# 5.5: Exercises- Matrix Methods for Dynamical Systems

Exercise (PageIndex{1})

Compute, without the aid of a machine, the Laplace transforms of (e^t) and (te^{-t}). Show ALL of your work.

Exercise (PageIndex{2})

Extract fromfib3.manalytical expressions for (x_2) and (x_{3})

Exercise (PageIndex{3})

Useeigto compute the eigenvalues of (B = egin{pmatrix} {2}&{-1} {-1}&{2} end{pmatrix}). Usedetto compute the characteristic polynomial of (B)rootsto compute the roots of this characteristic polynomial. Compare these to the results ofeig. How does Matlab compute the roots of a polynomial? (typehelp rootsfor the answer).

Exercise (PageIndex{4})

Adapt the Backward Euler portion offib3.mso that one may specify an arbitrary number of compartments, as infib1.m. Submit your well documented M-file along with a plot of (x_{1}) and (x_{10}) versus time (on the same well labeled graph) for a nine compartment fiber of length (l = 1cm).

Exercise (PageIndex{5})

Derive (frac{ ilde{x}(t)- ilde{x}(t-dt)}{dt} = B ilde{x}(t)+g(t)) from ( extbf{x}' = B extbf{x}+ extbf{g}), by working backwards toward (x(⁢0)). Along the way you should explain why

(frac{(frac{I}{d(t)}-B)^{-1}}{d(t)} = (I-d(t)B)^{-1})

## 5.5: Exercises- Matrix Methods for Dynamical Systems

Citation
Calin Belta. "Formal Methods for Dynamical Systems". Talk or presentation, 28, September, 2015.

Abstract
In control theory, complex models of physical processes, such as systems of differential equations, are usually checked against simple specifications, such as stability and set invariance. In formal methods, rich specifications, such as languages and formulae of temporal logics, are checked against simple models of software programs and digital circuits, such as finite transition graphs. With the development and integration of cyber physical and safety critical systems, there is an increasing need for computational tools for verification and control of complex systems from rich, temporal logic specifications. The formal verification and synthesis problems have been shown to be undecidable even for very simple classes of infinite-space continuous and hybrid systems. However, provably correct but conservative approaches, in which the satisfaction of a property by a dynamical system is implied by the satisfaction of the property by a finite over-approximation (abstraction) of the system, have received a lot of attention in recent years. The focus of this talk is on discrete-time linear systems, for which it is shown that finite abstractions can be constructed through polyhedral operations only. By using techniques from model checking and automata games, this allows for verification and control from specifications given as Linear Temporal Logic (LTL) formulae over linear predicates in the state variables. The usefulness of these computational tools is illustrated with various examples.

Posted by Sadigh Dorsa on 5 Oct 2015.
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## Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples The dynamical systems method (DSM) is a powerful computational method for solving operator equations. With this book as their guide, readers will master the application of DSM to solve a variety of linear and nonlinear problems as well as ill-posed and well-posed problems. The authors offer a clear, step-by-step, systematic development of DSM that enables readers to grasp the method's underlying logic and its numerous applications.

Dynamical Systems Method and Applications begins with a general introduction and then sets forth the scope of DSM in Part One. Part Two introduces the discrepancy principle, and Part Three offers examples of numerical applications of DSM to solve a broad range of problems in science and engineering. Additional featured topics include:

General nonlinear operator equations

Operators satisfying a spectral assumption

Newton-type methods without inversion of the derivative

Numerical problems arising in applications

Stable numerical differentiation

Stable solution to ill-conditioned linear algebraic systems

Throughout the chapters, the authors employ the use of figures and tables to help readers grasp and apply new concepts. Numerical examples offer original theoretical results based on the solution of practical problems involving ill-conditioned linear algebraic systems, and stable differentiation of noisy data.

Written by internationally recognized authorities on the topic, Dynamical Systems Method and Applications is an excellent book for courses on numerical analysis, dynamical systems, operator theory, and applied mathematics at the graduate level. The book also serves as a valuable resource for professionals in the fields of mathematics, physics, and engineering.  #### Buy Both and Save 25%!

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Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts. Wiley, 2015. p. 137-191.

Research output : Chapter in Book/Report/Conference proceeding › Chapter

T1 - Data-Driven Methods for Dynamical Systems

T2 - Quantifying Predictability and Extracting Spatiotemporal Patterns

N2 - This chapter reviews two examples of applied mathematics techniques for data analysis in dynamical systems. The two examples are: (1) Methods for quantifying predictability and model error based on data clustering and information theory and (2) nonlinear Laplacian spectral analysis (NLSA) algorithms for extracting spatiotemporal patterns from high-dimensional data. The chapter highlights these techniques with applications to climate atmosphere ocean science (CAOS), in particular, predictability assessment and Markov modeling of circulation regimes in a simple ocean model and extraction of modes of organized convection in the tropics from infrared brightness temperature satellite data. A common theme in these methods has been the coarse-grained geometry of the data. The machinery of discrete exterior calculus and spectral graph theory was combined with delay-coordinate mappings of dynamical systems to extract spatiotemporal modes of variability which are describable in terms of low-dimensional sets of diffusion eigenfunctions.

