Conditional Lyapunov Exponents Research Articles

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

Domestic and Global Causes for Exchange Rate Volatility: Evidence From Turkey

Understanding which factors influence exchange rate movements is important for understanding economic development, trade patterns, investment decisions, and for designing economic policies. In this study, we allow for monetary, domestic and global financial variables to assess the relevant importance of each of the variables to exchange rate volatility in the case of Turkey. The paper investigates the dynamics of exchange rate volatility of the Turkish lira in a complementary perspective by employing both Generalized autoregressive conditional heteroskedasticity (GARCH) method and Lyapunov exponents over the period from March 1, 2019 to November 11, 2021. Firstly, decomposing the impact of domestic and global financial variables, the results of the GARCH model indicate that the exchange rate volatility is affected by Volatility index (VIX) and Credit default swaps (CDS). This result suggests that the exchange rate shocks experienced are mainly caused by global factors, therefore policymakers should focus on volatility spillovers caused by global financial markets. Secondly, detected positive maximal Lyapunov exponent shows that complexity in foreign exchange markets has been increased, market expectations lead to multiple-equilibria and diverging volatility eventually will generate recurrent spikes in currency value. These complementary findings have important implications for interventions of Central banks and preventing systemic risks, as well as portfolio and risk management. JEL Classification: C32, G14, G15.

The conditional Lyapunov exponents and synchronisation of rotating turbulent flows

The synchronisation between rotating turbulent flows in periodic boxes is investigated numerically. The flows are coupled via a master–slave coupling, taking the Fourier modes with wavenumber below a given value $k_m$ as the master modes. It is found that synchronisation happens when $k_m$ exceeds a threshold value $k_c$ , and $k_c$ depends strongly on the forcing scheme. In rotating Kolmogorov flows, $k_c\eta$ does not change with rotation in the range of rotation rates considered, $\eta$ being the Kolmogorov length scale. Even though the energy spectrum has a steeper slope, the value of $k_c\eta$ is the same as that found in isotropic turbulence. In flows driven by a forcing term maintaining constant energy injection rate, synchronisation becomes easier when rotation is stronger. Here, $k_c\eta$ decreases with rotation, and it is reduced significantly for strong rotations when the slope of the energy spectrum approaches $-3$ . It is shown that the conditional Lyapunov exponent for a given $k_m$ is reduced by rotation in the flows driven by the second type of forcing, but it increases mildly with rotation for the Kolmogorov flows. The local conditional Lyapunov exponents fluctuate more strongly as rotation is increased, although synchronisation occurs as long as the average conditional Lyapunov exponents are negative. We also look for the relationship between $k_c$ and the energy spectra of the Lyapunov vectors. We find that the spectra always seem to peak at approximately $k_c$ , and synchronisation fails when the energy spectra of the conditional Lyapunov vectors have a local maximum in the slaved modes.

Generalized Function Projective Synchronization of Two Different Chaotic Systems with Uncertain Parameters

This study proposes a new approach to realize generalized function projective synchronization (GFPS) between two different chaotic systems with uncertain parameters. The GFPS condition is derived by converting the differential equations describing the synchronization error systems into a series of Volterra integral equations on the basis of the Laplace transform method and convolution theorem, which are solved by applying the successive approximation method in the theory of integral equations. Compared with the results obtained by constructing Lyapunov functions or calculating the conditional Lyapunov exponents, the uncertain parameters and the scaling function factors considered in this paper have fewer restrictions, and the parameter update laws designed for the estimation of the uncertain parameters are simpler and easier to realize physically.

A theoretical framework for nonlinear dynamics in finance systems: Implications for stability, control and synchronization

It has been widely observed that most deterministic dynamical systems go into chaos for some values of their parameters. One of the most popular and widely used criteria is the conditional Lyapunov exponents, which constitute average measurements of expansion or shrinkage of small displacements along the synchronized trajectory. The Lyapunov characteristic exponents play a crucial role in the description of the behaviour of dynamical systems as they can be used to analyse the stability limit sets and to check sensitive dependence on initial conditions, that is, the presence of chaotic attractors. In this paper, Lyapunov stability theory and linear matrix inequalities (LMI) are employed to design control functions for the respective, control, and synchronization of the chaotic and hyperchaotic finance systems. The designed linear matrix inequalities (LMI) nonlinear controllers are capable of stabilizing the chaotic and hyperchaotic finance systems at any position as well as controlling it to track any trajectory that is a smooth function of time. The respective chaotic attractors were found to have a moderate value of the largest Lyapunov exponents (0.874959 s^(-1) and 0.650847 s^(-1)) with associated (Lyapunov) dimensions of 1.00 and 2.00 for the chaotic and hyperchaotic finance systems respectively. Based on Lyapunov stability theory and linear matrix inequalities (LMI), some necessary and sufficient criteria for stable synchronous behaviour are obtained and an exact analytic estimate of the threshold coupling, k_th, for complete chaos synchronization is derived. Finally, numerical simulation results are presented to validate the feasibility of the theoretical analysis.

