Heteroclinic Cycles Research Articles

Global dynamics of a special class of planar sector-wise linear systems

In this article, we study the global dynamics of a special class of planar sector-wise linear differential systems with two subsystems being the same except for their equilibriums. Taking the position of one equilibrium as the bifurcation parameter, we provide a complete analysis about sliding cycle bifurcation and sliding homoclinic bifurcation. Moreover, we obtain the existence of all important separatrix orbits and their dependence on the bifurcation parameter, including sliding heteroclinic orbit, heteroclinic cycle, limit cycle, sliding homoclinic cycle and sliding cycle. For more information see https://ejde.math.txstate.edu/Volumes/2024/57/abstr.html

Understanding the role of B cells in CAR T-cell therapy in leukemia through a mathematical model.

Chimeric antigen receptor T (CAR T) cell therapy has been proven to be successful against a variety of leukemias and lymphomas. This paper undertakes an analytical and numerical study of a mathematical model describing the competition of CAR T, leukemia, tumor, and B cells. Considering its significance in sustaining anti-CD19 CAR T-cell stimulation, a B-cell source term is integrated into the model. Through stability and bifurcation analyses, the potential for tumor eradication, contingent on the continuous influx of B cells, has been revealed, showing a transcritical bifurcation at a critical B-cell input. Additionally, an almost heteroclinic cycle between equilibrium points is identified, providing a theoretical basis for understanding disease relapse. Analyzing the oscillatory behavior of the system, the time-dependent dynamics of CAR T cells and leukemic cells can be approximated, shedding light on the impact of initial tumor burden on therapeutic outcomes. In conclusion, the study provides insights into CAR T-cell therapy dynamics for acute lymphoblastic leukemias, offering a theoretical foundation for clinical observations and suggesting avenues for future immunotherapy modeling research.

Ghost Channels and Ghost Cycles Guiding Long Transients in Dynamical Systems.

Dynamical descriptions and modeling of natural systems have generally focused on fixed points, with saddles and saddle-based phase-space objects such as heteroclinic channels or cycles being central concepts behind the emergence of quasistable long transients. Reliable and robust transient dynamics observed for real, inherently noisy systems is, however, not met by saddle-based dynamics, as demonstrated here. Generalizing the notion of ghost states, we provide a complementary framework that does not rely on the precise knowledge or existence of (un)stable fixed points, but rather on slow directed flows organized by ghost sets in ghost channels and ghost cycles. Moreover, we show that the appearance of these novel objects is an emergent property of a broad class of models typically used for description of natural systems.

High frequency forcing of an attracting heteroclinic cycle

This article is concerned with the effect of time-periodic forcing on a vector field exhibiting an attracting heteroclinic network. We show that as the forcing frequency tends to infinity, the dynamics reduces to that of a network under constant forcing, the constant being the average value of the forcing term. We also show that under small constant forcing the network breaks up into an attracting periodic solution that persists for periodic forcing of high frequency.

Homoclinic behavior around a degenerate heteroclinic cycle in a Lorenz-like system

In this work, we analyze a degenerate heteroclinic cycle that appears in a Lorenz-like system when one of the involved equilibria changes from real saddle to saddle-focus. First, from a theoretical model based on the construction of a Poincaré return map, we demonstrate that an infinite number of homoclinic connections arise from the point of the parameter plane where the degenerate heteroclinic cycle appears. The subsequent numerical study not only illustrates the presence of the first homoclinic orbits in the infinite succession but also allows to find other important local and global organizing centers of codimension two (Bogdanov–Takens bifurcations, degenerate homoclinic and heteroclinic connections, T-points) and three (triple-zero bifurcation, doubly-degenerate heteroclinic cycles, degenerate T-points).