AB - This chapter reviews two examples of applied mathematics techniques for data analysis in dynamical systems. The two examples are: (1) Methods for quantifying predictability and model error based on data clustering and information theory and (2) nonlinear Laplacian spectral analysis (NLSA) algorithms for extracting spatiotemporal patterns from high-dimensional data. The chapter highlights these techniques with applications to climate atmosphere ocean science (CAOS), in particular, predictability assessment and Markov modeling of circulation regimes in a simple ocean model and extraction of modes of organized convection in the tropics from infrared brightness temperature satellite data. A common theme in these methods has been the coarse-grained geometry of the data. The machinery of discrete exterior calculus and spectral graph theory was combined with delay-coordinate mappings of dynamical systems to extract spatiotemporal modes of variability which are describable in terms of low-dimensional sets of diffusion eigenfunctions.

Every Monday 13:00 for (on average) 45 minutes we want to interact, starting 19.4.:
- discussion of exercises/homework
BigBlueButton (Mondays, 13:00) (for external participants without login)

Mainly analytical problems on classical and quantum dynamical systems.
Will be presented by students in live video conference.
Exercise 1 (solution)
Exercise 2 (solution)
Exercise 3 (solution)
Exercise 4
Exercise 5 (solution by multiple (20) artists)
Exercise 6
Monday 7.6.: Homework H3
Monday 14.6.: Discussion on "chaos in quantum mechanics"
Monday 21.6.: Homework H4
Monday 28.6.: Discussion on "regular/chaotic eigenfunctions"
Monday 5.7.: Discussion on "Floquet states, quasienergies"
Monday 12.7.: Homework H5, H6

## Example¶

In a typical application one would compute the phonon modes separately as those need very different convergence settings. (phonon.py)

The corresponding calculation of the effective potential changes can be done simultaneously. (elph.py)

The last line in the above script constructs the electron-phonon matrix in terms of LCAO orbitals (and cell repetitions) and saves it as elph.supercell_matrix.dzp.pckl .

After both calculations are finished the final electron-phonon matrix can be constructed with a ‘simple’ script. (construct_matrix.py)

Class for calculating the electron-phonon coupling in an LCAO basis.

The derivative of the effective potential wrt atomic displacements is obtained from a finite difference approximation to the derivative by doing a self-consistent calculation for atomic displacements in the +/- directions. These calculations are carried out in the run member function.

The subsequent calculation of the coupling matrix in the basis of atomic orbitals (or Bloch-sums hereof for periodic systems) is handled by the calculate_matrix member function.

Initialize with base class args and kwargs.

atoms (Atoms) – The atoms to work on.

calc (GPAW) – Calculator for the supercell finite displacement calculation.

supercell (tuple, list) – Size of supercell given by the number of repetitions (l, m, n) of the small unit cell in each direction.

name (str) – Name to use for files (default: ‘elph’).

delta (float) – Magnitude of displacements.

calculate_forces (bool) – If true, also calculate and store the dynamical matrix.

Zero matrix element inside/beyond the specified cutoffs.

This method is not tested.

cutmax (float) – Zero matrix elements for basis functions with a distance to the atomic gradient that is larger than the cutoff.

cutmin (float) – Zero matrix elements where both basis functions have distances to the atomic gradient that is smaller than the cutoff.

Calculate el-ph coupling in the Bloch basis for the electrons.

This function calculates the electron-phonon coupling between the specified Bloch states, i.e.:

In case the omega_ql keyword argument is not given, the bare matrix element (in units of eV / Ang) without the sqrt prefactor is returned.

Phonon frequencies and mode vectors must be given in ase units.

kpts (ndarray or tuple) – k-vectors of the Bloch states. When a tuple of integers is given, a Monkhorst-Pack grid with the specified number of k-points along the directions of the reciprocal lattice vectors is generated.

qpts (ndarray or tuple) – q-vectors of the phonons.

c_kn (ndarray) – Expansion coefficients for the Bloch states. The ordering must be the same as in the kpts argument.

u_ql (ndarray) – Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the phonons. Again, the ordering must be the same as in the corresponding qpts argument.

omega_ql (ndarray) – Vibrational frequencies in eV.

kpts_from (List[int] or int) – Calculate only the matrix element for the k-vectors specified by their index in the kpts argument (default: all).

spin (int) – In case of spin-polarised system, define which spin to use (0 or 1).

