Scalar Ordinary Differential Equations Research Articles

Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

AbstractThe global bifurcation diagrams for two different one-parametric perturbations ($$+\lambda x$$ + λ x and $$+\lambda x^2$$ + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation $$x'=f(t,x)$$ x ′ = f ( t , x ) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation $$+\lambda x$$ + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work.

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations

Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor.

Estimating rate-induced tipping via asymptotic series and a Melnikov-like method* *CK was partly supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges partial support of the EU within the TiPES project funded the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. 820970. IPL was partly supported by MICIIN/FEDER project RTI2018-096523-B-100 and by European Union’s

The paper deals with the study of rate-induced tipping in asymptotically autonomous scalar ordinary differential equations. We prove that, in such a tipping scenario, a solution which limits at a hyperbolic stable equilibrium of the past limit-problem loses uniform asymptotic stability and coincides with a solution which limits at a hyperbolic unstable equilibrium of the future limit-problem. We use asymptotic series to approximate such pairs of solutions and characterize the occurrence of a rate-induced tipping by using only solutions calculable on finite time intervals. Moreover, we show that a Melnikov-inspired method employing the asymptotic series allows to asymptotically approximate the tipping point.

Non-Iterative Simulation Methods for Virtual Analog Modelling

The simulation of nonlinear components is central to virtual analog simulation. In audio effects, circuit elements often include devices such as diodes and transistors, mostly operating in the strongly nonlinear regime. Mathematical models are of the form of systems of nonlinear ordinary differential equations (ODEs), and traditional integrators, such as the trapezoid and midpoint methods, can be employed as solvers. These methods are fully implicit, and require the solution of a nonlinear algebraic system at each time step, introducing further complications regarding the existence and uniqueness of the solution, as well as the choice of halting conditions for the iterative root finder. On the other hand, fast explicit methods such as Forward Euler, or explicit Runge-Kutta methods, are prone to unstable behaviour at standard audio sample rates, even at moderate amplitudes. For these reasons, in this work a family of linearly-implicit schemes is presented. These schemes take the form of a perturbation expansion, making the construction of higher-order schemes possible. Compared with classic implicit designs, the proposed methods have the advantage of efficiency, since the update is computed in a single iteration, through the solution of a linear system of equations. Furthermore, the existence and uniqueness of the update are proven by simple inspection of the update matrix. Compared to classic explicit designs, the proposed schemes display stable behaviour at standard audio sample rates. In the case of a single scalar ODE, sufficient conditions for numerical stability can be derived, imposing constraints on the choice of the sampling rate. Several theoretical results are provided, as well as numerical examples for typical stiff equations used in virtual analog modelling.

Elementary method for obtaining general solutions to systems of ordinary differential equations

An analytical method is proposed for finding the general solution of a system of ordinary differential equations (ODEs). The general solution is expressed as a series which in some cases can be summed to give an expression in closed form. A sufficient condition for the series to converge is derived. Illustrative examples are given for scalar first-order ODEs (Riccati, Abel, homogeneous, Bernoulli, linear, separable) and for higher order ODEs (Airy, linear oscillator, Lienard, van der Pol). The method relies only on a calculus background.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/58/abstr.html

A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras

The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries.

Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this problem in the case of Painleve equations, we use previously constructed invariants of a family of equations whose right-hand sides have a cubic nonlinearity in the first derivative. We obtain the conditions for point equivalence to the fifth Painleve equation if two or three of its parameters are equal to zero.

Complete Classification of Scalar Fourth-Order Ordinary Differential Equations and Linearizing Algorithms

Complete Classification of Scalar Fourth-Order Ordinary Differential Equations and Linearizing Algorithms

Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation

We consider the equivalence problem for projective-type scalar second-order ordinary differential equations with respect to invertible point changes of variables. Invariants of the equivalence transformation group of this family of equations are used to find necessary conditions for equivalence to the sixth Painleve equation.

The Explicit Fatunla’s Method for First-Order Stiff Systems of Scalar Ordinary Differential Equations: Application to Robertson Problem

Ordinary differential equations (ODEs) are among the most important mathe-matical tools used in producing models in the physical sciences, biosciences, chemical sciences, engineering and many more fields. This has motivated re-searchers to provide efficient numerical methods for solving such equations. Most of these types of differential models are stiff, and suitable numerical methods have to be used to simulate the solutions. This paper starts with a sur-vey on the basic properties of stiff differential equations. Thereafter, we present the explicit one-step algorithm proposed by Fatunla to solve stiff systems of first-order scalar ODEs. As an illustrative example, we consider the Robertson problem (RP) which is known to be stiff. The results obtained with the explicit Fatunla method (EFM) are compared with those computed by the solver RADAU which is based on implicit Runge-Kutta methods. Our results are in good agreement with the latter ones.

The multiplier problem of the calculus of variations for scalar ordinary differential equations

In the long-standing inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after being multiplied with the multiplier, becomes the Euler–Lagrange equation for the Lagrangian. An answer to this problem for the case of a scalar ordinary differential equation of order $$2n, n\ge 2,$$ is proposed.

Conditional symmetries of nonlinear third-order ordinary differential equations

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

Structure of Lie point and variational symmetry algebras for a class of odes

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

Algebraic properties of first integrals for systems of second-order ordinary differential equations of maximal symmetry

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary differential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I1, I2 and their quotient I2/I1 are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system.

A point symmetry based method for transforming ODEs with three-dimensional symmetry algebras to their canonical forms

We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

Generating functions and existence of contact symmetries of third order scalar ordinary differential equations

Analyzing the structure of generating functions associated to the contact symmetry condition of third order scalar ordinary differential equations, constraints guaranteeing that the ODE does not admit contact symmetries are obtained. Further, inspecting the subalgebras generated by contact symmetries having a specific form, an alternative procedure to generate third order equations with irreducible contact symmetry algebra sp(4) is developed.

Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in variable step-size formulation

This paper presents the construction of a new family of explicit schemes for the numerical solution of initial-value problems of ordinary differential equations (ODEs). The one-parameter family is constructed by considering a suitable rational approximation to the theoretical solution, resulting a family with second-order convergence and A-stable. Imposing that the principal term in the local truncation error vanishes, we obtain an expression for the parameter value in terms of the point (xn, yn) on each step. With this approach, the resulting method has third order convergence maintaining the characteristic of A-stability. Finally, combining this last method with other of order two in order to get an estimation for the local truncation error, an implementation in variable step-size has been considered. The proposed method can be used in a wide range of problems, for solving numerically a scalar ODE or a system of first order ODEs. Several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature.

Complexity of parametric initial value problems for systems of ODEs

We study the approximate solution of initial value problems for parameter dependent finite or infinite systems of scalar ordinary differential equations (ODEs). Both the deterministic and the randomized setting is considered, with input data from various smoothness classes. We study deterministic and Monte Carlo multilevel algorithms and derive convergence rates. Moreover, we prove their optimality by showing matching (in some limit cases up to logarithmic factors) lower bounds and settle this way the complexity. Comparisons between the deterministic and randomized setting are given, as well.