Two-dimensional Ordinary Differential Equation Research Articles

Global dynamics of a simple model for wild and sterile mosquitoes.

There are known methods to manage the population dynamics of wild and sterile mosquitoes by releasing genetically engineered sterile mosquitoes. Even if a two-dimensional system of ordinary differential equations is considered as a simple mathematical model for developing release strategies, fully understanding the global behavior of the solutions is challenging, due to the fact that the probability of mating is ratio-dependent. In this paper, we combine a geometric approach called the time-scale transformation and blow-up technique with the center manifold theorem to provide a complete understanding of dynamical systems near the origin. Then, the global behavior of the solution of the two-dimensional ordinary differential equation system is classified in a two-parameter plane represented by the natural death rate of mosquitoes and the sterile mosquito release rate. We also offer a discussion of the sterile mosquito release strategy. In addition, we obtain a better exposition of the previous results on the existence and local stability of positive equilibria. This paper provides a framework for the mathematical analysis of models with ratio-dependent terms, and we expect that it will theoretically withstand the complexity of improved models.

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

НЕЙРОННА МЕРЕЖА ТИПУ АВТОКОДУВАЛЬНИК ДЛЯ ВКЛАДЕННЯ ОДНОВИМІРНИХ ЧАСОВИХ РЯДІВ

The problem of time series embedding is a universal one. It is the main prerequisite when it comes to modeling of dynamical processes using systems of autonomous ordinary differential equations (ODEs) because they have hard requirements for the dimensionality of the problem. One-dimensional ODE can only exhibit 3 types of behavior while two-dimensional ODE can exhibit 9. This is why it is important to increase the dimensionality of the problem before starting the modeling to allow for wider range of possible behaviors in the final model. One way to increase the dimensionality is to delay-embed the time series data but this approach can be extended to allow the use of an autoencoder neural network that would associate a higher-dimensional vector to each point in the time series and will allow the modeling to be performed in higher dimension.

Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs

We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka–Volterra (LV) model describing predator–prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.

Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines

In this paper, we analyze the problem of finding the minimum dimension n such that an analytic map/ordinary differential equation over R n can simulate a Turing machine in a way that is robust to perturbations. We show that one-dimensional analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional C ∞ ordinary differential equation on the compact sphere S 2 .

Radially symmetric stationary solutions for certain chemotaxis systems in higher dimensions: A geometric approach

This paper reports results on the existence, shapes, and asymptotic behavior of positive radially symmetric stationary solutions for several models of chemotaxis in higher dimensions. The systems treated in this paper are the simplest parabolic-elliptic Keller-Segel model and the simplest attraction-repulsion chemotaxis system, in which a positive-valued (resp. negative-valued) solution of one is a negative-valued (resp. positive-valued) solution of the other from symmetry. In particular, the construction of functions satisfying equations that diverge at the endpoints of finite intervals is an interesting result. The key to the discussion is to derive a scalar equation by using a transformation on the averaged mass for the equation satisfied by the radially symmetric stationary solution and to investigate the infinity dynamics as geometric information for the two-dimensional ordinary differential equations derived from it. To achieve this, we use a method that combines classical results from the continuous dynamical systems theory and Poincaré-Lyapunov compactification. In addition, the results for singular solutions are discussed in light of the results of previous studies.

Sufficient conditions and criteria for the pulse-shaped explosion related to equilibria in a class of nonlinear systems

The pulse-shaped explosion (PSE) is a novel sharp transition behavior which can give rise to bursting oscillations. Exploring a general theory applied to PSE is an important problem in the research of PSE. In this paper, we propose a two-dimensional ordinary differential equation (ODE), which can represent a class of dynamical systems with multiple-frequency slow excitations, and contribute to the theoretical understanding of the generation of PSE. Based on the fast–slow analysis, we firstly transform the ODE with multiple-frequency slow excitations into the one with a single slow variable, and thus we can obtain the fast subsystem, based on which the sufficient conditions for the existence of PSE related to equilibria are given. Then, we analyze the sufficient conditions for PSE by considering the behavior of the equilibrium theoretically. Besides, the criteria of different PSE patterns related to equilibria are developed to investigate different manifestations of PSE. Finally, the validity of the theoretical analysis is showed by numerical results of several concrete examples satisfying the class of ODE given in this paper. Our results provide the sufficient conditions and criteria for PSE involved in the proposed ODE family, and facilitate one to generate any desired dynamical systems exhibiting PSE within the framework of the suggested equation form.

Stability indices of non-hyperbolic equilibria in two-dimensional systems of ODEs

We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.