A novel criterion for unpredictable motions

We demonstrate the extension of unpredictable motions in coupled autonomous systems with skew product structure in the case that generalized synchronization takes place. Sufficient conditions for the existence of unpredictable motions in the dynamics of the response system are provided. The theoretical results are exemplified for coupled autonomous systems in which the drive is a hybrid dynamical system and the response is a Lorenz system. The auxiliary system approach and conditional Lyapunov exponents are utilized to detect the presence of generalized synchronization.

Chaotic synchronization of mutually coupled systems-arbitrary proportional linear relations.

Considering a system combining two generalized Boolean transformations, we found that depending on the parameters, we can generate generalized synchronization such that the two chaotic orbits have arbitrary proportional linear relations. We rigorously determined its synchronization conditions by the explicit computing conditional Lyapunov exponent using the ergodic property and stable property of the Cauchy distribution. We found that a phenomenon similar to chaotic synchronization occurs even when the synchronization conditions are not strictly satisfied, which exhibits some degree of structural stability of chaotic synchronization. Our model can be further extended to systems with more degrees of freedom and, in the future, can be applied to reservoir computing.

Attenuation of Line Spectra and Vibration Based on a Chaotification Method and Global Analysis

The chaotification method can be used to weaken the line spectra component of the ship's radiated noise. The main obstacle that plagues the engineering realization of this innovative concept is how to realize the small-amplitude chaotic motion of the nonlinear vibration isolation system (NVIS) in a large parameter range and how to maintain the chaotic state under variable operating conditions. In this paper, the dimensionless dynamic equation of the NVIS with a flexible base is derived. A control method similar to generalized synchronization is used to realize the continuous chaos of the system. The maximum Lyapunov exponent, the conditional Lyapunov exponent, and disturbances are introduced to verify the effectiveness of the control method and its stability and robustness in a large range of parameters. Aiming at the coexistence of multiple attractors, a new global analysis method is proposed, which is used to study the distribution of basin of attraction (BA). According to the changes of BA before and after the system is controlled, the BA is partitioned. Initial conditions from different partitions are selected to study the attenuation effect of line spectra and vibration, and the favorable initial conditions for realizing the small-amplitude chaotic motion of the system are given. The method may be extended to the global analysis of other controlled systems.

Information theory, synchronization and topological order in complete dynamical networks of discontinuous maps

This paper is dedicated to the study of information measures, synchronization and a topological order in complete dynamical networks of discontinuous piecewise linear maps with different slopes. It stands out that the networks topologies are characterized by circulant matrices and the conditional Lyapunov exponents are explicitly determined. Some properties of the mutual information rate and the Kolmogorov–Sinai entropy, depending on the synchronization interval, are discussed. A topological order between the complete dynamical networks is presented, which is characterized by the monotony of the network topological entropy. It is proved that if the network topological entropy increases, then the mutual information rate and the Kolmogorov–Sinai entropy increase or decrease, according to the variation of the coupling parameter. Furthermore, various types of computer simulations show the experimental applications of these results and techniques.

Deep time-delay reservoir computing: Dynamics and memory capacity.

The deep time-delay reservoir computing concept utilizes unidirectionally connected systems with time-delays for supervised learning. We present how the dynamical properties of a deep Ikeda-based reservoir are related to its memory capacity (MC) and how that can be used for optimization. In particular, we analyze bifurcations of the corresponding autonomous system and compute conditional Lyapunov exponents, which measure generalized synchronization between the input and the layer dynamics. We show how the MC is related to the systems' distance to bifurcations or magnitude of the conditional Lyapunov exponent. The interplay of different dynamical regimes leads to an adjustable distribution between the linear and nonlinear MC. Furthermore, numerical simulations show resonances between the clock cycle and delays of the layers in all degrees of MC. Contrary to MC losses in single-layer reservoirs, these resonances can boost separate degrees of MC and can be used, e.g., to design a system with maximum linear MC. Accordingly, we present two configurations that empower either high nonlinear MC or long time linear MC.

