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In mathematicsMathieu functions are solutions of Mathieu's differential equation. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.

They are sometimes also referred to as cosine-elliptic and sine-ellipticor Mathieu functions of the first kind. Closely related are the modified Mathieu functionswhich are solutions of Mathieu's modified differential equation. A common normalization,  which will be adopted throughout this article, is to demand. Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory.

The central result is Floquet's theorem :. An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form.

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. See Ince's Theorem above. These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly. Mathieu functions of the first kind can be represented as Fourier series : .

By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. Moreover, in this particular case, an asymptotic analysis  shows that one possible choice of fundamental solutions has the property. There are several ways to represent Mathieu functions of the second kind. A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.

There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.

Examples of identities obtained in this way are . Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.

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There also exist integral relations between functions of the first and second kind, for instance: . Thus, the modified Mathieu functions decay exponentially for large real argument. In addition, the splitting of the characteristic numbers in quantum mechanics called eigenvalues corresponding to even and odd periodic Mathieu functions is calculated from their boundary conditions.

In quantum mechanics this provides the splitting of the eigenvalues into energy bands. Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1 the analysis of partial differential equations in elliptic geometries, and 2 dynamical problems which involve forces that are periodic in either space or time.

Examples within both categories are discussed below. Mathieu functions arise when separation of variables in elliptic coordinates is applied to 1 the Laplace equation in 3 dimensions, and 2 the Helmholtz equation in either 2 or 3 dimensions.In mathematicsMathieu functions are solutions of Mathieu's differential equation.

They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry. They are sometimes also referred to as cosine-elliptic and sine-ellipticor Mathieu functions of the first kind. Closely related are the modified Mathieu functionswhich are solutions of Mathieu's modified differential equation. A common normalization,  which will be adopted throughout this article, is to demand.

Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory. The central result is Floquet's theorem :. An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form.

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. See Ince's Theorem above. These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly. Mathieu functions of the first kind can be represented as Fourier series : . By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index.

Moreover, in this particular case, an asymptotic analysis  shows that one possible choice of fundamental solutions has the property. There are several ways to represent Mathieu functions of the second kind. A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series. There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions.

Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type. Examples of identities obtained in this way are .

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions. There also exist integral relations between functions of the first and second kind, for instance: .Updated 23 Sep Alexey Arefiev Retrieved April 17, Learn About Live Editor. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance.

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Select web site. Plots for real and imaginary parts of the eigenvalues have been added. A stability diagram has been added in addition to the plot of the growth rate.Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems.

Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time.

In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Am Nat — Brassil CE Can environmental variation generate positive indirect effects in a model of shared predation?

Caswell H Matrix population models: construction, analysis, and interpretation, 2nd edn. Sinauer, Sunderland. Chesson P Multispecies competition in variable environments. Theor Popul Biol — Ecol Monogr — Drazin PG Nonlinear systems. Cambridge University Press, Cambridge. SIAM, Philadelphia.

Ferriere R, Gatto M Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Gonzalez A, Holt RD The inflationary effects of environmental fluctuations in source-sink systems.

Math Probl Eng — Grimshaw R Nonlinear ordinary differential equations. CRC, Ann Arbor. Grover JP Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model. Grover JP, Holt RD Disentangling resource and apparent competition: realistic models for plant-herbivore communities.

J Theor Biol — Hastings A Population biology: concepts and models. Springer, Berlin Heidelberg New York. Hastings A, Powell T Chaos in a three-species food chain.

Ecology — Ann Zool Fenn — Koch AL Competition coexistence of two predators utilizing the same prey under constant environmental conditions.Dispersion curves play a relevant role in nondestructive testing. They provide estimations of the elastic and geometrical parameters from experiments and offer a better perspective to explain the wave field behavior inside bodies. They are obtained by different methods.

The Floquet-Bloch theory is presented as an alternative to them. The method is explained in an intuitive manner; it is compared to other frequently employed techniques, like searching root based algorithms or the multichannel analysis of surface waves methodology, and finally applied to fit the results of a real experiment. The Floquet-Bloch strategy computes the solution on a unit cell, whose influence is studied here.

It is implemented in commercially finite element software and increasing the number of layers of the system does not bring additional numerical difficulties. The lateral unboundedness of the layers is implicitly taken care of, without having to resort to artificial extensions of the modelling domain designed to produce damping as happens with perfectly matched layers or absorbing regions.

The study is performed for the single layer case and the results indicate that for unit cell aspect ratios under 0. The method is finally used to estimate the elastic parameters of a real steel slab.

Floquet-Bloch hereafter F-B theory provides a strategy to analyze the behavior of systems with a periodic structure. The mathematical description of these ideas, in the context of quantum mechanics, can be found in [ 34 ]. In the literature dealing with wave propagation problems in mechanical systems the theory is referred to as Floquet-Bloch theory or, simply, Floquet theory.

In layered systems, due to the heterogeneity of the relevant elastic properties, to particular geometric features, or to both, only certain wave modes can physically propagate inside the structure [ 5 ]. Each of these modes can be identified by a determined—generally nonlinear—function relating the time frequency and the spatial frequency or wave number.

