A. Krawiecki

Volatility clustering and scaling for financial time series due to attractor bubbling.

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time heat bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit, the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters, the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.

Stochastic resonance as a model for financial market crashes and bubbles

A bistable model of a financial market is considered, aimed at modelling financial crashes and bubbles, based on the Ising model with thermal-bath dynamics and long-range interactions, subject to a weak external information-carrying signal and noise. In the ordered phase, opposite stable orientations of magnetization correspond to the growing and declining market before and after the crash or bubble, and jumps of magnetization direction correspond to crashes and bubbles. It is shown that the influence of an information-carrying signal, assumed to be too weak to induce magnetization jumps, can be enhanced by the external noise via the effect of stochastic resonance. It is argued that in real stock markets the arrival of a piece of information, considered a posteriori to be the cause for a crash or bubble, can be enhanced in a similar way, thus leading to price return whose value is unexpectedly large in comparison with relatively weak importance of this piece of information.

Generalizations of the concept of marginal synchronization of chaos

Abstract Generalizations of the concept of marginal synchronization between chaotic systems, i.e. synchronization with zero largest conditional Lyapunov exponent, are considered. Generalized marginal synchronization in drive–response systems is defined, for which the function between points of attractors of different systems is given up to a constant. Auxiliary system approach is shown to be able to detect this synchronization. Marginal synchronization in mutually coupled systems which can be viewed as drive–response systems with the response system influencing the drive system dynamics is also considered, and an example from solid-state physics is analyzed. Stability of these kinds of synchronization against changes of system parameters and noise is investigated. In drive–response systems generalized marginal synchronization is shown to be rather sensitive to the changes of parameters and may disappear either due to the loss of stability of the response system, or as a result of the blowout bifurcation. Nonlinear coupling of the drive system to the response system can stabilize marginal synchronization.

DYNAMICAL PHASE TRANSITION IN THE ISING MODEL ON A SCALE-FREE NETWORK

Dynamical phase transition in the Ising model on a Barabasi–Albert network under the influence of periodic magnetic field is studied using Monte-Carlo simulations. For a wide range of the system sizes N and the field frequencies, approximate phase borders between dynamically ordered and disordered phases are obtained on a plane h (field amplitude) versus T/Tc (temperature normalized to the static critical temperature without external field, Tc∝lnN). On these borders, second- or first-order transitions occur, for parameter ranges separated by a tricritical point. For all frequencies of the magnetic field, position of the tricritical point is shifted toward higher values of T/Tc and lower values of h with increasing system size, i.e. the range of critical parameters corresponding to the first-order transition is broadened.

MICROSCOPIC SPIN MODEL FOR THE STOCK MARKET WITH ATTRACTOR BUBBLING ON REGULAR AND SMALL-WORLD LATTICES

A multi-agent spin model for changes of prices in the stock market based on the Ising-like cellular automaton with interactions between traders randomly varying in time is investigated by means of Monte Carlo simulations. The structure of interactions has topology of a small-world network obtained from regular two-dimensional square lattices with various coordination numbers by randomly cutting and rewiring edges. Simulations of the model on regular lattices do not yield time series of logarithmic price returns with statistical properties comparable with the empirical ones. In contrast, in the case of networks with a certain degree of randomness for a wide range of parameters the time series of the logarithmic price returns exhibit intermittent bursting typical of volatility clustering. Also the tails of distributions of returns obey a power scaling law with exponents comparable to those obtained from the empirical data.

Log-periodic oscillations and noise-free stochastic multiresonance due to self-similarity of fractals

The origin of log-periodic oscillations around the power-law trend of the escape probability from a precritical attractor and of the noise-free stochastic multiresonance, found in numerical simulations in chaotic systems close to crises is discussed. It is shown that multiple maxima of the spectral power amplification vs. the control parameter result from a fractal structure of a precritical attractor colliding with a possibly fractal basin of attraction at the crisis point. Qualitative explanation of the multiresonance, based on a concept of fractal self-similarity, or discrete-scale invariance, is given and compared with numerical results and analytic theory using a simple geometric models of the colliding fractal sets. 2003 Elsevier Science Ltd. All rights reserved.

Ferromagnetic transition in a simple variant of the Ising model on multiplex networks

Abstract Multiplex networks consist of a fixed set of nodes connected by several sets of edges which are generated separately and correspond to different networks (“layers”). Here, a simple variant of the Ising model on multiplex networks with two layers is considered, with spins located in the nodes and edges corresponding to ferromagnetic interactions between them. Critical temperatures for the ferromagnetic transition are evaluated for the layers in the form of random Erdos–Renyi graphs or heterogeneous scale-free networks using the mean-field approximation and the replica method, from the replica symmetric solution. Both methods require the use of different “partial” magnetizations, associated with different layers of the multiplex network, and yield qualitatively similar results. If the layers are strongly heterogeneous the critical temperature differs noticeably from that for the Ising model on a network being a superposition of the two layers, evaluated in the mean-field approximation neglecting the effect of the underlying multiplex structure on the correlations between the degrees of nodes. The critical temperature evaluated from the replica symmetric solution depends sensitively on the correlations between the degrees of nodes in different layers and shows satisfactory quantitative agreement with that obtained from Monte Carlo simulations. The critical behavior of the magnetization for the model with strongly heterogeneous layers can depend on the distributions of the degrees of nodes and is then determined by the properties of the most heterogeneous layer.

STOCHASTIC RESONANCE IN THE ISING MODEL ON A BARABÁSI–ALBERT NETWORK

Stochastic resonance is investigated in the Ising model with ferromagnetic coupling on a Barabasi–Albert network, subjected to weak periodic magnetic field. Spectral power amplification as a function of temperature shows strong dependence on the number of nodes, which is related to the dependence of the critical temperature for the ferromagnetic phase transition, and on the frequency of the periodic signal. Double maxima of the spectral power amplification evaluated from the time-dependent magnetization are observed for intermediate frequencies of the periodic signal, which are also dependent on the number of nodes. In the thermodynamic limit, the height of the maxima decreases to zero and stochastic resonance disappears. Results of numerical simulations are in qualitative agreement with predictions of the linear response theory in the mean-field approximation.

Stochastic resonance in spatiotemporal on–off intermittency

Abstract Stochastic resonance is investigated in a generic system with spatiotemporal on–off intermittency: a chain of coupled logistic maps with a time-dependent control parameter, driven by a spatiotemporal periodic signal. Spatiotemporal correlation function between the periodic signal and the output signal, reflecting the occurrence of laminar phases and chaotic bursts, has a maximum as a function of the mean value of the control parameter. For a given period and length of the periodic signal the height of this maximum can be increased by choosing an optimum coupling strength between maps. It is argued that the obtained result can be interpreted as an example of noise-free (dynamical) stochastic resonance in a system with spatiotemporal intermittency.

STOCHASTIC RESONANCE IN COUPLED THRESHOLD ELEMENTS WITH INPUT SIGNALS SHIFTED IN PHASE

Stochastic resonance in a system of two coupled threshold elements (neurons) forming a small artificial neural network is investigated. The elements have either antisymmetric or logistic (binary) response function and are driven by periodic signals and independent noises. Periodic signals at their inputs have equal amplitudes and frequencies but are shifted in phase. Depending on the response function and the phase shift, enhancement of stochastic resonance in individual elements, characterized by the output signal-to-noise ratio, and stochastic resonance with a spatiotemporal input signal, characterized by the correlation function between the input and output signals, are observed for proper coupling between elements.