Calculate gradient of effective potential and projector coefs.

This function loads the generated pickle files and calculates finite-difference derivatives.

calculate_supercell_matrix ( dump = 0 , name = None , filter = None , include_pseudo = True ) [source] ¶

Calculate matrix elements of the el-ph coupling in the LCAO basis.

This function calculates the matrix elements between LCAOs and local atomic gradients of the effective potential. The matrix elements are calculated for the supercell used to obtain finite-difference approximations to the derivatives of the effective potential wrt to atomic displacements.

Dump supercell matrix to pickle file (default: 0).

0: Supercell matrix not saved

1: Supercell matrix saved in a single pickle file.

2: Dump matrix for different gradients in separate files. Useful

for large systems where the total array gets too large for a single pickle file. Allows restart.

name (str) – User specified name of the generated pickle file(s). If not provided, the string in the name attribute is used.

filter (str) – Fourier filter atomic gradients of the effective potential. The specified components ( normal or umklapp ) are removed (default: None).

include_pseudo (bool) – Include the contribution from the psedupotential in the atomic gradients. If False , only the gradient of the effective potential is included (default: True).

Fourier filter atomic gradients of the effective potential.

This method is not tested.

V1t_xG (ndarray) – Array representation of atomic gradients of the effective potential in the supercell grid.

components (str) – Fourier components to filter out ( normal or umklapp ).

Calculate the el-ph coupling in the electronic LCAO basis.

For now, only works for Gamma-point phonons.

This method is not tested.

u_l (ndarray) – Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the phonons.

omega_l (ndarray) – Vibrational frequencies in eV.

Load supercell matrix from pickle file.

basis (str) – String specifying the LCAO basis used to calculate the supercell matrix, e.g. ‘dz(dzp)’.

name (str) – User specified name of the pickle file.

Dump supercell matrix to pickle file (default: 0).

0: Supercell matrix not saved by calculate_supercell_matrix

1: Supercell matrix was saved in a single pickle file.

2: Dumped matrix for different gradients in separate files.

Store lcao basis info for atoms in reference cell in attribute.

args (tuple) – If the LCAO calculator is not available (e.g. if the supercell is loaded from file), the load_supercell_matrix member function provides the required info as arguments.

Set LCAO calculator for the calculation of the supercell matrix.

© Copyright 2021, GPAW developers. Last updated on Wed, 07 Jul 2021 06:15:03.

## Conclusion

In summary, our comprehensive evaluation revealed that even state-of-the-art MCMC algorithms have problems to sample efficiently from many posterior distributions arising in systems biology. Problems arose in particular in the presence of non-identifiabilities and chaotic regimes. The examples provided in manuscripts presenting new algorithms are often not representative and a more thorough assessment on benchmark collections should be required (as is common practice in other fields). The presented study provides a basis for future developments of such benchmark collections allowing for a rigorous assessment of novel sampling algorithms. In this study, we already used six benchmark problems with common challenges to provide practical guidelines for the selection of sampling algorithms, adaptation and initialization schemes. Furthermore, the presented results highlight the need to address chain exploration quality by taking into account multiple MCMC runs which can be compared with each other before calculating effective sample sizes. The availability of the code will simplify the extension of the methods and the extension of the benchmark collection.

## 5.5: Exercises- Matrix Methods for Dynamical Systems

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ISSN 1088-6826(online) ISSN 0002-9939(print)

An abstract existence theorem at resonance

Authors: L. Cesari and R. Kannan
Journal: Proc. Amer. Math. Soc. 63 (1977), 221-225
MSC: Primary 47H15
DOI: https://doi.org/10.1090/S0002-9939-1977-0448180-3
MathSciNet review: 0448180
Full-text PDF Free Access

Abstract: By Schauder’s fixed point theorem and alternative method (bifurcation theory) an abstract existence theorem at resonance for operational equations is proved which contains as particular cases rather different existence theorems for ordinary and partial differential equations as those of Lazer and Leach and of Landesman and Lazer.

