| 1. |
- Dahne, Joel, 1994-
(författare)
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Computer Assisted Studies in Fluid Mechanics and Spectral Geometry
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains four papers in the area of partial differential equations. The first two papers are related to spectral geometry and the last two papers to fluid mechanics. A common theme of the papers is that they make use of computer assisted methods in the proofs.Paper I concerns the computation of very precise enclosures of eigenvalues of the Laplace-Beltrami operator on spherical triangles. The interest in these eigenvalues comes from a connection with the combinatorial problem of studying discrete random walks.Paper II gives a concrete counterexample to Payne's nodal line conjecture. The conjecture is concerned with the existence of bounded planar domains for which the second eigenfunction of the Dirichlet Laplacian has a nodal line that doesn't touch the boundary. The paper gives an explicit domain for which it is proved that the nodal line doesn't touch the boundary.Paper III and IV both prove the existence of a certain type of solution known as a highest cusped traveling wave. Paper III deals with the Burgers-Hilbert equation and Paper IV with a family of fractional Korteweg-de Vries equations. The existence is asserted by constructing an explicit approximation of the solution and proving the existence of an exact solution nearby with the use of a fixed point formulation. The proof not only establishes the existence, but also determines the precise asymptotic behavior of the waves near the cusp.
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| 2. |
- Dahne, Joel, et al.
(författare)
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Enclosing all zeros of a system of analytic functions
- 2019
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 348, s. 513-522
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Tidskriftsartikel (refereegranskat)abstract
- We present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral. We compare this to a simpler, also rigorous, method based on bisection. The latter is determined to be more efficient in the examples considered. This is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method. (C) 2018 Elsevier Inc. All rights reserved.
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| 3. |
- Galias, Zbigniew, et al.
(författare)
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On the Existence of the Double Scroll Attractor for the Chua's Circuit with a Smooth Nonlinearity
- 2018
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Ingår i: 2018 IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS (ISCAS). - : IEEE. - 9781538648810
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Konferensbidrag (refereegranskat)abstract
- In simulations of the Chua's circuit with a smooth nonlinearity for certain parameter values one observes the double scroll attractor. This attractor contains an unstable equilibrium, and typical trajectories belonging to the attractor may pass arbitrarily close to this equilibrium. In consequence, it is impossible to compute trajectories over the whole attractor using standard rigorous numerical integration procedures. This is due to the existence of trajectories which spend arbitrarily long time in a neighborhood of the equilibrium. In this work, a method to find enclosures of trajectories passing arbitrarily close to an unstable fixed point of spiral type is presented. This method is used to prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity.
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| 4. |
- Galias, Zbigniew, et al.
(författare)
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Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor
- 2019
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 266:5, s. 2408-2434
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.
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| 5. |
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| 6. |
- Gennemark, Peter, 1974, et al.
(författare)
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Optimal Design in Population Kinetic Experiments by Set-Valued Methods
- 2011
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Ingår i: AAPS Journal. - : Springer Science and Business Media LLC. - 1550-7416. ; 13:4, s. 495-507
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Tidskriftsartikel (refereegranskat)abstract
- We propose a new method for optimal experimental design of population pharmacometric experiments based on global search methods using interval analysis; all variables and parameters are represented as intervals rather than real numbers. The evaluation of a specific design is based on multiple simulations and parameter estimations. The method requires no prior point estimates for the parameters, since the parameters can incorporate any level of uncertainty. In this respect, it is similar to robust optimal design. Representing sampling times and covariates like doses by intervals gives a direct way of optimizing with rigorous sampling and dose intervals that can be useful in clinical practice. Furthermore, the method works on underdetermined problems for which traditional methods typically fail.
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| 7. |
- Johnson, Tomas, 1979-, et al.
(författare)
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A note on the convergence of parametrised non-resonant invariant manifolds
- 2011
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Ingår i: Qualitative Theory of Dynamical Systems. - : Springer Science and Business Media LLC. - 1575-5460 .- 1662-3592. ; 10:1, s. 107-121
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Tidskriftsartikel (refereegranskat)abstract
- Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.
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| 8. |
- Johnson, Tomas, 1979-, et al.
(författare)
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A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five
- 2009
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Ingår i: Dynamical systems. - : Informa UK Limited. - 1468-9367 .- 1468-9375. ; 24:2, s. 237-247
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Tidskriftsartikel (refereegranskat)abstract
- We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles, bifurcating from the given system for general (Lienard type) quartic perturbations, are constructed. We also discuss how to construct perturbations yielding a given configuration, and how to validate the correctness of such a candidate perturbation.
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| 9. |
- Johnson, Tomas, 1979-, et al.
(författare)
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An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7
- 2010
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - : World Scientific Publishing. - 0218-1274. ; 20:5, s. 1451-1458
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Tidskriftsartikel (refereegranskat)abstract
- The limit cycle bifurcations of a Z(2) equivariant planar Hamiltonian vector field of degree 7 under Z(2) equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.
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| 10. |
- Johnson, Tomas, 1979-, et al.
(författare)
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An improved lower bound on the number of limit cycles bifurcating from a quintic Hamiltonian planar vector field under quintic perturbation
- 2010
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - : World Scientific Publishing. - 0218-1274. ; 20:1, s. 63-70
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Tidskriftsartikel (refereegranskat)abstract
- The limit cycle bifurcations of a equivariant quintic planar Hamiltonian vector field under equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.
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