These relationships are called dispersion curves default and, as they summarize all the oscillatory behavior of the system, their calculation is of paramount importance in NDE applications [ 6 ]. Vibrations occur also in objects with periodic structure [ 7 ]. These problems usually admit a separation between the time and the spatial dependent parts of the solution.

For instance, the Helmholtz equation is a known example of equation describing the spatial behavior [ 8 ]. There, the physical periodic structure of the studied object translates into spatial periodicity of its coefficients.

Therefore, the F-B theory has been applied to obtain the dispersive properties of different mechanical periodic systems [ 8 — 12 ]. Many relevant structures can be assumed to be layered systems of infinite extent, for example, [ 1314 ] in civil engineering constructions, [ 1516 ] in optics, or [ 17 ] in electromagnetics. Therefore, theoretical methods and experimental techniques to obtain their dispersion curves have been devised.

From the theoretical side, different matrix techniques have been developed to address the calculation. They involve numerical computational methods whose complexity increases with the number of layers in the system [ 1819 ] or more recently [ 6 ]. In laboratory experiments or field work, the dispersion curves can be obtained using, for example, the multichannel analysis of surface waves MASW method. The MASW procedure involves collecting equally spaced measures of vibration along a profile on the system surface using, for example, accelerometers.

The resulting 2D space-time discrete image is Fourier-transformed to the frequency-wave number domain and then processed to build the dispersion curves [ 2021 ].

The method has some drawbacks inherent to the Fourier transform limitations which will be discussed later. The MASW has been applied successfully in the characterization of pavement systems [ 13 ], as a seismic data acquisition technique [ 20 ] or for geotechnical characterization [ 22 ].Exponents arising in the study of solutions of a linear ordinary differential equation invariant with respect to a discrete Abelian group cf.

The simplest example is a periodic ordinary differential equation. The solution called the Floquet solution, or the Bloch solution in physics can be represented as. Floquet solutions play a major role in any considerations involving periodic ordinary differential equations, similar to exponential-polynomial solutions in the constant-coefficient case. This approach to periodic ordinary differential equations was developed by G.

Floquet [a4]. One can find detailed description and applications of this theory in many places, for instance in [a3] and [a7]. One can find discussion of this matter in [a5] and [a2]. Consider now the case of a partial differential equation periodic with respect to several variables.

Transfer of Floquet theory to the case of spatially periodic partial differential equations is possible, but non-trivial. In some cases an equation can be periodic with respect to an Abelian group whose action is not just translation. This does not guarantee periodicity of the equation itself. This group is non-commutative in general [a8]so the standard Floquet theory does not apply. However, under a rationality condition, the group is commutative and a version of magnetic Floquet theory is applicable [a8].

## Floquet exponents Daleckii, M. Krein, "Stability of solutions of differential equations in Banach space"Transl. Eastham, "The spectral theory of periodic differential equations"Scottish Acad. Press [a4] G. Floquet, "Sur les equations differentielles lineaires a coefficients periodique" Ann.

Ecole Norm. Reed, B. Press [a7] V. Yakubovich, V. Zak, "Magnetic translation group" Phys. Encyclopedia of Mathematics. This article was adapted from an original article by P. See original article. Category : TeX done. This page was last modified on 19 Augustat Press Updated 13 Jul This toolbox is aimed at researchers familiar with AUTO, but also engineers that would like to apply these techniques. One of the biggest reasons why Dynamical Systems Theory is not being applied widely in an engineering context, is mainly due to the lack of bifurcation software that integrates with relative ease with existing toolsets.

We hope that it would be useful teaching tool and can help popularise the methods amongst the engineering community. Ample examples are also needed for a person new to the field, hence more aerospace examples will follow in future releases. At this stage we are still in the process of adding several engineering examples to the toolbox. Feel free to develop some examples for inclusion into the toolbox. There are template files that you can use for inclusion of your own examples.

Intel Visual Fortran 9. GCC 4. Extensive use of objects. Can be run in the new mode, or still with all the old AUTO files. Robust error checking. Similar notations to that of AUTO. A person familiar with AUTO should find it straightforward to pick up the new toolsets.

### Floquet-Bloch Theory and Its Application to the Dispersion Curves of Nonperiodic Layered Systems

Ample documentation. Templates files for people willing to contribute their own examples for inclusion into the demos. Limit Cycles are at least an order of magnitude slower. We had to make a trade-off between robustness and speed. We therefore decided that if we want to popularise the methods, then the code should work, and people should not have to struggle with decoding it. No ample enginering examples yet. Download the toolbox and unzip. Run the program installdynasys.

If you have admin rights keep the default values and install. If you do not have admin rights, install the toolbox to a directory where you have access rights. A startup. Type dynasysroot and dynasyshelproot at the command line.

13 Blochs Theorem and Brillouin Zones

If these commands are working, it should indicate where the toolbox components were installed. If not, something has gone wrong. Check that the paths are correctly defined. The Dynamical Systems Toolbox should appear on the menu. If not, either the paths were not defined correctly, or the info. We have only managed to compile on Windows with Intel Fortran 9. Also now possible to use on Linux with gcc 4. I am not able to make many updates because I am trying to finish my PhD, hence assume that the software will not be frequently updated.