Role of unstable periodic orbits in bubbling weak generalized synchronization

In this paper, we analyzed generalized synchronization (GS) from the viewpoint of unstable periodic orbits (UPOs) and Lyapunov vectors (LVs) in Lorenz system driven by a Rössler system. First, we found a new bifurcation route via crisis, a rapid increase in the support of the attractor, from strong GS to weak GS that possesses the power-law parameter sensitivity. We then showed that UPOs with positive conditional Lyapunov exponents (CLEs) are embedded within the attractor after the crisis. We also found the bubbling phenomenon in the response system, where a trajectory intermittently moves away from the synchronization manifold. Furthermore, as a result of the LV analysis, we found that non-hyperbolicity near UPOs with positive CLEs becomes strong in the weak GS regime.

Driven-dissipative dynamics of atomic ensembles in a resonant cavity: Quasiperiodic route to chaos and chaotic synchronization

We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity — presence of two or more incommensurate frequencies. This is known as the Ruelle–Takens–Newhouse route to chaos. The equations of motion have a Z2-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equations spontaneously break this symmetry. To understand the emergence and subsequent properties of various attractors, we concurrently study the mean-field trajectories, Poincaré sections, maximum and conditional Lyapunov exponents, and power spectra. Using Floquet analysis, we show that quasiperiodicity is born out of non-Z2-symmetric oscillations via a supercritical Neimark–Sacker bifurcation. Changing the detuning between the level spacings in the two ensembles and the repump rate results in the synchronization of the two chaotic ensembles. In this regime, the chaotic intensity fluctuations of the light radiated by the two ensembles are identical. Identifying the synchronization manifold, we understand the origin of synchronized chaos as a tangent bifurcation intermittency of the Z2-symmetric oscillations. At its birth, synchronized chaos is unstable. The interaction of this attractor with other attractors causes on–off intermittency until the synchronization manifold becomes sufficiently attractive. We also show coexistence of different phases in small pockets near the boundaries.

Synchronized chaotic swinging of parametrically driven pendulums

We study the problem of complete synchronization of a pendulums' array attached to a common, externally forced structure. The presented analysis is focused on chaotic synchronization of the swinging, parametric pendulums. We propose a simplified criterion of synchronization for an arbitrary number of parametrically driven oscillators, which is based on the idea of a so-called autonomous driver decomposition and conditional Lyapunov exponents. According to this proposal, complete synchronization of any number of parametric oscillators driven via a common transmitter of an external signal (the structure) can be analyzed using a three degrees of freedom system in a master-slave configuration. The reduced system consists of an equivalent synchronous oscillator coupled with its virtual replica via the linking structure. It is shown that introduction of the intermediate structure allows chaotic synchronization of parametric pendulums, even in the presence of parameter's mismatch. The idea of an intermediate transmitter and the methodology of verification of stability of the synchronous state can be applied for any set of dynamical systems with a common external drive.

The reservoir's perspective on generalized synchronization.

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across a range of dynamical relationships including generalized synchronization. For a drive-response setup, a temporal representation of the synchronized state is discussed as an alternative to the known instantaneous form. The reservoir has access to both representations through its fading memory property, each with advantages in different dynamical regimes. We also extract signatures of the maximal conditional Lyapunov exponent in the performance of variations of the reservoir topology. Moreover, the reservoir model reproduces different levels of consistency where there is no synchronization. In a bidirectional coupling setup, high reconstruction accuracy is achieved despite poor observability and independent of generalized synchronization.

Effects of periodically modulated coupling on amplitude death in nonidentical oscillators

The effects of periodically modulated coupling on amplitude death (AD) in two coupled nonidentical oscillators are explored. The AD domain is significantly influenced by tuning the modulation amplitude and frequency of the coupling. There is an optimal value of the modulation amplitude of the coupling with which the largest AD domain is observed in the parameter spaces. The AD domain is enlarged (shrunk) with the decrease of the modulation frequency for a given small (large) modulation amplitude. The mechanism of AD in the presence of periodic modulation in the coupling is investigated based on the local conditional Lyapunov exponent. The stability of the AD state can be availably characterized by the conditional Lyapunov exponents of the coupled system. The transition process from oscillating to the AD state is clearly verified by the fact that the conditional Lyapunov exponent transits from positive to negative. Our results are helpful to many potential applications in neuroscience and dynamical control in engineering.

