1,721,016 research outputs found
Are almost all trajectories dense in a chaotic set?
No full textChaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke "pizza machine" (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof. Copyright (c) 2024 The Authors
Tree-based algorithms for the stability of discrete-time switched linear systems under arbitrary and constrained switching
No full textWe present a direct approach to study the stability of discrete-time switched linear systems that can be applied to arbitrary switching, as well as when switching is constrained by a switching automaton. We explore the tree of possible matrix products, by pruning the subtrees rooted at contractions and looking for unstable repeatable products. Generically, this simple strategy either terminates with all contracting leafs-showing the system's asymptotic stability-or finds the shortest unstable and repeatable matrix product. Although it behaves in the worst case as the exhaustive search, we show that its performance is greatly enhanced by measuring contractiveness w.r.t. sum-of-squares polynomial norms, optimized to minimize the largest expansion among the system's modes
Optimal initial perturbations in a boundary layer with wall actuation
No full textWaves of span-wise velocity at the surface of the flow body, the wall, are known to be very effective in reducing the friction drag in turbulent channels and boundary layers. They can also delay the laminar-turbulent transition. To investigate this interesting property, in this work, we add velocity perturbations within a 3D Blasius boundary layer with wall actuation by means of a standing sinusoidal wave in the stream-wise direction. We look for the initial velocity perturbation pattern able to trigger the maximum energy gain in a given target time. The Navier-Stokes equations act as a constraint in the optimization problem. The results are strongly affected by the actuating parameters, namely the amplitude and wave-length of the sinusoidal profile, in terms of the energy gain and also of the space travelled by initial velocity perturbations during the target time. Opposite behaviours arise, such as an energy gain/loss whenever the actuating wave-length is greater/smaller of the space travelled by the perturbation
Optimal chemotherapy counteracts cancer adaptive resistance in a cell-based, spatially-extended, evolutionary model
No full textMost aggressive cancers are incurable due to their fast evolution of drug resistance. We model cancer growth and adaptive response in a simplified cell-based (CB) setting, assuming a genetic resistance to two chemotherapeutic drugs. We show that optimal administration protocols can steer cells resistance and turned it into a weakness for the disease. Our work extends the population-based model proposed by Orlando et al (2012 Phys. Biol.), in which a homogeneous population of cancer cells evolves according to a fitness landscape. The landscape models three types of trade-offs, differing on whether the cells are more, less, or equal effective when generalizing resistance to two drugs as opposed to specializing to a single one. The CB framework allows us to include genetic heterogeneity, spatial competition, and drugs diffusion, as well as realistic administration protocols. By calibrating our model on Orlando et al's assumptions, we show that dynamical protocols that alternate the two drugs minimize the cancer size at the end of (or at mid-points during) treatment. These results significantly differ from those obtained with the homogeneous model - suggesting static protocols under the pro-generalizing and neutral allocation trade-offs - highlighting the important role of spatial and genetic heterogeneities. Our work is the first attempt to search for optimal treatments in a CB setting, a step forward toward realistic clinical applications
A simple tree-based algorithm for deciding the stability of discrete-time switched linear systems
No full textWe re-evaluate the direct approach to study the stability of discrete-time switched systems with finitely many linear modes. We explore the tree of possible matrix products by pruning the paths leading to contractions and looking for unstable repeatable products. Generically, this simple strategy either terminates with all contracting leafs-showing the asymptotic stability of the system-or finds the shortest unstable matrix product. Although it behaves in the worst-case as the exhaustive search, we show that its performance is greatly enhanced by measuring contractiveness w.r.t. sum-of-squares polynomial norms, optimized to minimize the largest expansion among the system's modes. We test our approach on several benchmark examples proposed to test switched Lyapunov function methods. Although these methods are mathematically elegant and efficient, they are more involved to apply and cannot decide the system's instability. Moreover, our algorithm is flexible. Adding/removing system's modes, setting mode-independent dwell-times, and constraining switching to finite automata can all be easily implemented and exploited in control design
Forecasting of noisy chaotic systems with deep neural networks
