Appearance of chaotic dynamics as a result of multi-frequency tori destruction is carried out on the example of a model of a multimode generator. Quasiperiodic bifurcations occurring with multi-frequency tori are discussed in the context of the Landau-Hopf scenario. Structure of the parameter space is studied, areas with various chaotic dynamics, including chaos and hyperchaos, are revealed. Scenarios of the development of chaotic dynamics are described, the features of chaotic signals of various types are revealed.
A concept of algebraic invariants and algebraically invariant solutions for autonomous ordinary differential equations and systems of autonomous ordinary differential equations is considered. A variety of known exact solutions of autonomous ordinary differential equations and a great number of traveling wave solutions of famous partial differential equations are in fact algebraically invariant solutions. A method, which can be used to find all irreducible algebraic invariants, is introduced. As an example, all irreducible algebraic invariants for the traveling wave reductions of the dispersive Kuramoto–Sivashinsky equation and the modified Kuramoto–Sivashinsky equation are classified. Novel solutions of the modified Kuramoto–Sivashinsky equation are obtained.
The dynamics of a non-autonomous oscillator in which the phase and frequency of the external force depend on the dynamical variable is studied. Such a control of the phase and frequency of the external force leads to the appearance of complex chaotic dynamics in the behavior of oscillator. A hierarchy of various periodic and chaotic oscillations is observed. The structure of the space of control parameters is studied. It is shown there are oscillatory modes similar to those of a non-autonomous oscillator with a potential in the form of a periodic function in the system dynamics, but there are also significant differences. Physical experiments of such systems are implemented.
Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., are challenging because of the complexity of these processes. Important related research questions include, first, how to determine the qualitative properties of the dynamics of these processes, namely, whether the process is stable, unstable, chaotic (deterministic), or stochastic; and second, how best to estimate its quantitative indicators including dimension, entropy, and correlation characteristics.
These questions can be studied both empirically and theoretically. In the empirical approach, researchers consider real data expressed in terms of time series, identify the patterns of their dynamics, and then forecast the short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamical models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models.
To implement these approaches, either numerical or analytical methods can be used. While numerical methods make it possible to study dynamical models, the possibility of obtaining reliable results using them is significantly limited due to the necessity of performing calculations only over finite time intervals, rounding-off errors in numerical methods, and the unbounded space of initial data sets. Analytical methods allow researchers to overcome these limitations and to identify the exact qualitative and quantitative characteristics of the dynamics of the process. However, effective analytical applications are often limited to low-dimensional models (in the literature, two-dimensional dynamical systems are most often studied).
In this paper, we develop analytical methods for the study of deterministic dynamical systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (to localize self-excited and hidden attractors and identify multistability), but also to overcome difficulties related to implementing reliable numerical analysis of quantitative indicators such as Lyapunov exponents and the Lyapunov dimension. We demonstrate the effectiveness of the proposed methods using the mid-size firm model suggested by Shapovalov.
In this work, we report the existence of extreme events in the well known Bonhoeffer-van der Pol (BVP) oscillator under the excitation of a periodically forced voltage. Extreme events refer to the sudden and random increase in the amplitude of one or more of the state variables of the dynamical system and arise because of the incidence of interior crises or the presence of discontinuous boundaries or intermittency. We have chosen this system because of the fact that it has spawned several systems modelling neuronal dynamics such as Hindmarsh-Rose (HR) and Hodgkin-Huxley (HH) models. Our investigations involve both laboratory experiments and numerical simulations. We have obtained time plots, phase portraits, Poincaré maps, bifurcation graphs, Lyapunov exponents and signal to noise ratio (SNR) to study the general dynamics and to confirm the presence of extreme events, we have used statistical measures such as phase slip analysis, distribution functions for both experimental and numerical data. To the best of our knowledge, we believe that it is for the first time that the occurrence of extreme event has been reported using both real time experimental and numerical studies on this forced BVP system.