Analysis of chimera states as drive-response systems

Chimera states are spatiotemporal segregations – stably coexisting coherent and incoherent groups – that can occur in systems of identical phase oscillators. Here we demonstrate that this remarkable phenomenon can also be understood in terms of Pecora and Carroll’s drive-response theory. By calculating the conditional Lyapunov exponents, we show that the incoherent group acts to synchronize the coherent group; the latter playing the role of a response. We also compare the distributions of finite-time conditional Lyapunov exponents to the characteristic distribution that was reported previously for chimera states. The present analysis provides a unifying explanation of the inherently frustrated dynamics that gives rise to chimera states.

Conditional Lyapunov exponent criteria in terms of ergodic theory

The conditional Lyapunov exponent is defined for investigating chaotic synchronization, in particular complete synchronization and generalized synchronization. We find that the conditional Lyapunov exponent is expressed as a formula in terms of ergodic theory. Dealing with this formula, we find what factors characterize the conditional Lyapunov exponent in chaotic systems.

An Improved Numerical Algorithm for the Fractional Differential Equations and Its Application in the Fractional-Order Nonlinear Systems

In this paper, an improved numerical algorithm for the fractional differential equations is proposed based on the variational iteration method. Using the improved numerical scheme, we investigate a new fractional-order hyperchaotic system, and find that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4. The lowest order we find to yield hyperchaos is 3.46 in this new fractional-order system. The existence of two positive Lyapunov exponents further verifies our results. Numerical results show that the proposed method has a faster speed and more accurate comparing with the traditional predictor-corrector algorithm. Based on the stability theory of the fractional-order system, a nonlinear controller is constructed to achieve synchronization fora class of nonlinear fractional-order systems using the pole placement technique. The nonlinear control method can synchronize two different fractional-order hyperchaotic systems. This synchronization scheme which is simple and theoretically rigorous enables synchronization of fractional-order hyperchaotic systems to be achieved in a systematic way and does not need to compute the conditional Lyapunov exponents. Simulation results are given to validate the effectiveness of the proposed synchronization method.

Conditional Lyapunov exponents and transfer entropy in coupled bursting neurons under excitation and coupling mismatch

This work has a twofold aim: (a) to analyze an alternative approach for computing the conditional Lyapunov exponent (λcmax) aiming to evaluate the synchronization stability between nonlinear oscillators without solving the classical variational equations for the synchronization error dynamical system. In this first framework, an analytic reference value for λcmax is also provided in the context of Duffing master-slave scenario and precisely evaluated by the proposed numerical approach; (b) to apply this technique to the study of synchronization stability in chaotic Hindmarsh-Rose (HR) neuronal models under uni- and bi-directional resistive coupling and different excitation bias, which also considered the root mean square synchronization error, information theoretic measures and asymmetric transfer entropy in order to offer a better insight of the synchronization phenomenon. In particular, statistical and information theoretical measures were able to capture similarity increase between the neuronal oscillators just after a critical coupling value in accordance to the largest conditional Lyapunov exponent behavior. On the other hand, transfer entropy was able to detect neuronal emitter influence even in a weak coupling condition, i.e. under the increase of conditional Lyapunov exponent and apparently desynchronization tendency. In the performed set of numerical simulations, the synchronization measures were also evaluated for a two-dimensional parameter space defined by the neuronal coupling (emitter to a receiver neuron) and the (receiver) excitation current. Such analysis is repeated for different feedback couplings as well for different (emitter) excitation currents, revealing interesting characteristics of the attained synchronization region and conditions that facilitate the emergence of the synchronous behavior. These results provide a more detailed numerical insight of the underlying behavior of a HR in the excitation and coupling space, being in accordance with some general findings concerning HR coupling topologies. As a perspective, besides the synchronization overview from different standpoints, we hope that the proposed numerical approach for conditional Lyapunov exponent evaluation could outline a valuable strategy for studying neuronal stability, especially when realistic models are considered, in which analytical or even Jacobian evaluation could define a laborious or impracticable task.

Determining the degree of synchronism for intermittent phase synchronization in human electroencephalography data

The phenomenon of intermittent phase synchronization during development of epileptic activity in human beings has been discovered based on EEG data. The presence of synchronous behavior phases has been detected both during spike-wave discharges and in the regions of background activity of the brain. The degree of synchronism in the intermittent phase-synchronization regime in both cases has been determined, and it has been established that spike-wave discharges are characterized by a higher degree of synchronism than exists in the regions of background activity of the brain. To determine the degree of synchronism, a modified method of evaluating zero conditional Lyapunov exponents from time series is proposed.