Autonomous ODEs Research Articles

Period-Increasing Indicates Loss of Stability

We first present a mathematical theory on period-increasing as an indicator of stability loss in a dynamical system of any dimension not less than two, and show that in two-dimensional autonomous ODEs, the period-increasing is closely associated with homoclinic or heteroclinic bifurcation. Then we demonstrate that the proposed period-increasing regime shift can occur in some ecological models.

Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates. Similarities for characterizing finite-time extinction and asymptotic behavior of compacton traveling waves to blow-up solutions are also shown.

Self-Crossing Memristive Pinched Hystereses in Autonomous Implicit Models

This paper shows that autonomous implicit ODEs (based on planar lemniscates) are interesting models, yielding pinched self-crossing hystereses that are intrinsic features of all memristive elements in nanoscale electronics. Each model has a folded saddle that allows for an oscillating trajectory to traverse different sides of singularity. The models considered in this paper are autonomous and therefore different from typical input–state–output models considered thus far. The models preserve the usual properties of memristive elements. For example, the area enclosed by the pinched hystereses decreases with the increased frequency of oscillations. The same-time instant zero-crossing property is also satisfied provided that certain conditions are met. Another novel aspect of this paper is the fact that the autonomous models are based on various planar lemniscates (of Gerono, Devil and Bernoulli) which can be nonlinearly transformed to model pinched hystereses of various shapes. The proposed models are differentiable and the use of the sign and absolute value terms, typical in modeling of memristive elements, is avoided. Several simulation results are included and two simple analog circuits having the pinched hysteretic characteristics of mem-inductors and mem-capacitors are proposed.

Steady-state analysis of second-order traffic models and application to traffic flow control

We investigate the problem of controller design after linearization around the steady-state solutions of second-order traffic models. At steady-state the traffic variables follow space-dependent autonomous ODEs. The steady-state solutions can be used to account for the heterogeneity of the road in space, which can be caused by different environmental factors. Proceeding by analogy with open-channel flow studies, where such states are relevant for operating a network with efficient control strategies, we analyse and compare steady-state solutions for different second-order traffic models. We show that the steady-state density profile remains in three different regions. The behaviour of the considered models is discussed, leading to proposed corrections of the models. We select one corrected model, which we use for the design and implementation of a boundary controller to illustrate the effectiveness of our approach.

Spherically Symmetric Solutions of the Full Compressible Euler Equations in $$R^N$$RN

This paper focuses on the multidimensional spherically symmetric full compressible Euler equations and constructs rigorously a family of global self-similar bounded weak solutions for all positive time to its initial value problem with constant initial data. The main approach is to reduce the full compressible Euler equations to an autonomous system of ordinary differential equations under the spherically symmetric and self-similar assumptions. We establish the detailed structures of solutions as well as their existence by analyzing carefully the properties of the integral curves of the autonomous ODE system.

Dynamics of T cell receptor distributions following acute thymic atrophy and resumption.

Naive human T cells are produced and developed in the thymus, which atrophies abruptly and severely in response to physical or psychological stress. To understand how an instance of stress affects the size and “diversity” of the peripheral naive T cell pool, we derive a mean-field autonomous ODE model of T cell replenishment that allows us to track the clone abundance distribution (the mean number of different TCRs each represented by a specific number of cells). We identify equilibrium solutions that arise at different rates of T cell production, and derive analytic approximations to the dominant eigenvalues and eigenvectors of the mathematical model linearized about these equilibria. From the forms of the eigenvalues and eigenvectors, we estimate rates at which counts of clones of different sizes converge to and depart from equilibrium values—that is, how the number of clones of different sizes ”adjusts” to the changing rate of T cell production. Under most physiological realizations of our model, the dominant eigenvalue (representing the slowest dynamics of the clone abundance distribution) scales as a power law in the thymic output for low output levels, but saturates at higher T cell production rates. Our analysis provides a framework for quantitatively understanding how the clone abundance distribution evolves under small changes in the overall T cell production rate.

Kernel-Based Discretization for Solving Matrix-Valued PDEs

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems

On the Arbitrarily Long-Term Stability of Conservative Methods

We show the arbitrarily long-term stability of conservative methods for autonomous ODEs. Given a system of autonomous ODEs with conserved quantities, if the preimage of the conserved quantities possesses a bounded locally finite neighborhood, then the global error of any conservative method with the uniformly bounded displacement property is bounded for all time when the uniform time step is taken sufficiently small. On finite precision machines, the global error still remains bounded and independent of time until some arbitrarily large time determined by machine precision and tolerance. The main result is proved using elementary topological properties for discretized conserved quantities which are equicontinuous. In particular, long-term stability is also shown using an averaging identity when the discretized conserved quantities do not explicitly depend on time steps. Numerical results are presented to illustrate the long-term stability result.

