Hahn decomposition theorem

Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to linear differential equations of the form

${\dot {x}}=A(t)x,\,$

with $\displaystyle A(t)$ a piecewise continuous periodic function with period $T$.

The main theorem of Floquet theory, Floquet's theorem, due to Template:Harvs, gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change $\displaystyle y=Q^{-1}(t)x$ with $\displaystyle Q(t+2T)=Q(t)$ that transforms the periodic system to a traditional linear system with constant, real coefficients.

In solid-state physics, the analogous result (generalized to three dimensions) is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix $\phi \,(t)$ is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix $\Phi (t)$ is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists $t_{0}$ such that $\Phi (t_{0})$ is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using $\Phi (t)=\phi \,(t){\phi \,}^{-1}(t_{0})$. The solution of the linear differential equation with the initial condition $x(0)=x_{0}$ is $x(t)=\phi \,(t){\phi \,}^{-1}(0)x_{0}$ where $\phi \,(t)$ is any fundamental matrix solution.

Floquet's theorem

Let ${\dot {x}}=A(t)x$ be a linear first order differential equation, where $x(t)$ is a column vector of length $n$ and $A(t)$ an $n\times n$ periodic matrix with period $T$ (that is $A(t+T)=A(t)$ for all real values of $t$). Let $\phi \,(t)$ be a fundamental matrix solution of this differential equation. Then, for all $t\in \mathbb {R}$,

$\phi (t+T)=\phi (t)\phi ^{-1}(0)\phi (T).\$

Here

$\phi ^{-1}(0)\phi (T)\$

is known as the monodromy matrix. In addition, for each matrix $B$ (possibly complex) such that

$e^{TB}=\phi ^{-1}(0)\phi (T),\$

there is a periodic (period $T$) matrix function $t\mapsto P(t)$ such that

$\phi (t)=P(t)e^{tB}{\text{ for all }}t\in \mathbb {R} .\$

Also, there is a real matrix $R$ and a real periodic (period-$2T$) matrix function $t\mapsto Q(t)$ such that

$\phi (t)=Q(t)e^{tR}{\text{ for all }}t\in \mathbb {R} .\$

In the above $B$, $P$, $Q$ and $R$ are $n\times n$ matrices.

Consequences and applications

This mapping $\phi \,(t)=Q(t)e^{tR}$ gives rise to a time-dependent change of coordinates ($y=Q^{-1}(t)x$), under which our original system becomes a linear system with real constant coefficients ${\dot {y}}=Ry$. Since $Q(t)$ is continuous and periodic it must be bounded. Thus the stability of the zero solution for $y(t)$ and $x(t)$ is determined by the eigenvalues of $R$.

The representation $\phi \,(t)=P(t)e^{tB}$ is called a Floquet normal form for the fundamental matrix $\phi \,(t)$.

The eigenvalues of $e^{TB}$ are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps $x(t)\to x(t+T)$. A Floquet exponent (sometimes called a characteristic exponent), is a complex $\mu$ such that $e^{\mu T}$ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since $e^{(\mu +{\frac {2\pi ik}{T}})T}=e^{\mu T}$, where $k$ is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise.

• Floquet theory is very important for the study of dynamical systems.
• Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.
• Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.

Floquet's theorem applied to Mathieu equation

Mathieu's equation is related to the wave equation for the elliptic cylinder.

Given $a\in \mathbb {R} ,q\in \mathbb {C}$, the Mathieu equation is given by

${\frac {d^{2}y}{dw^{2}}}+(a-2q\cos 2w)y=0.$

The Mathieu equation is a linear second-order differential equation with periodic coefficients.

One of the most powerful results of Mathieu's functions is the Floquet's Theorem [1, 2]. It states that solutions of Mathieu equation for any pair (a, q) can be expressed in the form

$y(w)=F_{\nu }(w)=e^{iw\nu }P(w)\,$

or

$y(w)=F_{\nu }(-w)=e^{-iw\nu }P(-w)\,$

where $\nu$ is a constant depending on a and q and P(.) is $\pi$-periodic in w.

The constant $\nu$ is called the characteristic exponent.

If $\nu$ is an integer, then $F_{\nu }(w)$ and $F_{\nu }(-w)$ are linear dependent solutions. Furthermore,

$y(w+k\pi )=e^{i\nu k\pi }y(w){\text{ or }}y(w+k\pi )=e^{-i\nu k\pi }y(w),\,$

for the solution $F_{\nu }(w)$ or $F_{\nu }(-w)$, respectively.

We assume that the pair (a, q) is such that $|\cosh(i\nu \pi )|<1$ so that the solution $y(w)$ is bounded on the real axis. General solution of Mathieu's equation ($q\in \mathbb {R}$, $\nu$ non-integer) is the form

$y(w)=c_{1}e^{iw\nu }P(w)+c_{2}e^{-iw\nu }P(-w),\,$

where $c_{1}$ and $c_{2}$ are arbitrary constants.

All bounded solutions −those of fractional as well as integral order− are described by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency.

Another very important property of Mathieu's functions is the orthogonality [3]:

If $a(\nu +2p,q)$ and $a(\nu +2s,q)$ are simple roots of

$\cos(\pi \nu )-y(\pi =0)=0,\,$

then:

$\int _{0}^{\pi }F_{\nu +2p}(w)F_{\nu +2s}(-w)\,dw=0,\qquad p\neq s,$

i.e.,

$\langle F_{\nu +2p}(w),F_{\nu +2s}(w)\rangle =0,\qquad p\neq s,$

where <·,·> denotes an inner product defined from 0 to π.

References

• C. Chicone. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999.
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Translation of Mathematical Monographs, 19, 294p.
• W. Magnus, S. Winkler. Hill's Equation, Dover-Phoenix Editions, ISBN 0-486-49565-5.
• N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973. ISBN 978-0-7011-1936-2.

External links

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