Explicit analytical solution of a differential equation model for solar heating systems

In the issue of the environmental protection, the role of the widespread application of solar heating systems is very important. Well-usable mathematical models or their solutions can help to promote this aim. In the literature, analytical solutions for the models of solar heating systems are rather rare, despite of their advantages (fast and easy usability, low computational demand, low cost).In the present paper, we intend to (partly) make up for this shortage with respect to a two-dimensional ordinary differential equation model of complete collector-storage systems. The worked out new analytical system solution provides easy- and fast-to-use explicit formulas (for direct algebraic substitutions) to determine both the temperature of the collector and the temperature of the storage.The analytical solution is validated with the measured data of a real solar heating system. As a part of the validation of the (two-dimensional) system solution, the analytical solution of the (one-dimensional) collector sub-model (which is already known and successfully used in the literature) is also validated separately with the measured data of a real solar collector. Based on the validation containing a relatively long time period of 19 days, the error of the new system solution is 3.4% in case of the collector and 2.0% in case of the storage, which means an outstanding precision in the field.

Traveling wave solutions for degenerate nonlinear parabolic equations

We consider the traveling wave solutions of the degenerate nonlinear parabolic equation $$u_{t}=u^{p}(u_{xx}+u)$$ which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation $$v_{\tau }=v^{p}(v_{xx}+v-v^{-p+1})$$ related with it. We first give a result on the whole dynamics on the phase space $${\mathbb {R}}^{2}$$ with including infinity about two-dimensional ordinary differential equation that introduced the traveling wave coordinates: $$\xi = x-ct$$ by applying the Poincare compactification and dynamical system approach. Second, we focus on the connecting orbits on it and give a result on the existence of the weak traveling wave solutions with quenching for $$c>0$$ and $$p\in 2{\mathbb {N}}$$ . Moreover, we give the detailed information about the asymptotic behavior of $$u(\xi )$$ , $$u'(\xi )$$ , $$v(\xi )$$ and $$v'(\xi )$$ for $$p\in 2{\mathbb {N}}$$ . In the case that $$p \in 2 {\mathbb {N}}+1$$ , it is too complicated to determine the dynamics near the singularities on the Poincare disk, however, we classify the connecting orbits and corresponding traveling wave solutions and obtain their asymptotic behavior.

LIMIT CYCLE BIFURCATIONS IN THE IN-PLANE GALLOPING OF ICED TRANSMISSION LINE*

In this paper, we establish a mathematical model to describe in-plane galloping of iced transmission line with geometrical and aerodynamical nonlinearities using Hamilton principle. After Galerkin Discretization, we get a two-dimensional ordinary differential equations system, further, a near Hamiltonian system is obtained by rescaling. By calculating the coefficients of the first order Melnikov function or the Abelian integral of the near-Hamiltonian system, the number of limit cycles and their locations are obtained. We demonstrate that this model can have at least 3 limit cycles in some wind velocity. Moreover, some numerical simulations are conducted to verify the theoretical results.

Hopf bifurcation with additive noise

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

Numerical Model Construction with Closed Observables

Performing analysis, optimization and control using simulations of many-particle systems is computationally demanding when no macroscopic model for the dynamics of the variables of interest is available. In case observations on the macroscopic scale can only be produced via legacy simulator code or live experiments, finding a model for these macroscopic variables is challenging. In this paper, we employ time-lagged embedding theory to construct macroscopic numerical models from output data of a black box, such as a simulator or live experiments. Since the state space variables of the constructed, coarse model are dynamically closed and observable by an observation function, we call these variables closed observables. The approach is an online-offline procedure, as model construction from observation data is performed offline and the new model can then be used in an online phase, independent of the original. We illustrate the theoretical findings with numerical models constructed from time series of a two-dimensional ordinary differential equation system, and from the density evolution of a transport-diffusion system. Applicability is demonstrated in a real-world example, where passengers leave a train and the macroscopic model for the density flow onto the platform is constructed with our approach. If only the macroscopic variables are of interest, simulation runtimes with the numerical model are three orders of magnitude lower compared to simulations with the original fine scale model. We conclude with a brief discussion of possibilities of numerical model construction in systematic upscaling, network optimization and uncertainty quantification.

Analysis of an emission model with electromagnetic coils using electric shock

In this paper, we investigate an emission model with two electromagnetic coils using two-dimensional ordinary differential equations. By carrying out the bifurcation analysis of the model, it is shown there are different kinds of bifurcation in this model, including Hopf bifurcation, and Bogdanov–Takens bifurcation. The obtained results may provide some new insights in electric firing models.

