1,720,975 research outputs found
Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov-Takens Bifurcation
No full textThis paper provides for the first time correct third-order homoclinic predictors in n-dimensional ODEs near a generic Bogdanov-Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincar\'e method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincar\'e) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.</p
Homoclinic solutions in finite and infinite dimensional systems
No full textThis thesis is concerned with higher-order asymptotics to the homoclinic orbit near the generic and transcritical codimension two Bogdanov-Takens bifurcation in infinite dimensional systems generated by delay differential equations (DDEs). First, we will obtain accurate homoclinic asymptotics in the normal form. Then we will perform the parameter-dependent center manifold reduction near the generic and transcritical Bogdanov-Takens points. To achieve this, we rigorously derive a method to translate asymptotics of solutions in the normal form for a local bifurcation, to asymptotics of solutions on the parameter-dependent center manifold. In particular, we allow for a time-reparametrization in the ho-mological equation, enabling us to consider orbital normal forms. The use of orbital normal forms turns out to be particularly useful when obtaining third-order homoclinic asymptotics near the transcritical Bogdanov-Takens bifurcations. Indeed, we show that, by using orbital normal forms, these asymptotics can be obtained through a simple transformation from the generic case. Additionally, a detailed comparison is provided between applying different Bogdanov-Takens normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincaré) to approximate the homoclinic solution near Bogdanov-Takens points. In particular, we will show that higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Next to the codimension two Bogdanov-Takens bifurcation points, we also perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurca-tions emanating from these codimension two bifurcation points. Furthermore, the known existence theorem of a smooth finite-dimensional parameter-dependent center manifold for delay differential equations is generalized to allow for the equilibrium under consideration to vanish, as is the case in the zero-Hopf and the generic Bogdanov-Takens bifurcation points. The proof is given at the abstract semigroup level using the framework of perturbation theory for dual semigroups. The non-hyperbolic cycles and homoclinic asymptotics are implemented in DDE-BifTool to start numerical continuation of these homoclinic curves. The homoclinic predictor in MatCont has been corrected as well. The effectiveness of the new predictors is demonstrated in numerous examples. In-depth treatments of the examples are also provided, as well as the MATLAB, Python, and Julia source code to reproduce the obtained results. Finally, we present a novel phenomenon in the study of the renormalization group (RG). Namely, we found Shilnikov homoclinic orbits in the RG flow of a quantum field theory, proving the existence of chaotic RG-behavior in the vicinity of a fixed point
Bifurcation Analysis of Bogdanov-Takens Bifurcations in Delay Differential Equations
No full textIn this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt-Poincar\'e method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool.</p
The role of frailty in shaping social contact patterns in Belgium, 2022-2023
No full textSocial contact data are essential for understanding the spread of respiratory infectious diseases and designing effective prevention strategies. However, many studies often overlook the heterogeneity in mixing patterns among older age groups and individual frailty levels, assuming homogeneity across these sub-populations. This shortcoming may undermine non-pharmaceutical interventions by not targeting specific contact behaviours, potentially reducing their effectiveness in controlling disease. To address this gap, we conducted a contact survey in Flanders, Belgium (June 2022-June 2023). We collected data from 5995 participants (overall response rates of 19.34%) who recorded 31,375 contacts with distinct individuals. Within this cohort, 14.50% were classified as frail, and 46.85% were classified as non-frail. On average, participants report 5.48 contacts daily, with a median of 4 contacts (IQR: 2-7). These contacts vary based on participants' age and frailty levels, influenced by the locations of their interactions. Using the collected data, we reconstructed frailty-dependent contact matrices and developed a contact-based mathematical model that integrates participants' and contactees' frailty levels to investigate how frailty levels affect transmission dynamics. Incorporating frailty levels into the mathematical model substantially alters the shape of epidemic curves and peak incidences. Such insights might provide useful insights for informing non-pharmaceutical interventions, indicating the potential benefit of similar data collection in different countries.Funding
Funding for this study [study number: 215366] was provided by GSK (GlaxoSmithKline). GSK was provided the opportunity to review a preliminary version of this publication for factual accuracy, but the authors are solely
responsible for final content and interpretation.
Acknowledgements
The authors gratefully acknowledge the IMI VITAL project for their valuable input and feedback during the development of the study protocol. We extend our sincere thanks to the Ipsos team for conducting the survey, collecting data, and facilitating the rapid progress of this study. We especially appreciate the exceptional project management support provided by Sarah Vercruysse. All important findings will be informed to the IMI VITAL WP3
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
No full textIn this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual
semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative.
The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more
generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful toolto study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models
Chaotic RG flow in tensor models
No full textWe study biantisymmetric tensor quantum field theories with O(N-1) x O(N-2) symmetry. Working in 4 - epsilon dimensions we calculate the beta functions up to second order in the coupling constants and analyze in detail the renormalization group (RG) flow and its fixed points. We allow N-1 and N-2 to assume general real values and treat them as bifurcation parameters. In studying the behavior of these models in a nonunitary regime in the space of N-1 and N-2 we find a point where a zero-Hopf bifurcation occurs. In the vicinity of this point, we provide analytical and numerical evidence for the existence of Shilnikov homoclinic orbits, which induce chaotic behavior in the RG flow of a subset of nearby theories. As a simple warm-up example for the study of chaotic RG flows, we also review the non-Hermitian Ising chain and show how, for special complex values of the coupling constant, its RG transformations are equivalent to the Bernoulli map.We are grateful to Igor R. Klebanov for insightful discussions and suggestions throughout the project. We are also grateful to A. Gorsky, A. Morozov, Yu. A. Kuznetsov, A. Milekhin, Y. Oz, A. Polyakov, Y. Wang, S. Dubovsky, and V. Rosenhaus for valuable discussions and comments. F. K. P. acknowledges support from Russian Science Foundation (Grant No. 20-71-10073)
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
No full textIn this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual
semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative.
The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more
generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful toolto study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
Full text linkIn this paper, we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf, and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models
Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs
Full text linkA recent work by the authors on the existence of a periodic smooth
finite-dimensional center manifold near a nonhyperbolic cycle in delay
differential equations motivates the derivation of periodic normal forms. In
this paper, we prove the existence of a special coordinate system on the center
manifold that will allow us to describe the local dynamics on the center
manifold near the cycle in terms of these periodic normal forms. To construct
the linear part of this coordinate system, we prove the existence of time
periodic smooth Jordan chains for the original and adjoint system. Moreover, we
establish duality and spectral relations between both systems by using tools
from the theory of delay equations and Volterra integral equations, dual
perturbation theory, duality theory and evolution semigroups.Comment: 53 pages, 1 figure. arXiv admin note: text overlap with
arXiv:2207.0248
Periodic Center Manifolds for DDEs in the Light of Suns and Stars
No full textIn this paper, we prove the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in classical delay differential equations by using the Lyapunov-Perron method. The results are based on the rigorous functional analytic perturbation framework for dual semigroups (sun-star calculus). The generality of the dual perturbation framework ensures that the results extend to a much broader class of evolution equations
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