Invariant manifolds in a reversible Hamiltonian system: The tentacle-like geometry

We study a one-parameter family of time-reversible Hamiltonian vector fields in R4, which has received great attention in the literature. On the one hand, it is due to the role it plays in the context of certain applications in the field of Physics or Engineering and, on the other hand, we especially highlight its relevance within the framework of generic unfoldings of the four-dimensional nilpotent singularity of codimension four.The system exhibits a bifocal equilibrium point for a range of parameter values. The associated two-dimensional invariant manifolds, stable and unstable, fold into the phase space in such a way that they produce intricate patterns. This entangled geometry has previously been called tentacular geometry.We consider a three-dimensional level set containing the bifocal equilibrium point to gain insight into the folding behavior of these invariant manifolds. Our method consists of describing the traces left by invariant manifolds when crossing an invariant cross section by the reversibility map. With this new approach, we provide a better understanding of how tentacular geometry evolves with respect to the parameter.Our techniques enables us to link the tentacular geometry on the cross section with the study of cocooning cascades of homoclinic tangencies. Indeed, we present a general theory to extend to R4 the phenomena related to cocoon bifurcation that classically develop within families of three-dimensional reversible vector fields. On the basis of these results, we conjecture the existence of heteroclinic cycles consisting of two orbits connecting the bifocus with a saddle node periodic orbit.

Creation of Three‐Scroll Hidden Conservative Lorenz‐Like Chaotic Flows

AbstractIn contrast to the various well‐known two‐scroll hidden Lorenz‐like attractors, little seems to be known about three‐scroll hidden conservative chaotic flows in the 3D Lorenz‐like system family whose Kaplan–Yorke dimension is approximately equal to . This technical note first proposes a three‐scroll chaotic system, i.e., , , , which generates three‐scroll hidden chaotic flows without equilibrium points when . Moreover, its Kaplan–Yorke dimension is . Then, for and , this system has another type of singularly degenerate heteroclinic cycle consisting of two different equilibria in the lines of semi‐hyperbolic equilibria and , i.e., and , with . With a small perturbation of , the kind of three‐scroll hidden conservative chaotic flows described above are created. Finally, five cycles are obtained when the proposed system undergoes Hopf bifurcation.

Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

Periodic bursting cycles on the edge to square duct turbulence

The long-time behaviour of trajectories embedded in the edge manifold between the laminar and turbulent basins of attraction is investigated in order to identify simple invariant solutions to the Navier-Stokes equations in straight ducts with square cross-section. With the aid of the iterative bisection procedure of Toh and Itano (J. Fluid Mech., vol. 481, 2003, pp. 67-76), we detect three travelling waves that represent attracting ‘edge states’ within the separatrix and saddles w.r.t. the full state space. All three travelling waves are presumably connected to solutions known in the community. In other situations, the edge states to which trajectories are attracted seem to be invariant periodic cycles that feature a rich dynamics: Along each cycle, the flow oscillates periodically between two flow states, in each of which streamwise vorticity is predominantly residing near one set of parallel walls, while the vortical activity associated to the other pair of opposing walls is markedly reduced. This ‘switching’ dynamics is accompanied by intermittent bursting-like high-dissipation excursions. While we cannot rigorously prove that the identified periodic cycle is a periodic orbit rather than a finite-precision approximation to a heteroclinic cycle, sound arguments for the former supposition are given. Interestingly, the different phases of the periodic cycle qualitatively resemble a similar ‘switching’ behaviour observed in (chaotic) marginal turbulence, which suggests that the found periodic cycles could help understanding the characteristic dynamics of duct turbulence at low Reynolds numbers.

Bifurcations of Sliding Heteroclinic Cycles in Three-Dimensional Filippov Systems

Global bifurcations with sliding have rarely been studied in three-dimensional piecewise smooth systems. In this paper, codimension-2 bifurcations of nondegenerate sliding heteroclinic cycle [Formula: see text] are investigated in three-dimensional Filippov systems. Two cases of sliding heteroclinic cycle are discussed: [Formula: see text] connecting two saddle-foci, [Formula: see text] connecting one saddle-focus and one saddle. It is proved that at most one sliding homoclinic or one sliding periodic orbit can bifurcate from [Formula: see text] under certain conditions at the eigenvalues of the equilibria, but they cannot coexist. The asymptotic stability of the sliding periodic orbit and the structural feature of the bifurcation curves of homoclinic orbits are further studied. Finally, two numerical examples corresponding to cases [Formula: see text] and [Formula: see text], respectively, are simulated to verify the theoretical results.

