1,329 research outputs found
Continuous approximations of a class of piece-wise continuous systems
In this paper we provide a rigorous mathematical foundation for continuous
approximations of a class of systems with piece-wise continuous functions. By
using techniques from the theory of differential inclusions, the underlying
piece-wise functions can be locally or globally approximated. The approximation
results can be used to model piece-wise continuous-time dynamical systems of
integer or fractional-order. In this way, by overcoming the lack of numerical
methods for diffrential equations of fractional-order with discontinuous
right-hand side, unattainable procedures for systems modeled by this kind of
equations, such as chaos control, synchronization, anticontrol and many others,
can be easily implemented. Several examples are presented and three comparative
applications are studied.Comment: IJBC, accepted (examples revised
Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach
In this article we provide homotopy solutions of a cancer nonlinear model
describing the dynamics of tumor cells in interaction with healthy and effector
immune cells. We apply a semi-analytic technique for solving strongly nonlinear
systems - the Step Homotopy Analysis Method (SHAM). This algorithm, based on a
modification of the standard homotopy analysis method (HAM), allows to obtain a
one-parameter family of explicit series solutions. By using the homotopy
solutions, we first investigate the dynamical effect of the activation of the
effector immune cells in the deterministic dynamics, showing that an increased
activation makes the system to enter into chaotic dynamics via a
period-doubling bifurcation scenario. Then, by adding demographic stochasticity
into the homotopy solutions, we show, as a difference from the deterministic
dynamics, that an increased activation of the immune cells facilitates cancer
clearance involving tumor cells extinction and healthy cells persistence. Our
results highlight the importance of therapies activating the effector immune
cells at early stages of cancer progression
Energy-optimal steering of transitions through a fractal basin boundary.
We study fluctuational transitions in a discrete dy- namical system having two co-existing attractors in phase space, separated by a fractal basin boundary. It is shown that transitions occur via a unique ac- cessible point on the boundary. The complicated structure of the paths inside the fractal boundary is determined by a hierarchy of homoclinic original sad- dles. By exploiting an analogy between the control problem and the concept of an optimal fluctuational path, we identify the optimal deterministic control function as being equivalent to the optimal fluctu- ational force obtained from a numerical analysis of the fluctuational transitions between two states
Optimal fluctuations and the control of chaos.
The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel-Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identied with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms
Exploring Transfer Function Nonlinearity in Echo State Networks
Supralinear and sublinear pre-synaptic and dendritic integration is
considered to be responsible for nonlinear computation power of biological
neurons, emphasizing the role of nonlinear integration as opposed to nonlinear
output thresholding. How, why, and to what degree the transfer function
nonlinearity helps biologically inspired neural network models is not fully
understood. Here, we study these questions in the context of echo state
networks (ESN). ESN is a simple neural network architecture in which a fixed
recurrent network is driven with an input signal, and the output is generated
by a readout layer from the measurements of the network states. ESN
architecture enjoys efficient training and good performance on certain
signal-processing tasks, such as system identification and time series
prediction. ESN performance has been analyzed with respect to the connectivity
pattern in the network structure and the input bias. However, the effects of
the transfer function in the network have not been studied systematically.
Here, we use an approach tanh on the Taylor expansion of a frequently used
transfer function, the hyperbolic tangent function, to systematically study the
effect of increasing nonlinearity of the transfer function on the memory,
nonlinear capacity, and signal processing performance of ESN. Interestingly, we
find that a quadratic approximation is enough to capture the computational
power of ESN with tanh function. The results of this study apply to both
software and hardware implementation of ESN.Comment: arXiv admin note: text overlap with arXiv:1502.0071
Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps
What input signals will lead to synchrony vs. desynchrony in a group of
biological oscillators? This question connects with both classical dynamical
systems analyses of entrainment and phase locking and with emerging studies of
stimulation patterns for controlling neural network activity. Here, we focus on
the response of a population of uncoupled, elliptically bursting neurons to a
common pulsatile input. We extend a phase reduction from the literature to
capture inputs of varied strength, leading to a circle map with discontinuities
of various orders. In a combined analytical and numerical approach, we apply
our results to both a normal form model for elliptic bursting and to a
biophysically-based neuron model from the basal ganglia. We find that,
depending on the period and amplitude of inputs, the response can either appear
chaotic (with provably positive Lyaponov exponent for the associated circle
maps), or periodic with a broad range of phase-locked periods. Throughout, we
discuss the critical underlying mechanisms, including slow-passage effects
through Hopf bifurcation, the role and origin of discontinuities, and the
impact of noiseComment: 17 figures, 40 page
Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes
In this work, we consider systems that are subjected to intermittent
instabilities due to external stochastic excitation. These intermittent
instabilities, though rare, have a large impact on the probabilistic response
of the system and give rise to heavy-tailed probability distributions. By
making appropriate assumptions on the form of these instabilities, which are
valid for a broad range of systems, we formulate a method for the analytical
approximation of the probability distribution function (pdf) of the system
response (both the main probability mass and the heavy-tail structure). In
particular, this method relies on conditioning the probability density of the
response on the occurrence of an instability and the separate analysis of the
two states of the system, the unstable and stable state. In the stable regime
we employ steady state assumptions, which lead to the derivation of the
conditional response pdf using standard methods for random dynamical systems.
The unstable regime is inherently transient and in order to analyze this regime
we characterize the statistics under the assumption of an exponential growth
phase and a subsequent decay phase until the system is brought back to the
stable attractor. The method we present allows us to capture the statistics
associated with the dynamics that give rise to heavy-tails in the system
response and the analytical approximations compare favorably with direct Monte
Carlo simulations, which we illustrate for two prototype intermittent systems:
an intermittently unstable mechanical oscillator excited by correlated
multiplicative noise and a complex mode in a turbulent signal with fixed
frequency, where multiplicative stochastic damping and additive noise model
interactions between various modes.Comment: 29 pages, 15 figure
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