Research

Sensitivity, Flexibility, and Robustness of

Dynamical Systems

M. Marhl, M. Perc,

Determining the flexibility of regular and chaotic attractors

Chaos, Slotitons & Fractals
28 (2006) 822-833

Abstract: We present an overview of measures that are appropriate for determining the flexibility of regular and chaotic attractors. In particular, we focus on those system properties that constitute its responses to external perturbations. We deploy a systematic approach, first introducing the simplest measure given by the local divergence of the system along the attractor, and then develop more rigorous mathematical tools for estimating the flexibility of the system’s dynamics. The presented measures are tested on the regular Brusselator and chaotic Hindmarsh–Rose model of an excitable neuron with equal success, thus indicating the overall effectiveness and wide applicability range of the proposed theory. Since responses of dynamical systems to external signals are crucial in several scientific disciplines, and especially in natural sciences, we discuss several important aspects and biological implications of obtained results.

M. Perc, M. Marhl

Chaos in temporarily destabilized regular systems with the slow passage effect

Chaos, Slotitons & Fractals
27 (2006) 395-403

Abstract: We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.

M. Perc and M. Marhl

Noise enhances robustness of intracellular Ca2+ oscillations

Physics Letters A
316 (2003) 304-310

M. Perc and M. Marhl

Sensitivity and flexibility of regular and chaotic calcium oscillations

Biophysical Chemistry
104 (2003) 509-522

Abstract: Sensitivity and flexibility are important properties of biological systems. These properties are here investigated for intracellular calcium oscillations. For a particular model, we comparatively investigate sensitivity and flexibility of regular and chaotic Ca2+ oscillations. For this model, we obtain two main results. First, sensitivity of the model system to parameter shifting does not depend on the complexity of Ca2+ oscillations. We observe, however, that both regular and chaotic Ca2+ oscillations are highly sensitive in regions close to bifurcation points. Second, also flexibility of Ca2+ oscillations does not significantly depend on the type of Ca2+ oscillations. Our results show that regular as well as chaotic Ca2+ oscillations in the studied model are highly flexible in regimes with weak dissipation. Both results are discussed in the sense of possible biological importance.

M. Marhl and S. Schuster

Under what conditions signal transduction pathways are highly flexible in response to external forcing? A case study on calcium oscillations

Journal of Theoretical Biology
224 (2003) 491-500

Abstract: Sensitivity and flexibility are typical properties of biological systems. These properties are here investigated in a model for simple and complex intracellular calcium oscillations. In particular, the influence of external periodic forcing is studied. The main point of the study is to compare responses of the system in a chaotic regime with those obtained in a regular periodic regime. We show that the response to external signals in terms of the range of synchronization is not significantly different in regular and chaotic Ca2+ oscillations. However, both types of oscillation are highly flexible in regimes with weak dissipation. Therefore, we conclude that dissipation of free energy is a suitable index characterizing flexibility. For biological systems this appears to be of special importance since for thermodynamic reasons, notably in view of low free energy consumption, dissipation should be minimized.

Elsevier	*M. Marhl, M. Perc,* Determining the flexibility of regular and chaotic attractors Chaos, Slotitons & Fractals 28 (2006) 822-833	Abstract: We present an overview of measures that are appropriate for determining the flexibility of regular and chaotic attractors. In particular, we focus on those system properties that constitute its responses to external perturbations. We deploy a systematic approach, first introducing the simplest measure given by the local divergence of the system along the attractor, and then develop more rigorous mathematical tools for estimating the flexibility of the system’s dynamics. The presented measures are tested on the regular Brusselator and chaotic Hindmarsh–Rose model of an excitable neuron with equal success, thus indicating the overall effectiveness and wide applicability range of the proposed theory. Since responses of dynamical systems to external signals are crucial in several scientific disciplines, and especially in natural sciences, we discuss several important aspects and biological implications of obtained results.

Elsevier	*M. Perc, M. Marhl* Chaos in temporarily destabilized regular systems with the slow passage effect Chaos, Slotitons & Fractals 27 (2006) 395-403	Abstract: We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.

Elsevier	*M. Perc and M. Marhl* Noise enhances robustness of intracellular Ca2+ oscillations Physics Letters A 316 (2003) 304-310	Abstract: We investigate responses of a model for intracellular Ca2+ oscillations to external pulsatile forcing in the presence of additive Gaussian noise. Our results show that noise makes the system less susceptible to external forcing and thus enhances robustness of Ca2+ oscillations. The results can be well explained by the local divergence of limit cycles in the phase space.

Elsevier	*M. Perc and M. Marhl* Sensitivity and flexibility of regular and chaotic calcium oscillations Biophysical Chemistry 104 (2003) 509-522	Abstract: Sensitivity and flexibility are important properties of biological systems. These properties are here investigated for intracellular calcium oscillations. For a particular model, we comparatively investigate sensitivity and flexibility of regular and chaotic Ca2+ oscillations. For this model, we obtain two main results. First, sensitivity of the model system to parameter shifting does not depend on the complexity of Ca2+ oscillations. We observe, however, that both regular and chaotic Ca2+ oscillations are highly sensitive in regions close to bifurcation points. Second, also flexibility of Ca2+ oscillations does not significantly depend on the type of Ca2+ oscillations. Our results show that regular as well as chaotic Ca2+ oscillations in the studied model are highly flexible in regimes with weak dissipation. Both results are discussed in the sense of possible biological importance.


Elsevier	*M. Marhl and S. Schuster* Under what conditions signal transduction pathways are highly flexible in response to external forcing? A case study on calcium oscillations Journal of Theoretical Biology 224 (2003) 491-500	Abstract: Sensitivity and flexibility are typical properties of biological systems. These properties are here investigated in a model for simple and complex intracellular calcium oscillations. In particular, the influence of external periodic forcing is studied. The main point of the study is to compare responses of the system in a chaotic regime with those obtained in a regular periodic regime. We show that the response to external signals in terms of the range of synchronization is not significantly different in regular and chaotic Ca2+ oscillations. However, both types of oscillation are highly flexible in regimes with weak dissipation. Therefore, we conclude that dissipation of free energy is a suitable index characterizing flexibility. For biological systems this appears to be of special importance since for thermodynamic reasons, notably in view of low free energy consumption, dissipation should be minimized.