A good question.

The simplest I can think of is a servo motor system with a friction term that changes with time (for example as the bearing lubricant warms up) . 

A more complex system is a missile or rocket where the mass decreases as fuel is consumed. 

Voltage controlled/variable gain amplifiers ... which are important in sound mixing and synthesizers.

James and David, many thanks!

One of the most interesting systems to consider is the Turing pattern formation mechanism. It is described briefly at

http://rocs.hu-berlin.de/complex_sys_2016/resources/PatternFormation.pdf

(The most relevant is section 1.3).

The original paper by A. Turing is available from Jstor

http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf

We apply this mechanism in biology, social sciences, economy, earth sciences and many other disciplines. Work with the Turing system is both entertaining and educational.

Janusz, the Turing pattern formation is interesting, but the equations in Section 1.3 are non-linear differential equations, not Linear Time-Varying (LTV). An LTV system can be put into the form 

\dot{x}(t) = A(t) x(t) 

where A is independent of the state x. I do not see how the equations

\dot{u} = u^2 w - u

\dot{w} = \beta - u^2 w

can be easily transformed into this form. Sometimes non-linear systems are approximated by linear time varying systems by linearizing along a solution trajectory. Such an approximation is sometimes called a quasi linear parameter varying (QLPV) system, linear parameter varying systems can be put into the form

\dot{x}(t) = A(p(t)) x(t)

where p(t) is a time-varying parameter. 

Dmitry, James, could in your opinion the equivalent resistance described by (10) here

https://www.researchgate.net/publication/264427786_On_hyperbolic_laws_of_capacitor_discharge_through_self-timed_digital_loads

be called Linear Time-Varying?

Article On hyperbolic laws of capacitor discharge through self-timed...

James,

Thank you for your comments, I agree with your observations.

The general problem considered by Turing in his original paper http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf

is indeed nonlinear (eq. 6.1). All further analysis is performed, however, with the linearized equations describing the system not very far from the equilibrium point (eq. 6.2).

@ Alexander Kushnerov

Dear sir. Which of the resulting differential equation of the final form do you have in mind? Does it govern the changes of  the current, of the voltage or of  some  parameters?   Please, note, that (10) is not a differential equation. Suggestion: consult the above answer by James Whidborne.

Instead of the expression t*exp(-2t), you could use t*exp(-2t/T), where T is a time constant. That would be physically relevant, and could be linearised around certain times t=t0 . 

One usually reduces(=specifies by exclusion) equations with invariance, e.g. scaling of time or translation; meaning that the functional expression do not change at a transformation. The word variant should mean the opposite, and linear variance should (simply) mean linear dependency, but maybe it is meant piecewise linear. Perhaps you could specify the question?

Dear Prof. Domsta,

The equation (10) is just the Ohm's law applied to V given in (4) and I given in (9). Both V and I in turn, are obtained as a solution of the non-linear differential equation (8). We have an electrical, but what is more important switching circuit realizing the hyperbola defined by (4). Imagine you have a big capacitor charged to some initial voltage and a small capacitor that can be in 2 states. 1) it is connected in parallel to the big one 2) it is shorted. If we switch this small capacitor with a fixed frequency, we get the regular (invariant) resistor and the voltage across the big capacitor will drop exponentially. If the paper the switching frequency depends on the voltage across the big capacitor. This gives a hyperbolic decay instead of exponential.    

Dear Dr. Kushnerov,

I wanted to omit the detailed relation between all the quantities, and therefore I have asked just for the differential equation and for the governed quantity. If it were  the voltage  V , then  the equation exposed in the paper has the form of  (9), which is

-- either a linear differential non-homogeneous equation  (the central equality)

-- or nonlinear (equality of the second and the fourth terms in (9) ).

None of them is of  LTV type. Thus, either  the definition of  LTV is not the same as in the answer by James Whidborne or you mean another quantity. Thank you in advance for any explanation of these doubts.

Kind regards, Joachim

Yes, Prof. Domsta, you're right, except of that I regard not (9), but (8), namely

\dot{V}(t) = -wV^2(t)

(w=constant) as a non-linear differential equation. Its solution is V(t)=1/(wt+1), while I(t)=CwV^2(t), and R(t)=V(t)/I(t)=(wt+1)/wC. So, the resistance R(t) is time varying, but simultaneously R(V)=1/wCV, that is R(V) depends not linearly on V. 

On one hand James Whidborne writes: "An LTV system can be put into the form 

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

but on the other hand, I do not know how the solution of this equation looks. Can it be like R(t)=(wt+1)/wC ?

P.S. We need to ask the administration of researchgate to embed Latex (MathJax) here.

Dear Alexander,

1.  You are right:

-- the resistance R is not linearly dependent on V,

-- is time varying.

But this is not the issue. The equation sought should be a linear differential equation for a chosen observed quantity ( i, V , R, C, or whatever else) replacing x in the equation

(*)  d{x(t)}/d{t} = A(t) x(t),   (the LHS denotes the derivative of x with respect to t )

AND  the coefficient A(t) should be independent of anything but t . In your model such a quatity and such an equation  is not distinguished yet.

2. There is also the possibility, that  any other variable is chosen for the independent variable (instead of  t ). Even, one can imagine an equation of the following form d{R(V)}/d{V} = Q(V) R(V) .  I am not developing this point of view.

3. The solution of  (*)  equals   x(t) = x(0) \exp[ \int_0^t \, A(u)\, du ]

4.   Function R(t)=(wt+1)/wC is a solution of the following differential linear non-homogeneous time independent equation:

d{R(t)}/d{t} = 1/C, with the initial condition R(0) = 1/wC

On the other hand, its inverse G=1/R = wC/(wt +1) satisfies the following differential linear homogeneous time varying equation:

d{G(t)}/d{t} = - [ w/(wt+1)] G(t), with initial condition G(0) = wC.

5. Other details are omitted for readability of the message:)

Regards, Joachim

Dear Joachim, dear Alexander,

I believe that the time-variant approximation R(t) proposed in the paper could be considered as a part of a more complex electrical circuit thus resulting in an LTV system. However, I'm not so good in electrical networks to propose a meaningful example.  

Dear Lena,

I'm pretty sure that the expression you proposed could be relevant (a coefficient, whose value is in turn governed by a DE). However, what I'm missing here is a physical context. I'd like to have a real physical system, which (under certain assumptions) can be modelled as an LTV system.

Thank you all for the interesting discussion!

Entirely linear, is the velocity at free fall, in an ideal model.

Invoking air resistance, and other details, (e.g. in the old James Bond movies, they appear to dive and slide in the air layers), it gets nonlinear and nonsmooth.

The present models have a terminal velocity. For example van Warren knows about this, and may provide ref litterature.

Dear Dmitry,

Let me suggest the following simplifying modification of the above examples:

d{V}/d{t} =  - V(t)  / T(t)  for  t >=0,  with  given V(0) , and e.g. periodic  T(t).

This is a model of  discharging a capacitor with intensity dependent on the season (e.g. of the day). Physically, this happens when the inner resistance of the capacitor depends on the outer temperature.  In order to avoid detailed reasoning one uses an a priori assumed form. Let me illustrate this with the assumption  that    T(t) = 1/(2 + cos t ). Then, by separation of variables (or, according to the previously given formula) you can get:

V(t) = V(0) \exp [ - 2t - sint ]

which clearly illustrates the perturbation of the exponential decay  V(0) \exp[ -2t]  by the POSITIVE  factor  \exp [-sin t]  varying between  1/e  and  e . In the next excercise you can demonstrate by the Fourier expansion the harmonics   a(n) cos(nt) + b(n) sin(nt), for every  n . All this is performable without  MathAppl ("by hands", only), which should be a value for itself.

The same equation in the following form

d{R}/d{t}= - A(t) R(t)  for t >=0, with given R(0)=1 ,  and   given  function  A(t)

found applications in the models of  aging, where  R(t) is the reliability , or probability that the modelled object is active at instant  t ,  and  A(t)  is named intensity of aging at  instant  t. Usual functions assumed in such problems  are  increasing polynomials. In the simplest case  A(t) =  a t^{p}   which leads to  R(t) =  \exp [ -  a t^{p+1} / p ]  known as an extended Weibull distribution of the time to the first failure. (More precisely,  not  R  but  F=1 - R  is a distribution function of the time.)

I think the above two examples may help in explaining the role of the time dependence of factors in arbitrary differential equations, not only in the linear ones.

Regards, Joachim

 Alexander, the solution to 

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

for a known initial condition, x(0)=x_0, can be found in many textbooks on dynamic system theory, e.g. Kailath or Zadeh & Desoer, but is not often easy to compute. Unlike linear time invariant systems, there is no general analytical expression that gives the solution. 

Dahleh et al have some nice notes on the MIT Opencourseware site:

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap11.pdf

The nonlinear system in your very interesting paper is (I think) a piece-wise linear system, hence you were able to obtain analytical solutions. 

Dear, Prof. Domsta thank you for providing different points of view on the same task. This will help us to understand better how we should modify our circuit to turn it to a LTV system. The examples from MIT provided by Prof. Whidborne are also quite helpful. We did analysis of an existing circuit. Synthesis seems to be much more difficult. As to non-linear capacitors, whose "inner resistance...depends on the outer temperature", the commercial multilayer ceramic capacitors behave so. Fig. 1 here shows the capacitance change, but ESR (equivalent series resistance) changes also 

https://www.maximintegrated.com/en/app-notes/index.mvp/id/5527