Ulrike Feudel

Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons.

We study global bifurcations of the chaotic attractor in a modified Hodgkin-Huxley model of thermally sensitive neurons. The control parameter for this model is the temperature. The chaotic behavior is realized over a wide range of temperatures and is visualized using interspike intervals. We observe an abrupt increase of the interspike intervals in a certain temperature region. We identify this as a homoclinic bifurcation of a saddle-focus fixed point which is embedded in the chaotic attractors. The transition is accompanied by intermittency, which obeys a universal scaling law for the average length of trajectory segments exhibiting only short interspike intervals with the distance from the onset of intermittency. We also present experimental results of interspike interval measurements taken from the crayfish caudal photoreceptor, which qualitatively demonstrate the same bifurcation structure. (c) 2000 American Institute of Physics.

Characterizing strange nonchaotic attractors

Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Second, we propose to characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown that phase sensitivity appears if there is a nonzero probability for positive local Lyapunov exponents to occur. (c) 1995 American Institute of Physics.

Multistability and the control of complexity

We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. (c) 1997 American Institute of Physics.

COMPLEX DYNAMICS IN MULTISTABLE SYSTEMS

The coexistence of several stable states for a given set of parameters has been observed in many natural and experimental systems as well as in theoretical models. This paper gives an overview over the wide range of applications in different disciplines of science. Furthermore, different system classes possessing multistability are analyzed in terms of the appearance of coexisting attractors and their basins of attraction. It is shown that multistable systems are very sensitive to perturbations leading to a noise-induced hopping process between attractors. The role of chaotic saddles in the escape from attractors in multistable systems is discussed.

Strange non-chaotic attractor in a quasiperiodically forced circle map

Abstract We show that in the quasiperiodically forced circle map strange non-chaotic attractors can appear for non-linearities far from the border of chaos. The destruction of a two-frequency quasiperiodic torus connected with the appearance of a strange non-chaotic attractor is described in detail. This strange non-chaotic attractor is characterized by logarithmically slow diffusion of the phase. It is shown that in this regime the high-order phase-locking states disappear and the rotation number varies rather smoothly with the parameters.

Extreme multistability: Attractor manipulation and robustness

The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.

Plankton blooms in vortices: the role of biological and hydrodynamic timescales

We study the interplay of hydrodynamic mesoscale structures and the growth of plankton in the wake of an island, and its interaction with a coastal upwelling. Our focus is on a mechanism for the emergence of localized plankton blooms in vortices. Using a coupled system of a kinematic flow mimicking the mesoscale structures behind the island and a simple three component model for the marine ecosystem, we show that the long residence times of nutrients and plankton in the vicinity of the island and the confinement of plankton within vortices are key factors for the appearance of localized plankton blooms

Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors

Abstract We study a simple mechanical system consisting of two rotors that possesses a large number (3000+) of coexisting periodic attractors. A complex fractal boundary separates these tiny islands of stability and their basins of attraction. Hence, the systems long term behavior is acutely sensitive to the initial conditions. This sensitivity combined with the systems many periodic sinks give rise to a rich dynamical behavior when the system is subjected to small amplitude noise. This dynamical behavior is of great utility, and this is demonstrated by using perturbations which are smaller than the noise level to gear and influence the dynamics toward a specific periodic behavior.

Birth of strange nonchaotic attractors due to interior crisis

Abstract We study the interior crisis in the period-3-window of the quasiperiodically forced logistic map. Two routes from quasiperiodicity to chaos involving strange nonchaotic attractors (SNA) are discovered: Along one route we observe a sudden widening of the SNA. This is similar to the interior crisis in chaotic systems. Along the other route we find a direct transition from an invariant curve to a strange nonchaotic attractor exactly at the interior crisis point. This is a new mechanism of the appearance of strange nonchaotic attractors. Beyond the interior crisis the temporal behavior can be described as a crisis-induced intermittency, whose scaling behavior is discussed.

Evidence of chaos in eco-epidemic models.

We study an eco-epidemic model with two trophic levels in which the dynamics are determined by predator-prey interactions as well as the vulnerability of the predator to a disease. Using the concept of generalized models we show that for certain classes of eco-epidemic models quasiperiodic and chaotic dynamics are generic and likely to occur. This result is based on the existence of bifurcations of higher codimension such as double Hopf bifurcations. We illustrate the emergence of chaotic behavior with one example system.