distinguishing the Lorenz system from the Chen system. In addition, it is numerically verified that both of the latter two systems can exhibit the coexistence of both a chaotic attractor and multiple limit cycles but the former two systems seem not to have this property. If DHB takes place simultaneously at three equilibria of ULTS, then this system has an invariant algebraic surface, and rigorously prove the existence of some global dynamics such as periodic orbit, center, homoclinic/heteroclinic orbits. Third, it is shown that a singularly degenerate heteroclinic cycle can exist in the case of b = 0 (where b is a parameter of ULTS, like that in the Lorenz system), and a chaotic attractor can be generated by perturbing this cycle for small b > 0. These results altogether indicate that the ULTS can exhibit complex dynamics, and provide a more reasonable classification for chaos in the 3D autonomous chaotic ODE systems that were developed based on the Lorenz system, in contrast to the previous studies.

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper, we consider Lienard systems of the form $$\frac{{dx}} {{dt}} = y, \frac{{dy}} {{dt}} = - \left( {x + bx^3 - x^5 } \right) + \varepsilon \left( {\alpha + \beta x^2 + \gamma x^4 } \right)y,$$ where b ∈ ℝ, 0 < |∈| ≪ 1, (α, β, γ) ∈ D ∈ ℝ3 and D is bounded. We prove that for |b| ≫ 1 (b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals Open image in new window is 2 (counting the multiplicity) and this upper bound is a sharp one.

Cooperation within triplets in the rock-paper-scissors game

We study a population involved in a cyclic game of three strategies – the rock-paper-scissors game – whose agents interact through groups of three individuals (triplets), considering the possibility that two weak agents cooperate and beat a strong one. In a wide range of parameters the system presents a stable heteroclinic cycle, which implies that in a finite population some of the strategies become extinct and others survive. We find that the cooperation within triplets only benefits the survival of the strategy if the cooperation probability is above a certain threshold. We study the survival probabilities of the different strategies as a function of the cooperation parameters through a analytic approximation and compare with simulations, obtaining a good agreement. Results are generalizable to other systems with heteroclinic cycles.

Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.