Does time-varying magnetic field induce time varying-electric field?

Question

As we all know that Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:
$$text{emf} = -N frac{dPhi}{dt}$$
So if we are applying a time-varying magnetic field(let $dB/dt =$ constant) on a stationary conducting coil then induced electric field across the coil work as a driving force to induce a current in that coil. According to the above formula, induced emf in a coil will be constant if $dB/dt =$ constant, But if the induced electric field is time-varying then induced emf also be time-varying? isn't it?
What I want to say is that I learned somewhere in the past that Statement: "A time-varying electric field can not exist without a corresponding time-varying magnetic field and vice versa", But According to Faraday's law, a linear time-varying magnetic field induces a static electric field, So, is that mean the above statement is wrong?
Or in other words,
Understand with 3 statements written below-
(1) Linearly time-varying electric field {i.e. $dE/dt =$ constant} is capable of inducing static Magnetic field only (not capable of inducing dynamic magnetic field).
(2) Linearly time-varying magnetic field {i.e. $dB/dt =$ constant} is capable of inducing a static electric field only (not capable of inducing dynamic electric field)
(3) "A time-varying electric field can not exist without a corresponding time-varying magnetic field and vice versa"
So, statements (1) and (2) can be understood and verified by
Faraday-Maxwell equation   $$oint Ecdot dl = - frac{dPhi}{dt}$$ Where $Phi =$ magnetic flux, verifies the statement (2), and
Ampere-Maxwell equation   $$oint B.ds =  mu_0I + mu_0epsilon_0 frac{dPhi}{dt}$$ Where $Phi=$ electric flux , verifies the statement (1).
But if statement (3) is Correct then it violates the other two,
Please tell me, about the validation of the 3rd statement.

Dvij D.C. · Accepted Answer

None of the three claims are correct.

A dynamic electric field can obviously exist without $frac{dmathbf{B}}{dt}$ being non-zero. In fact, it can exist without even $mathbf{B}$ being non-zero. The Faraday-Maxwell equation only implies that the curl of the electric field would be zero without a magnetic field. A dynamic electric field can exist without a magnetic field if the current density is non-zero as can be seen by the Ampere-Maxwell equation. For an explicit counter-example, see this post and Section $18.2$ from the link therein. 
A dynamic magnetic field can obviously exist without $frac{dmathbf{E}}{dt}$ being non-zero. A dynamic magnetic field simply requires the curl of the electric field to be non-zero, as can be seen by the Faraday-Maxwell equation. 
The third claim is doubly incorrect for it's simply the intersection of the first two claims.

Harish Chandra Rajpoot · Answer

It depends on how magnetic field $B$ or magnetic flux $Phi$ varies with time $t$ i.e. linearly varying or non-linearly varying with time $t$

Case-1: If the magnetic field $B$ is varying linearly with time i.e. $B=at+b$ (Assuming area of coil $A$ is constant with time $t$) then $$frac{dPhi}{dt}=frac{d(Bcdot A)}{dt}=Afrac{dB}{dt}=aA=text{constant}implies text{emf}=text{constant}$$ 
Thus a magnetic field varying linearly with time $t$ induces a constant electric field $E$ as the induced emf is constant.
Case-2: If the magnetic field $B$ is varying non-linearly with time say $B=at^2+bt+c$ (it may also be a sinusoidal function $B=asin(omega t)$ of time $t$) then $$frac{dPhi}{dt}=frac{d(Bcdot A)}{dt}=Afrac{dB}{dt}=A(2at+b)ne text{constant}implies text{emf}ne text{constant}$$ 
Thus a magnetic field varying non-linearly with time $t$ will induce a time-varying electric field $E$ as the induced emf is time-varying i.e. $text{emf}=f(t)$.

Does time-varying magnetic field induce time varying-electric field?

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