Full text linkRecurrent neural networks have recently proved the state-of-the-art approach in forecasting complex oscillatory time series on a multi-step horizon. Researchers in the field investigated different machine learning techniques and training approaches on dynamical systems with different degrees of complexity. Still, these analyses are usually limited to noise-free chaotic time series. This paper extends the analysis from a deterministic to a noisy environment, by considering both observation and structural noise. Observation noise is evaluated by adding different levels of artificially-generated random values on deterministic processes obtained from the simulation of four archetypal chaotic systems. A case of structural noise is implemented through a time-varying version of the logistic map, which exhibits a slow structural change of the system's dynamic that makes the system non-stationary. Finally, a time series of ozone concentration in Northern Italy is considered to test the theoretical findings on a real-world case study in which both forms of noise play a significant role. Recurrent neural networks formed by LSTM cells are compared with two benchmark feed-forward architectures. LSTM trained without the standard teacher forcing approach, i.e., with training that replicates the setting used in inference mode, proved to have the best performance in compensating the stochasticity generated by the observation noise and reproducing the structural non-stationarity of the process
Optimal control of two cytotoxic drug maximum tolerated dose steers and exploits cancer adaptive resistance in a cell-based framework
No full textAggressive cancers are typically incurable because of drug resistance development. We model cancer growth and adaptive genetic response in a cell-based (CB) setting, displaying how optimal maximum tolerated dose administration protocols of two drugs counteract drug resistance and turn it into an exploitable weakness. Our CB model is a spatial extension of the population-based model proposed by Orlando et al. [1], where a homogeneous population of cancer cells evolves according to a fitness landscape. To make a first feasibly test of the optimal drug administration control problem in the CB framework, we add only the elements we consider most relevant for describing cancer growth and evolution: phenotypic heterogeneity, spatial competition, and drugs diffusion, as well as realistic administration protocols. We calibrate our model on Orlando et al.'s one and find that dynamical protocols switching between the two drugs minimize the cancer size at the end of (or at mid-points during) treatment. These results differ from those of Orlando and colleagues, which suggest static protocols under generalizing and neutral allocation trade-offs
Stabilizing switching automata for discrete-time switched linear systems
No full textWe develop an algorithm for the automatic generation of a switching automaton that stabilizes a given discrete-time switched linear system. The algorithm iteratively checks the stability of the system constrained by the current automaton and modifies the automaton's graph, until a stabilizing solution, or the empty graph, is reached. Stability is checked by means of a recently developed direct algorithm, that in case of instability provides a graph's cycle with unstable corresponding matrix product. Our heuristic to modify the switching graph limits the number of consecutive repetitions of the unstable periodic path. It is based on an ergodic result that tightly lower bounds the system's Constrained Joint Spectral Radius - the largest long-term average growth rate of the system's state - by looking at the matrix products along cycles of the switching graph. By only limiting the execution of unstable cycles, termination is not yet proved, though it is granted by a second, more constraining heuristic, that simply cuts the cycle if a prescribed length is exceeded. Our procedure can start either from the graph allowing arbitrary switching or from an application-specific graph under which the system is unstable
AUTOopt: An AUTO driver for boundary-value optimization problems
No full textAUTOopt is a python script that automatically generates the equation and command files for the software package AUTO to solve boundary-value optimization problems. Though the method of successive continuations to find local extrema of an objective functional over an ODE boundary-value problem has been proposed more than 25 years ago (in the early version of AUTO), the burden of writing the equations for the adjoint variables of the optimization problem and that of coding the script to organize the sequence of continuations have been left on the user. We finally make this powerful feature accessible, by fully automatizing the generation of the Fortran code for the optimization problem and of the python script to drive the sequence of AUTO runs. Not even the first-derivatives of the user problem, involved in the equations for the adjoint variables, are required. If not provided by the user, they are approximated by finite differences. On the other hand, to improve accuracy, the user can also provide the second-derivatives, that are used to pass AUTO the first-derivatives of the optimization problem. Several examples are illustrated
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