The generalized logistic equation is used to interpret the COVID-19 epidemic data in several countries: Austria, Switzerland, the Netherlands, Italy, Turkey and South Korea. The model coefficients are calcu- lated: the growth rate and the expected number of infected people, as well as the exponent indexes in the generalized logistic equation. It is shown that the dependence of the number of the infected people on time is well described on average by the logistic curve (within the framework of a simple or general- ized logistic equation) with a determination coefficient exceeding 0.8. At the same time, the dependence of the number of the infected people per day on time has a very uneven character and can be described very roughly by the logistic curve. To describe it, it is necessary to take into account the dependence of the model coefficients on time or on the total number of cases. Variations, for example, of the growth rate can reach 60%. The variability spectra of the coefficients have characteristic peaks at periods of sev- eral days, which corresponds to the observed serial intervals. The use of the stochastic logistic equation is proposed to estimate the number of probable peaks in the coronavirus incidence.
We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial equations. Moreover, we demonstrate that an autonomous first integral for one of two non-locally related equations can be constructed in the parametric form from the general solution of the other equation. In order to illustrate our results, we construct two integrable subfamilies of the considered family of equations, that are non-locally equivalent to two equations from the Painlevé–Gambier classification. We also discuss several particular members of these subfamilies, including a traveling wave reduction of a nonlinear diffusion equation, and construct their invariant curves and first integrals.
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.
We propose a robust universal approach to identify multiple network dynamical states, including stationary and travelling chimera states based on an adaptive coherence measure. Our approach allows automatic disambiguation of synchronized clusters, travelling waves, chimera states, and asynchronous regimes. In addition, our method can determine the number of clusters in the case of cluster synchronization. We further couple our approach with a new speed calculation method for travelling chimeras. We validate our approach by an example of a ring network of type II Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections where we identify a rich repertoire of coherent and wave states. We propose that the method is robust for the networks of phase oscillators and extends to a general class of relaxation oscillator networks.
The dynamics of two-component solitons with a small spatial displacement of the high-frequency (HF) component relative to the low-frequency (LF) one is investigated in the framework of the Zakharov-type system. In this system, the evolution of the HF field is governed by a linear Schrödinger equation with the potential generated by the LF field, while the LF field is governed by a Korteweg-de Vries (KdV) equa- tion with an arbitrary dispersion-nonlinearity ratio and a quadratic term accounting for the HF feedback on the LF field. The oscillation frequency of the soliton’s HF component relative to the LF one is found analytically. It is shown that the solitons are stable against small perturbations. The analytical results are confirmed by numerical simulations.
Localized stationary solutions of nonlinear nonlocal Whitham equation with resonance dispersion relation are considered. The existence of exponentially localized smooth and singular solitons, bound states of the solitons and localized solutions with oscillating asymptotics is recognized. The velocity spectra of solitons, in contrast to all other known Whitham equations, appear to be discrete. Asymptotic “quantization rules” for calculation of discrete spectra are obtained.
We consider reaction–diffusion equation in perforated domain, with rapidly oscillating coefficient in boundary conditions. We do not assume any Lipschitz condition for the nonlinear function in the equa- tion, so, the uniqueness theorem for the corresponding initial boundary value problem may not hold for the considered reaction-diffusion equation. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion equation with a “strange term”(potential).
Cyclicality and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global attractor and study limiting dynamics of the model. We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically. With the help of delayed feedback control (DFC), the possibility of transition from irregular limiting dynamics to regular periodic dynamics is shown to solve the problem of reliable forecasting. At the same time, we demonstrate the complexity and ambiguity of applying numerical procedures to calculate the Lyapunov dimension along different trajectories of the global attractor, including unstable periodic orbits (UPOs).
We present an analytical description of the class of unsteady vortex surface waves generated by non- uniformly distributed, time-harmonic pressure. The fluid motion is described by an exact solution of the equations of hydrodynamics generalizing the Gerstner solution. The trajectories of the fluid particles are circumferences. The particles on a free surface rotate around circumferences of the same radii, with the centers of the circumferences lying on different horizons. A family of waves has been found in which a variable pressure acts on a limited section of the free surface. The law of external pressure distribution includes an arbitrary function. An example of the evolution of a non-uniform wave packet is considered. The wave and pressure profiles, as well as vorticity distribution are studied. It is shown that, in the case of a uniform traveling wave of external pressure, the Gerstner solution is valid but with a different form of the dispersion relation. A possibility of observing the studied waves in laboratory and in the real ocean is discussed.