Autonomous ODEs Research Articles

Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's

Explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous ODEs are derived. They include second-order coefficients for the Bogdanov--Takens bifurcation, third-order coefficients for the cusp and fold-Hopf bifurcations, and coefficients of the fifth-order terms for the generalized Hopf (Bautin) and double Hopf bifurcations. The formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the Jacobian matrix of the right-hand sides and its transpose, as well as multilinear functions from the Taylor expansion of the right-hand sides at the critical equilibrium. The normal form coefficients for the fold-Hopf bifurcation in the "new" Lorenz model are computed using the derived formulas, proving the existence of a nontrivial invariant set in the system.

Simplest dissipative chaotic flow

Numerical examination of third-order, autonomous ODEs with one dependent variable and quadratic nonlinearities has uncovered what appears to be the algebraically simplest example of a dissipative chaotic flow, x ⃛ + A x ̈ − x ⃛ 2 + x = 0 . This system exhibits a period-doubling route to chaos for 2.017 < A < 2.082 and is approximately described by a one-dimensional quadratic map.

A Predator–Prey Model with Ivlev's Functional Response

The uniqueness of limit cycles is proved for a two-dimensional predator–prey system with a functional response of Ivlev type. The system of planar autonomous ODE's is transformed to a Liénard system to which a modified theorem of Zhang is applied.

Existence of h 2-expansion for some extrapolation methods for solving autonomous higher order ODEs

Extrapolation formulae for solving autonomous dth order ordinary differential equations directly are derived and proven to have h 2-expansion. Some numerical results for directly solving second-order equations using the extrapolation formulae are presented and compared with the corresponding reduction method.

Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps

We present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase-parameter space. The invariant sets involved in our study are equilibrium points and limit cycles of autonomous ODEs, periodic solutions of time-periodic nonautonomous ODEs, fixed points and periodic orbits of iterated maps. The more general concept of a continuation strategy is also discussed. It allows the analysis of various singularities of generic systems and of their mutual relationships. The approach is extended to codimension three singularities. We introduce several bifurcation functions and show how to use them to construct well-posed continuation problems. The described continuation technique is supported by an interactive graphical program called LOCBIF. We discuss briefly the concepts of the LOCBIF interface and give some examples of typical applications.

Hopf bifurcation in a reaction-diffusion system governed by the fitz-hugh-nagumo equations

A combination of analytical and numerical methods is proposed to deal with the Hopf bifurcation in a reaction-diffusion system modeled by the Fitz-Hugh-Nagumo (FHN) equations. Finite-differencing is applied to the diffusion term to transform the FHN equations into an autonomous ODE system. Perturbation techniques based on Poincaré normal form and center manifold theory are used to trace a bifurcate periodic solution branch, from the onset up to a reasonable extent. Fourier spectral method and continuation techniques are used to trace the subsequent development of this branch. The capability of this approach is demonstrated by examples in which the pursued periodic solution becomes highly unstable, and with period tending towards infinity.

A note on multistep methods and attracting sets of dynamical systems

We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.

Homoclinic and heteroclinic bifurcations of Vector fields

We study a bifurcation of homoclinic and heteroclinic orbits in a two or more parameter family of autonomous ODEs, where the unperturbed system has two heteroclinic orbits joined at a common saddle point. Under some assumptions on eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system, we can show that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system. A bifurcation diagram is given for such families. Some homoclinic bifurcations are also treated including the one producing a twice-rounding homoclinic orbit.

Constrained systems, characteristic surfaces, and normal forms

ODEs with a small parameter e multiplying derivatives, which have been studied as a singular perturbation problem, are formulated in a coordinate-free manner, as a pair of a vector field and a tensor field, in order to treat them as a bifurcation problem. This formulation is an extension of the classical interpretation of autonomous ODEs as vector fields, and it is called a constrained system. A method to obtain normal forms for constrained systems is given as well as several results of computing them. Unfoldings of these normal forms include a large part of typical behaviors observed in such ODEs with e. The local classification of characteristic surfaces is also discussed.

On a Class of Superlinear Sturm–Liouville Problems with Arbitrarily Many Solutions

We derive multiplicity results for autonomous superlinear ODE’s of the form $\begin{gathered} - u''(x) = g(u(x)) + t,\qquad x \in (0,\pi ),\quad t \in R, \hfill u(0) = u(\pi ) = 0, \hfill \end{gathered} $ with $g'( - \infty ) < + \infty $ and $g'( + \infty ) = + \infty $.We show that for any given $n \in N$ there exist at least n solutions of the problem if t is sufficiently negative. The proof is carried out by using variational methods jointly with a rearrangement argument.

Computation of periodic solutions of nonlinear odes

The paper proposes a new Gauss-Newton algorithm for the computation of periodic orbits in autonomous nonlinear ODEs. On the basis of Floquet theory, the new algorithm is shown to converge quadratically in a neighbourhood of the solution. Nontrivial examples are included.

Optimality of Euler-integral information for solving a scalar autonomous ode

We study the minimum error of algorithms for solving a scalar autonomous ODE. Information is defined asn evaluations of functionals of the right hand sidef. These functionals can be linear or nonlinear and continuous. Euler-integral information is defined and its optimality is proved in a class of regularf.