Growth Rates of Sublinear Functional and Volterra Differential Equations

This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymptotically so), and the nonlinear dependence grows less rapidly than any linear function. We impose extra regularity properties on a function asymptotic to this nonlinear function, rather than on the nonlinearity itself. The main result of the paper demonstrates that the growth rate of the solution can be found by determining the rate of growth of a trivial functional differential equation (FDE) with the same nonlinearity and all its associated measure concentrated at zero; the trivial FDE is nothing other than an autonomous nonlinear ODE. We also supply direct asymptotic information about the solution of the FDE under additional conditions on the nonlinearity, and exploit the theory of regular variation to sharpen and extend the results.

On the existence of canards in a nonlinear fluid system manifesting oscillatory behaviour

In an earlier study dealing with a nonlinear fluid oscillator governed by two autonomous ODEs, the solutions were found to display some aberrant characteristics adjacent to the boundaries of the oscillatory regime in parameter space. It was argued that this behaviour indicated the presence of canards. In the present study it is formally proved that this indeed is the case, and some numerical examples illustrating the phenomenon as well as its effects are presented.

Continuous and discrete one dimensional autonomous fractional ODEs

In this paper, we study 1D autonomous fractional ODEs $D_c^{\gamma}u=f(u), 0< \gamma <1$, where $u: [0,\infty)\mapsto\mathbb{R}$ is the unknown function and $D_c^{\gamma}$ is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate that as the memory effect becomes stronger ($\gamma\to 0$), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Gr\"onwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

Chaos based adaptive double-image encryption scheme using hash function and S-boxes

Most current symmetric cryptographic algorithms (symmetric ciphers and hash functions) are considered to be relatively secure against attacks by quantum computers. Here we propose an adaptive color double-image encryption scheme based on autonomous ODE chaotic system and SHA-512. The double-image is diffused firstly by chaotic sequences with bitwise exclusive or operation, and an algorithm is designed to transform two 512-bit hash values into two S-Boxes, to substitute for the pixels of diffused double-image respectively. In each encryption process, the initial values of chaotic system are from random signal, they are true random numbers. Simulation results demonstrate that the scheme is suitable and effective for color double-image encryption.

Smooth Dynamics Becomes Hybrid in the Limit

We show the appearance of an essentially nonlocal dynamics describing the limit behavior of trajectories of a class of dynamical systems defined by classical autonomous ODEs with smooth right-hand sides containing a small parameter and becoming discontinuous in the formal limit. The limit dynamics is shown to be described by an explicitly constructed Nerode--Kohn hybrid dynamical system consisting of a continuous plant (an ODE) and a finite state machine which are interacting and producing hybrid dynamics with possible memory effects. We remark, however, that, from the “statistical” point of view, the limit behavior of an ensemble of trajectories can still be described by an ODE, with possibly time-dependent and discontinuous right-hand side depending on the chosen ensemble.

Topology trivialization transition in random non-gradient autonomous ODEs on a sphere

We calculate the mean total number of equilibrium points in a system of N random autonomous ODEs introduced by Cugliandolo et al [] to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac–Rice framework. Analysing the asymptotic behaviour for large N we confirm that the phenomenon of ‘topology trivialization’ revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random ‘magnetic field’ term in dynamical equations we find a ‘phase transition’ from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.

A CRITERION TO UNIFORM STABILITY FOR FUNCTIONAL PERTURBED DIFFERENTIAL EQUATIONS

In this paper we consider a class of non autonomous ODEs with a functional perturbation. For the unperturbed equation a Lyapunov function bounded by two quadratic forms is known. The Lipschitzean rate of the vector field along with some additional re- quirements to the derivatives of the Lyapunov function guarantee existence of uniform stable solutions. A sufficient condition that guarantees uniform stability of the zero-solution to the equation under consideration is discussed.

Optimal Consumption Until Ruin for an Endowment Described by an Autonomous ODE for an Infinite Time Horizon

We give an algorithmic solution of the optimal consumption problem [Formula: see text], where Ct denotes the accumulated consumption until time t, and τ denotes the time of ruin. Moreover, the endowment process Xt is modeled by [Formula: see text]. We solve the problem by showing that the function provided by the algorithm solves the Hamilton-Jacobi (HJ) equation in a viscosity sense and that the same is true for the value function of the problem. The argument is finished by a uniqueness result. It turns out that one has to change the optimal strategy at a sequence of endowment values, described by a free boundary value problem. Finally we give an illustrative example.

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

On the Approximation of Stable and Unstable Fiber Bundles of (Non)Autonomous ODEs — A Contour Algorithm

We propose an algorithm for the approximation of stable and unstable fibers that applies to autonomous as well as to nonautonomous ODEs. The algorithm is based on computing the zero-contour of a specific operator; an idea that was introduced in [Hüls, 2006] for discrete time systems. We present precise error estimates for the resulting contour algorithm and demonstrate its efficiency by computing stable and unstable fibers for a (non)autonomous pendulum equation in two space dimensions. Our second example is the famous three-dimensional Lorenz system for which several approximations of the two-dimensional Lorenz manifold are calculated. In both examples, we observe equally well performance for autonomously and nonautonomously chosen parameters.

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.

Existence of periodic orbits in nonlinear oscillators of Emden–Fowler form

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden–Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden–Fowler equation, are discussed in the same context.