Approximations for acoustically excited bubble cluster dynamics

In this paper the effect of interaction on the expansion of a bubble in a regular monodisperse cluster is investigated. By a geometric construction a two-dimensional ordinary differential equation with an exact expression for first-order bubble interactions is derived for an n-bubble model. An approximate equation is derived for the rapid expansion of the bubble which can be solved yielding an analytic expression for the collapse of a bubble which undergoes inertial cavitation. It is then demonstrated that the maximum volume of a bubble in a cluster is considerably less than that of a single bubble. This result is of significance as typically the dispersion relationship, the wave speed and the co-efficient of attenuation are calculated using a single bubble model and summed for the total number of bubbles to yield the void fraction. Furthermore it is shown that the maximum radius of a bubble in the cluster is considerably smaller than that of a single bubble, yet the duration of the collapse phase is only weakly dependent on the number of bubbles. Hence, it is conjectured that the likelihood of fragmentation due to Rayleigh–Taylor instability is reduced. The results from the analysis are in good agreement with full numerical simulations of multi-bubble dynamics, as well as experimental observations

Effects of Time Delay and Noise on Asymptotic Stability in Human Quiet Standing Model

A human quiet standing stability is discussed in this paper. The model under consideration is proposed to be a delayed differential equation (DDE) with multiplicative white noise perturbation. The method of the center manifold is generalized to reduce a delayed differential equation to a two-dimensional ordinary differential equation, to study delay-induced instability or Hopf bifurcation. Then, the stochastic average method is employed to obtain the Itô equation. Thus, the top Lyapunov exponent is calculated and the necessary and sufficient condition of the asymptotic stability in views of probability one is obtained. The results show that the exponent is related to not only the strength of noise but also the delay, namely, the reaction speed of brain. The effect of the strength of noise on the human quiet standing losing stability is weak for a small delay. With the delay increasing, such effect becomes stronger and stronger. A small change in the strength of noise may destabilize the quiet standing for a large delay. It implies that a person with slow reaction is easy to lose the stability of his/her quiet standing.

On embedded bifurcation structure in some discretized vector fields

In this paper, we study a dynamic structure of discretized vector fields obtained from the Brusselator, which is described by two-dimensional ordinary differential equations (ODEs). We found that a bifurcation structure of the logistic map is embedded in the discretized vector field. The embedded bifurcation structure was unraveled by the dynamical orbits that eventually converge to a fixed point. We provide a detailed mathematical analysis to explain this phenomenon and relate it to the solution of the original ODEs.

G-CSF Control of Neutrophils Dynamics in the Blood

White blood cell neutrophil is a key component in the fast initial immune response against bacterial and fungal infections. Granulocyte colony stimulating factor (G-CSF) which is naturally produced in the body, is known to control the neutrophils production in the bone marrow and the neutrophils delivery into the blood. In oncological practice, G-CSF injections are widely used to treat neutropenia (dangerously low levels of neutrophils in the blood) and to prevent the infectious complications that often follow chemotherapy. However, the accurate dynamics of G-CSF neutrophil interaction has not been fully determined and no general scheme exists for an optimal G-CSF application in neutropenia. Here we develop a two-dimensional ordinary differential equation model for the G-CSF-neutrophil dynamics in the blood. The model is built axiomatically by first formally defining from the biology the expected properties of the model, and then deducing the dynamic behavior of the resulting system. The resulting model is structurally stable, and its dynamical features are independent of the precise form of the various rate functions. Choosing a specific form for these functions, three complementary parameter estimation procedures for one clinical (training) data set are utilized. The fully parameterized model (6 parameters) provides adequate predictions for several additional clinical data sets on time scales of several days. We briefly discuss the utility of this relatively simple and robust model in several clinical conditions.

Effect of regenerative process on the sample stability of a multiple delay differential equation

In this paper, explicit analytical expressions for the stability behaviour of a single degree of freedom machining model have been derived. The regenerative chatter due to chip thickness and feed rate variations, a delay in the spring force and the probabilistic property of the displacement process of the chatter induced by the earlier tool cuts in the undeformed chip thickness have been taken into account. A characteristic equation for the linearized stability at equilibrium machining is presented, and regions of stable and unstable machining for multiple fixed time delays are captured in the phase plane of two model parameters. By a combined use of the classical Hopf bifurcation theorem and the centre manifold, equations governing stochastic chattering, which are infinite in character, are reduced to two-dimensional ordinary differential equations. The integral averaging method and the Lyapunov exponent have been employed to explicitly derive the required analytical expressions.

All possible chaotic dynamics can be approximated in three dimensions

It is known that neither one- nor two-dimensional ordinary differential equations can exhibit chaotic phenomena, while such behaviour is possible in three-dimensional systems. Here it is shown that the solutions of any dissipative equation can be approximated arbitrarily closely by the trajectories of an appropriate three-dimensional system. This class includes the Navier-Stokes equations, showing that the dynamical system which may describe fully developed turbulence can be approximated using just three degrees of freedom.