International Journal of Bifurcation and Chaos

Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing.

We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some stationary state. In this paper it is shown that the interconnection symmetry, though ensuring complete stability, is not in the general case sufficient to guarantee that complete stability is robust with respect to sufficiently small perturbations of the interconnections. To this end, a class of third-order CNN's with competitive (inhibitory) interconnections between distinct neurons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This result is of obvious relevance in view of CNN's implementation, since perfect interconnection symmetry in unattainable in hardware (e.g. VLSI) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leonard model of the voting paradox, a special Volterra–Lotka model of three competing species. Finally, it is shown that the results in this paper can also be viewed as an extension of previous results by Zou and Nossek for a two-cell CNN with opposite-sign interconnections between distinct neurons. Such an extension has a significant interpretation in the framework of a general theorem by Smale for competitive dynamical systems.

This paper presents analytical criteria for local activity in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin–Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the long-lasting action and pace-maker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin–Huxley CNN's supply a possible explanation for why a heart with a normal heart-rate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin–Huxley CNN by a computer. As a second application, we present a smoothed Chua's circuit (SCC) CNN. The bifurcation diagrams of the SCC CNN's show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixed-cell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

This paper deals with bifurcation structures related to families of fractional harmonics (or ultra-subharmonics) solutions generated by the Duffing–Rayleigh equation with a nonsymmetrical periodic external force. It presents some results on fractional reducible harmonics and their bifurcations. In particular a new type of contact of fold curves with Neimark's bifurcation curves (improperly called "Hopf bifurcation"), due to such fractional harmonics, is described. This contact is different from the one related to an Arnold's tongue. The corresponding configuration is called pocket of a reducible harmnonics.

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

In this paper, Lyapunov exponents are successfully calculated and employed to design a sinusoidally excited heated plate system representing hypersonic skin panels on transatmospheric vehicles, for the prevention of chaotic oscillations. The single mode equation of motion governing the vibrations of the heated plate system is of Duffing form. The coefficients of the equation are obtained as functions of the design variables. The boundaries of regular and chaotic regions of motion are defined in various two-dimensional parametric spaces defined by the values of the largest Lyapunov exponent at each point. The critical buckling temperatures that transform the potential form from single-well to double-well are discussed for various plate aspect ratios. It is shown that finding chaos-free aspect ratios is possible, and chaos can be eliminated or delayed at the design stage simply by choosing the appropriate size and aspect ratio for the panel. The influence of the thickness of panels on the chaotic response of the system is investigated. The results of this research have direct impact on shape optimization considerations for skin panels subject to high temperatures, and on the design of various other dynamical systems that are influenced by a combination of thermal loads and external excitation.

Assuming a good embedding and additive noise, the traditional approach to time-series embedding prediction has been to predict pointwise by (usually linear) regression of the k-nearest neighbors; no good mathematics has been previously developed to appropriately select the model (where to truncate Taylor's series) to balance the conflict between noise fluctuations of a small k, and large k data needs of fitting many parameters of a high ordered model. We present a systematic approach to: (1) select the statistically significant neighborhood for a fixed (usually linear) model, (2) give an unbiased estimate of predicted mean response together with a statement of quality of the prediction in terms of confidence bands.

The control process for a chaotic system as proposed by Ott, Grebogi and Yorke (OGY) is investigated for systems with a one-dimensional Poincaré map. The calculation of the Poinaré map for the controlled system turns out to be an instructive method to analyze the mechanism of OGY control. We introduce the notion of a region of immediate capturing, characterizing the maximum control region that guarantees success of OGY control. Concerning the mean time to achieve control, we show that the simple probability argument in OGY's original paper has to be refined. The exponent of OGY's asymptotic power law is not affected by this detail, but the scale factor is incorrect. For the case of one-dimensional maps we derive a modified law for the dependence of the mean time to achieve control on the size of the control region. In contrast to OGY's prediction this formula is also valid for large control bounds and fits well with numerical calculations. The revised statistical argument is corroborated by investigating the time evolution of the density during control. This approach allows an alternative determination of the mean time to achieve control. The evolution of a stationary density shape is demonstrated, which notably differs from the invariant density even for moderate control bounds. This observation explains minor deviations from the modified law that can be found especially for large control bounds.

In this paper we show that unbounded chaotic trajectories are easily observed in the iteration of maps which are not defined everywhere, due to the presence of a denominator which vanishes in a zero-measure set. Through simple examples, obtained by the iteration of one-dimensional and two-dimensional maps with denominator, the basic mechanisms which are at the basis of the existence of unbounded chaotic trajectories are explained. Moreover, new kinds of contact bifurcations, which mark the transition from bounded to unbounded sets of attraction, are studied both through the examples and by general theoretical methods. Some of the maps studied in this paper have been obtained by a method based on the Schröoder functional equation, which allows one to write closed analytical expressions of the unbounded chaotic trajectories, in terms of elementary functions.

High-dimensional chaos has been an area of growing recent investigation. The questions of how dynamical systems become high-dimensionally chaotic with multiple positive Lyapunov exponents, and what the characteristic features associated with the transition are, remain less investigated. In this paper, we present one possible route to high-dimensional chaos. By this route, a subsystem becomes chaotic with one positive Lyapunov exponent via one of the known routes to low-dimensional chaos, after which the complementary subsystem becomes chaotic, leading to additional positive Lyapunov exponents for the whole system. A characteristic feature of this route is that the additional Lyapunov exponents pass through zero smoothly. As a consequence, the fractal dimension of the chaotic attractor changes continuously through the transition, in contrast to the transition to low-dimensional chaos at which the fractal dimension changes abruptly. We present a heuristic theory and numerical examples to illustrate this route to high-dimensional chaos.

We consider a nonlocal eigenvalue problem which arises in the study of stability of point-condensation solutions in the Gierer–Meinhardt system and generalized Gray–Scott system. We give some sufficient conditions for stability and instability. The conditions are new and can be applied to the study of stability of single point-condensation solutions.

Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

This paper uses the skew tent map as a basic endomorphism to generate pseudo-random numbers of several familiar statistical distributions. For a special value of the parameter a in the skew tent map, we obtain the F-distribution and the Z-distribution in [Golubentsev & Anikin, 1998].

The heartbeat time series commonly used in diagnostics is investigated. Since this series is not stationary, the difference series is considered and in the framework of stochastic processes its stationarity and mixing properties are tested. For this kind of data, affected by noise and low precision, a notion of correlation exponent, in some sense related to the correlation dimension, is proposed. The data analysis shows the existence of a profile of the correlation exponents, which evidentiates the amounts of randomness and of determinism in the heartbeat dynamics.

We study the phenomena of synchronization in a system of two coupled oscillators. The oscillators, identical when decoupled, model forced pendula and are coupled with a linear torsional spring. Using the idea of energy transfer between the two oscillators, we derive nonlinear control laws which stabilize the in-phase motion. Physically, the additional control terms act through a controlled transfer the energy between the two oscillators. Thus the mechanical energy of the system remains constant as the oscillators are synchronized. Trajectories of the controlled system are required to remain on the constant energy surface of the Hamiltonian of the uncontrolled system, which represents the mechanical energy of the coupled oscillator system.