Stability of cycles and survival in a jungle game with four species

The Jungle Game is used in population dynamics to describe cyclic competition among species that interact via a food chain. The dynamics of the Jungle Game supports a heteroclinic network whose cycles represent coexisting species. The stability of all heteroclinic cycles in the network for the Jungle Game with four species determines that only three species coexist in the long-run, interacting under cyclic dominance as a Rock–Paper–Scissors Game. This is in stark contrast with other interactions involving four species, such as cyclic interaction and intraguild predation. We use the Jungle Game with four species to determine the success of a fourth species invading a population of Rock–Paper–Scissors players.

Wake dynamics in buoyancy-driven flows: Steady-state-Hopf-mode interaction with O(2) symmetry revisited.

We present a detailed mathematical study of a truncated normal form relevant to the bifurcations observed in wake flow past axisymmetric bodies, with and without thermal stratification. We employ abstract normal form analysis to identify possible bifurcations and the corresponding bifurcation diagrams in parameter space. The bifurcations and the bifurcation diagrams are interpreted in terms of symmetry considerations. Particular emphasis is placed on the presence of attracting robust heteroclinic cycles in certain parameter regimes. The normal form coefficients are computed for several examples of wake flows behind buoyant disks and spheres, and the resulting predictions compared with the results of direct numerical flow simulations. In general, satisfactory agreement is obtained.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems

Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems

Complex localization mechanisms in networks of coupled oscillators: Two case studies.

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

Pseudo and true singularly degenerate heteroclinic cycles of a new 3D cubic Lorenz-like system

In contrast to the coexistence of infinitely many pseudo and true singularly degenerate heteroclinic cycles with nearby two-scroll hyperchaotic Lorenz-like attractors coined in four-dimensional systems, this note reports the ones with nearby bifurcated chaotic attractors in a new 3D cubic Lorenz-like system. Except for the rabbit head/boat shape two-scroll attractors, there exist globally exponentially asymptotically stable parabolic type equilibria and a pair of heteroclinic orbits. In addition, numerical simulations also validate the correctness of the theoretical analysis.

A double-zero bifurcation in a Lorenz-like system

The Lorenz system presents a double-zero bifurcation (a double-zero eigenvalue with geometric multiplicity two). However, its study by means of standard techniques is not possible because it occurs for a non-isolated equilibrium. To circumvent this difficulty, we add in the third equation a new term, Dz2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Dz^2$$\\end{document}. In this Lorenz-like system, the analysis of the double-zero bifurcation of the equilibrium at the origin guarantees, for certain values of the parameters, the existence of a heteroclinic cycle between the two equilibria located on the z-axis. The numerical continuation in parameter space of the locus of heteroclinic connections allows to detect various degeneracies of codimension two and three, some of which have not been previously studied in the literature. These bifurcations are organizing centers of the complicated dynamics exhibited by this system. Furthermore, studying how the bifurcation sets evolve when D tends to zero, we are able to explain, in the Lorenz system, the origin of several global connections which are related to T-point heteroclinic loops.

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of ‘defensive alliances’ observed in previous numerical experiments. We further describe the structure of the heteroclinic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition.

Finite switching near heteroclinic networks

We address the level of complexity that can be observed in the dynamics near a robust heteroclinic network. We show that infinite switching, which is a path towards chaos, does not exist near a heteroclinic network such that the eigenvalues of the Jacobian matrix at each node are all real. Furthermore, for a path starting at a node that belongs to more than one heteroclinic cycle, we find a bound for the number of such nodes that can exist in any such path. This constricted dynamics is in stark contrast with examples in the literature of heteroclinic networks such that the eigenvalues of the Jacobian matrix at one node are complex.

Correction to: Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Correction to: Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation