Simple Dynamical System Research Articles

Evolution of Duffing's equation solution obtained with approximated impulse process excitation

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Mode-locking dynamics of hair cells of the inner ear

We explore mode locking of spontaneous oscillations of saccular hair cell bundles to periodic mechanical deflections. A simple dynamic systems framework is presented that captures the main features of the experimentally observed behavior in the form of an Arnold tongue. We propose that the phase-locking transition can proceed via different bifurcations. At low stimulus amplitudes F, the transition to mode locking as a function of the stimulus frequency ω has the character of a saddle-node bifurcation on an invariant circle. At higher stimulus amplitudes, the mode-locking transition has the character of a supercritical Andronov-Hopf bifurcation.

Impacts of multi-scale solar activity on climate. Part II: Dominant timescales in decadal-centennial climate variability

Part II of this study detects the dominant decadal-centennial timescales in four SST indices up to the 2010/2011 winter and tries to relate them to the observed 11-yr and 88-yr solar activity with the sunspot number up to Solar Cycle 24. To explore plausible solar origins of the observed decadal-centennial timescales in the SSTs and climate variability in general, we design a simple one-dimensional dynamical system forced by an annual cycle modulated by a small-amplitude single- or multi-scale “solar activity.” Results suggest that nonlinear harmonic and subharmonic resonance of the system to the forcing and period-doubling bifurcations are responsible for the dominant timescales in the system, including the 60-yr timescale that dominates the Atlantic Multidecadal Oscillation. The dominant timescales in the forced system depend on the system’s parameter setting. Scale enhancement among the dominant response timescales may result in dramatic amplifications over a few decades and extreme values of the time series on various timescales. Three possible energy sources for such amplifications and extremes are proposed. Dynamical model results suggest that solar activity may play an important yet not well recognized role in the observed decadal-centennial climate variability. The atmospheric dynamical amplifying mechanism shown in Part I and the nonlinear resonant and bifurcation mechanisms shown in Part II help us to understand the solar source of the multi-scale climate change in the 20th century and the fact that different solar influenced dominant timescales for recurrent climate extremes for a given region or a parameter setting. Part II also indicates that solar influences on climate cannot be linearly compared with non-cyclic or sporadic thermal forcings because they cannot exert their influences on climate in the same way as the sun does.

Elementary bifurcations for a simple dynamical system under non-Gaussian Lévy noises

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies.A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian α-stable Lévy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by solving a nonlocal Fokker-Planck equation numerically. This allows numerically investigating phenomenological bifurcation or P-bifurcation, for stochastic differential equations with non-Gaussian Lévy noises.

A Physics-Based Friction Model and Integration to a Simple Dynamical System

Dynamical modeling and simulation of mechanical structures containing jointed interfaces require reduced-order fretting models for efficiency. The reduced-order models in the literature compromise both accuracy and the physical basis of the modeling procedure, especially with regard to interface contact and friction modeling. Recently, physics-based fretting models for nominally flat-on-flat contacts, including roughness effects, have been developed and validated on individual (isolated) mechanical lap joints (Eriten et al., 2011, “Physics-Based Modeling for Fretting Behavior of Nominally Flat Rough Surfaces,” Int. J. Solids Struct., 48(10), pp. 1436-1450). These models follow a “bottom up” modeling approach; utilizing the micromechanics of sphere-on-flat fretting contact (asperity scale), and statistical summation to model flat-on-flat contacts at the macroscale. Since these models are physical, the effects of surface roughness, contact conditions, and material properties on fretting and dynamical response of the jointed interfaces can be studied. The present work illustrates an example of how the physics-based models can be incorporated into studies of the dynamics of jointed structures. A comparison with friction models existing in the literature is also provided.

Digital Implementation of a Pseudo-Random Noise Generator using Chaotic Maps

Random Number Generators (RNG) are an important component in modern digital secure communication systems and other applications. Chaotic systems have proven to be a good source of random signals for specific values of its control parameter. Iterated one-dimensional maps are simple dynamical systems suitable for chaos generation. The random or pseudorandom iteration of four variants of the scaled and discretized Bernoulli map is used to design a pseudo-random number generator (PRNG), which can produce binary sequences with a known statistical distribution. The values of the control parameter for which the system of multiple maps produces digital noise are determined by mathematical tools such as bifurcation diagrams, entropy function and the Lyapunov exponents. The proposed chaotic system is implemented in a compact digital FPGA architecture suitable for digital applications where PRNG's were needed.

Non-classical large deviations for a noisy system with non-isolated attractors

We study the large deviations of a simple noise-perturbed dynamical system havingcontinuous sets of steady states, which mimic those found in some partial differentialequations related, for example, to turbulence problems. The system is a two-dimensionalnonlinear Langevin equation involving a dissipative, non-potential force, which has theessential effect of creating a line of stable fixed points (attracting line) touching a line ofunstable fixed points (repelling line). Using different analytical and numericaltechniques, we show that the stationary distribution of this system satisfies, inthe low-noise limit, a large deviation principle containing two competing terms:(i) a ‘classical’ but sub-dominant large deviation term, which can be derivedfrom the Freidlin–Wentzell theory of large deviations by studying the fluctuationpaths or instantons of the system near the attracting line, and (ii) a dominantlarge deviation term, which does not follow from the Freidlin–Wentzell theory, asit is related to fluctuation paths of zero action, referred to as sub-instantons,emanating from the repelling line. We discuss the nature of these sub-instantons, andshow how they arise from the connection between the attracting and repellinglines. We also discuss in a more general way how we expect these to arise in moregeneral stochastic systems having connected sets of stable and unstable fixedpoints, and how they should determine the large deviation properties of thesesystems.

Oscillators and relaxation phenomena in Pleistocene climate theory

Ice sheets appeared in the northern hemisphere around 3 Ma (million years) ago and glacial–interglacial cycles have paced Earth's climate since then. Superimposed on these long glacial cycles comes an intricate pattern of millennial and sub-millennial variability, including Dansgaard–Oeschger and Heinrich events. There are numerous theories about these oscillations. Here, we review a number of them in order to draw a parallel between climatic concepts and dynamical system concepts, including, in particular, the relaxation oscillator, excitability, slow–fast dynamics and homoclinic orbits. Namely, almost all theories of ice ages reviewed here feature a phenomenon of synchronization between internal climate dynamics and astronomical forcing. However, these theories differ in their bifurcation structure and this has an effect on the way the ice age phenomenon could grow 3 Ma ago. All theories on rapid events reviewed here rely on the concept of a limit cycle excited by changes in the surface freshwater balance of the ocean. The article also reviews basic effects of stochastic fluctuations on these models, including the phenomenon of phase dispersion, shortening of the limit cycle and stochastic resonance. It concludes with a more personal statement about the potential for inference with simple stochastic dynamical systems in palaeoclimate science.

Direct interaction with no correlation: An experimental pitfall in neural systems

Neural systems continuously optimize how organisms process their environment and are highly dynamic. Building predictive models of these systems is challenging due to the large number of their elements. Therefore, in experimental and descriptive neurobiology the researcher typically does not seek to catalogue all variables that affect one another, but rather tries to isolate variables that interact directly. Because of methodological limitations, observed variables are often measured near equilibrium. The presented analysis demonstrates that statistical tests performed on such equilibrium values are fundamentally incapable of detecting direct interactions in a large subset of simple dynamical systems. Some of these problems can be avoided by using explicit statistical models that include time as a variable.

Damage detection from the variation of parameter matrices estimated by incomplete FRF data

It is not easy to experimentally obtain the FRF (Frequency Response Function) matrix corresponding to a full set of DOFs (degrees of freedom) for a dynamic system. Utilizing FRF data measured at specific positions, with DOFs less than that of the system, as constraints to describe a damaged system, this study identifies parameter matrices such as mass, stiffness and damping matrices of the system, and provides a damage identification method from their variations. The proposed parameter identification method is compared to Lee and Kim's method and Fritzen's method. The validity of the proposed damage identification method is illustrated in a simple dynamic system. An accurate dynamic finite element model of a structure is very important for structural design and analysis. The modes of vibration and frequency response of the finite element model are compared to experimental measurements to test its accuracy. If a discrepancy between the two is found, the analytical model should be modified to satisfy the experimental measurements, and then the model should be updated for subsequent simulation and design studies. An FRF matrix of the full set of DOFs measured by experiments is used to predict parameter matrices of the dynamic system. The FRFs have been used directly to update condensed analytical models for obtaining the proper modal model. A single FRF measured at several frequencies, along with a correlated analytical model of the structure in its original state, is used for updating structural parameters. There have been many attempts to update the unknown physical parameters directly from the FRFs (Mottershead and Stanway 1986). It has been reported (Lee and Shin 2002) that FRF data provide more information than modal data, as the latter are extracted from a very limited frequency range related to resonance. Friswell and Penny (1990) proposed an approach to reduce the model order so that the stated estimation process reduces to a least squares problem based on the FRFs. Fanning and Carden (2004) presented a method for detecting added mass in structural systems from a single FRF measured at several frequencies in an identification algorithm. Cha and

Chaos Models And The Monetary Dynamics Of Hyperinflation

Modern methods of qualitative analysis of dynamic systems go back nearly a century to Poincare (1880, 1892). Since the classic work of Smale (1967), it has become clear that very complicated, or chaotic, trajectories (time path) can easily arise in certain dynamic systems and that such complicates trajectories can persist when small perturbations of the underlying systems occur. Such a phenomenon, referred to as chaos, a case that is emphatically not pathological, is essentially one in which a dynamic mechanism that is very simple, and, above all, deterministic yields a time path so complicated that it will invariably pass all the standard tests of randomness. The seminal contribution of Lorenz (1963), Li and Yorke (1975), May (1976), Stefan (1977), amongst others have greatly facilitated an exploration of the pertinence of such complicated dynamics, arising in simple first order dynamic non-linear systems, to a variety of fields, including physics, biology, ecology and of late economics. In the context of the above literature and the development thereafter we intend to build the model that comprises of four equations. These specific: (1) The demand for real balances, (2) The money- inflation link, (3) the government budget deficit, and (4) the inflation tax revenue. The reduced form of the model is seen to yield a three parameter system whose phase diagram for the inflation rate (expressed in terms of a transcendental equation) produces solutions which are capable of generating stable, cyclic, or chaotic behavior.

BCP - A Benchmark for Teaching Structural Dynamical Systems

AbstractNew object-oriented modelling techniques for physical systems along with more complex systems of interest lead to increasing requirements on modern simulators regarding performance and stability as well as state event modelling and event handling in combination with hybrid and structural dynamic systems. To see how modelling and simulation environments deal with such systems and to compare new approaches to hybrid modelling, a new benchmark is defined using three case studies with modelling and simulation tasks. The first case study investigates the classic bouncing ball example with some extensions. The second example discusses an electrical circuit - it can be used to compare the simulation results of different implementation techniques (e.g. physical modelling vs. classical DAE form) in one simulator. The third case study is a definition of a pendulum on a rope with free falling phase, representing another simple structural dynamic mechanical system. These three parts offer a spectrum of questions for testing basic features for handling hybrid and structural dynamic physical systems and they represent minimum requirements on hybrid simulators regarding state events.

A model for grand minima and geomagnetic reversals

AbstractWe suggest a simple dynamical system which mimics a nonlinear dynamo which is able to provide (in specific domains of its parametric space) the temporal evolution of solar magnetic activity cycles as well as evolution of geomagnetic field including its polarity reversals. A qualitative explanation for the physical nature of both phenomena is presented and discussed.

Stability in a system subject to noise with regulated periodicity

The stability of a simple dynamical system subject to multiplicative one-side pulse noise with hidden periodicity is investigated both analytically and numerically. The stability analysis is based on the exact result for the characteristic functional of the renewal pulse process. The influence of the memory effects on the stability condition is analyzed for two cases: (i) the dead-time-distorted poissonian process, and (ii) the renewal process with Pareto distribution. We show that, for fixed noise intensity, the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity.

Input–output properties of the Page–Hinkley detector

We consider the stochastic input–output properties of a simple non-linear dynamical system, the so-called Page–Hinkley detector, playing a key role in change detection, and also in queuing theory. We show that for L -mixing inputs with negative expectation the output process of this system is L -mixing. The result is applied to get an upper bound for the false alarm rate. The proof is then adapted to get a similar result for the case of random i.i.d. inputs. Possible extensions and open problems are given in the discussion.

Assessing Manual Pursuit Tracking in Parkinson’s Disease Via Linear Dynamical Systems

Quantitative assessment of motor performance is important for diseases of motor control, such as Parkinson's disease (PD). Manual tracking tasks are well suited for motor assessment, as they can be performed concomitantly with brain mapping techniques. Here we propose utilizing second-order linear dynamical systems to assess manual pursuit tracking performance. With the desired trajectory as the input, and the subject's actual motor response as the output, a linear model characterized by natural frequency and damping ratio is identified for each subject. We applied this method to 10 PD subjects (on and off L: -dopa medication) and 10 normal subjects performing a multi-frequency sinusoidal tracking task. Model parameters were more sensitive than overall tracking errors in determining significant differences between groups. The effect of L: -dopa medication was to reduce the damping ratio and make the range in natural frequency across individuals approach that of normal subjects. We interpret the changes in damping ratio and natural frequency as possibly related to suppression of compensatory cerebellar activity and/or improvement of motor program selection, and the changes in natural frequency as an implicit strategy to maintain settling time in the face of reduce damping ratio. Our results suggest that simple linear dynamical system models can be a powerful method to assess tracking performance in Parkinson's disease because of the additional insight they can provide into neurological processes.

Signal Amplification by Sensitive Control of Bifurcation Topology

We describe a novel amplification scheme based on inducing dynamical changes to the topology of a bifurcation diagram of a simple nonlinear dynamical system. We have implemented a first bifurcation-topology amplifier using a coupled pair of parametrically driven high-frequency nanoelectromechanical systems resonators, demonstrating robust small-signal amplification. The principles that underlie bifurcation-topology amplification are simple and generic, suggesting its applicability to a wide variety of physical, chemical, and biological systems.

Application of impulse damper in control of a chaotic friction-induced vibration

Friction-induced vibration is an important phenomenon with adverse effects on many dynamic systems involving friction. In this study, a very simple and well-known one-dimensional friction-induced dynamic system is considered in which the novel PZT stack impulse damper is incorporated into the system. It has been shown that by appropriately tuning the damping parameters, the chaotic behavior is removed quickly and efficiently. It has also been demonstrated that the system is sensitive to parameter change, and minimal modification of these parameters can revert the chaotic or periodic motion.

Time‐dependent solution of molecular quantum systems using multiwavelet

AbstractWe solve the time‐dependent Schrödinger equation (TDSE), time‐dependent Hartree‐Fock equations (TDHF), and time‐dependent Density Functional Theory DFT(TDDFT) [Runge and Gross, Phys Rev Lett 1984, 52, 997] equations of general electronic/muonic systems using a multiresolution multiwavelet (MRMW) basis set. The stable solution of these timedependent equations with relatively large time step is obtained using the Cayley formalism with the MRMW basis set. The Cayley operator corresponding to each axis is applied to the direct‐product basis set or a subset of the direct‐product basis set to obtain high efficiency for multidimensional cases. The method is tested by applications to the simulation of electron and muon dynamics of simple molecular systems in the framework of Ehrenfest dynamics employing classical point charge nucleus approximation. Stable solutions of the equations for the quantum and classical particles are obtained. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Advising a Bus Company on Number of Needed Buses: How High-School Physics Students' Deal With a "Complex Problem"?.

Since 2003, international project PISA evaluates 15-year old students in solving problems that include “decision taking”, “analysis and design of systems” and “trouble-shooting”. This article presents the results of a pilot research conducted with 215 students from first to fourth grade of a high school in Sarajevo (Bosnia and Herzegovina). The students, in an imaginary, real-life like scenario, had to advice a bus company about number of buses needed for operating a 24-hour service between two cities. Research instrument was structured in a way, which permits exploring how students deal with a “complex problem” (in PISA terminology it would be problem type “analysis and design of systems”) in real-life setting. Negative research results show that students mainly (1) use either verbal or visual reasoning mode; (2) have poor “sense-making” approach in analyzing a simple dynamical system and (3) have underdeveloped an important competency for working in knowledge-based economy (decision-taking based on evidence and arguments). A positive result is that majority of students found interesting the problem in question. Some brief recommendations about the presence and role of “complex problem” in physics teaching are offered in conclusions.