• Lamberto Cesari, Alternative methods in nonlinear analysis, International Conference on Differential Equations (Proc., Univ. Southern California, Los Angeles, Calif., 1974) Academic Press, New York, 1975, pp. 95–148. MR 0430884
• Lamberto Cesari, Nonlinear oscillations in the frame of alternative methods, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 29–50. MR 0636951 ---, Functional analysis and nonlinear differential equations, Dynamical Systems (Cesari, Kannan and Schuur, editors), Dekker, New York, 1976.
• Lamberto Cesari, An abstract existence theorem across a point of resonance, Dynamical systems (Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., 1976) Academic Press, New York, 1977, pp. 11–26. MR 0467420
• Lamberto Cesari, Nonlinear oscillations across a point of resonance for nonselfadjoint systems, J. Differential Equations 28 (1978), no. 1, 43–59. MR 477909, DOI https://doi.org/10.1016/0022-0396%2878%2990079-7
• Lamberto Cesari, Nonlinear problems across a point of resonance for nonselfadjoint systems, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 43–67. MR 499091
• Djairo Guedes de Figueiredo, The Dirichlet problem for nonlinear elliptic equations: a Hilbert space approach, Partial differential equations and related topics (Program, Tulane Univ., New Orlenas, La., 1974) Springer, Berlin, 1975, pp. 144–165. Lecture Notes in Math., Vol. 446. MR 0437924
• E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609–623. MR 0267269
• A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl. (4) 82 (1969), 49–68. MR 249731, DOI https://doi.org/10.1007/BF02410787
• Stephen A. Williams, A connection between the Cesari and Leray-Schauder methods, Michigan Math. J. 15 (1968), 441–448. MR 236791
• S. A. Williams, A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem, J. Differential Equations 8 (1970), 580–586. MR 267267, DOI https://doi.org/10.1016/0022-0396%2870%2990031-8
L. Cesari, The alternative method in nonlinear analysis, Internat. Conf. on Differential equations (H. Antosiewicz, editor), Academic Press, New York, 1975, pp. 95-148. ---, Nonlinear oscillations in the frame of alternative methods, Internat. Conf. on Dynamical Systems, Vol. 1 (Providence, R.I.), Academic Press, New York, 1976, pp. 29-50. ---, Functional analysis and nonlinear differential equations, Dynamical Systems (Cesari, Kannan and Schuur, editors), Dekker, New York, 1976. ---, An abstract existence theorem across a point of resonance, Internat. Sympos. on Dynamical Systems (Gainesville, Fla., March 24-26, 1976), Academic Press, New York (to appear). ---, Nonlinear oscillations across a point of resonance for nonselfadjoint systems, J. Differential Equations (to appear). ---, Nonlinear problems across a point of resonance for nonselfadjoint systems, Nonlinear Analysis, A volume in honor of E. H. Rothe, Academic Press, New York (to appear). D. G. De Figueiredo, The Dirichlet problem for nonlinear elliptic equations: a Hilbert space approach, Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446, Springer-Verlag, Berlin and New York, 1975, pp. 144-165. E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/70), 609-623. MR 42 #2171. A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura Appl. (4) 82 (1969), 49-68. MR 40 #2972. S. A. Williams, A connection between the Cesari and Leray-Schauder methods, Michigan Math. J. 15 (1968), 441-448. MR 38 #5085. ---, A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem, J. Differential Equations 8 (1970), 580-586. MR 42 #2169.

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H15

## Comprehensive benchmarking of Markov chain Monte Carlo methods for dynamical systems

Background: In quantitative biology, mathematical models are used to describe and analyze biological processes. The parameters of these models are usually unknown and need to be estimated from experimental data using statistical methods. In particular, Markov chain Monte Carlo (MCMC) methods have become increasingly popular as they allow for a rigorous analysis of parameter and prediction uncertainties without the need for assuming parameter identifiability or removing non-identifiable parameters. A broad spectrum of MCMC algorithms have been proposed, including single- and multi-chain approaches. However, selecting and tuning sampling algorithms suited for a given problem remains challenging and a comprehensive comparison of different methods is so far not available.

Results: We present the results of a thorough benchmarking of state-of-the-art single- and multi-chain sampling methods, including Adaptive Metropolis, Delayed Rejection Adaptive Metropolis, Metropolis adjusted Langevin algorithm, Parallel Tempering and Parallel Hierarchical Sampling. Different initialization and adaptation schemes are considered. To ensure a comprehensive and fair comparison, we consider problems with a range of features such as bifurcations, periodical orbits, multistability of steady-state solutions and chaotic regimes. These problem properties give rise to various posterior distributions including uni- and multi-modal distributions and non-normally distributed mode tails. For an objective comparison, we developed a pipeline for the semi-automatic comparison of sampling results.

Conclusion: The comparison of MCMC algorithms, initialization and adaptation schemes revealed that overall multi-chain algorithms perform better than single-chain algorithms. In some cases this performance can be further increased by using a preceding multi-start local optimization scheme. These results can inform the selection of sampling methods and the benchmark collection can serve for the evaluation of new algorithms. Furthermore, our results confirm the need to address exploration quality of MCMC chains before applying the commonly used quality measure of effective sample size to prevent false analysis conclusions.

Keywords: Benchmark collection Markov chain Monte Carlo Ordinary differential equation Parameter estimation Sampling analysis Systems biology.

## SIAM Journal on Applied Dynamical Systems

We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high-dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).