Is resonance a general property of second-order differential equations?

I have read at this site as an answer at a question about how antennas work but that is not important

The resonant frequency of an antenna is determined by its constitution. Mathematically speaking, this is a general property of second order differential equations but in down-to-earth terms any AC circuit with some inductors and capacitors in it has a resonant frequency etc etc

What is this general property?

• What is the second derivative of sin(x) or cos(x)? Commented Feb 13, 2023 at 16:59
• @JonCuster i dont understand Commented Feb 13, 2023 at 17:46
• For an equation like ${d^{2} x \over d^{2}t} + x(t) = 0$ finding an answer using sin() and cos() is trivial, immediately pointing to a preferred frequency based on specifics of the problem. Commented Feb 13, 2023 at 17:53
• I don't know the answer, but there's a nice example in infectious disease modeling. The SIR system of (two) first-order ordinary differential equations, describing the dynamics of disease spread (populations of: Susceptible, Infectious, and Recovered individuals) can be shown to have a natural oscillation frequency that depends on the parameters. If one parameter is moduated in time at a frequency close to the natural frequency, a resonance phenomena occurs. This is a very simple explanation for flu seasonality (PNAS 2004). Commented Feb 15, 2023 at 19:26
• Commented Feb 17, 2023 at 5:17

Consider the second order differential equation \begin{align} f'' - \alpha^2 f = C \cos(\omega t) \end{align} with $$\alpha$$ a real constant. (Note the sign of the second term, which makes this equation different from the equation of motion for a driven harmonic oscillator.) The particular solution to the inhomogeneous equation is \begin{align} f_I(t) = -\frac{C}{\alpha^2 + \omega^2}\cos(\omega t) \end{align} Assuming we want "resonance" to mean something like "the amplitude of the system's motion becomes relatively large if the system is driven near a natural frequency scale of the system," this system does not exhibit resonance. The constant $$\alpha$$ provides a natural frequency scale of the undriven (homogeneous) system, but nothing special happens when the driving frequency $$\omega$$ is equal to this natural frequency. Instead, the amplitude of the motion is maximized when $$\omega = 0$$, i.e. for a constant driving force.

The point of this example is to show that no, resonance is not a general feature of second-order differential equations.

• +1 from me. A couple minor comments: The lack of resonance is easier to see when the constant $\alpha$ is real. For imaginary $\alpha$ we do see resonance. Also, it might be helpful to point out that sign of the $\alpha^2 f$ term is opposite of the typical mass-spring case.
– hft
Commented Feb 15, 2023 at 19:40
• @hft Thanks. I wasn't even thinking about the possibility of complex $\alpha$. I have edited to specify this.
– d_b
Commented Feb 15, 2023 at 20:30
• Do you have any similar examples where the motion for the homogeneous equation is guaranteed to be bounded? Currently, your $f_{hom} \sim e^{\alpha t}$ at large positive $t$ for generic initial conditions (a measure zero subset of initial conditions gives a instead decay at large positive $t$). Commented Feb 20, 2023 at 21:43
• @user196574 I don't, and my suspicion is such examples will oscillate and will tend to exhibit resonance. I would also be interested to see an example of a second-order system with bounded solutions that doesn't show some sort of resonance.
– d_b
Commented Feb 20, 2023 at 23:23
• @d_b Actually, now that I think about it, the generic case with bounded solutions probably doesn't suffer large resonances. The reason is that if the amplitude of oscillation grows enough, the drive will become off-resonant since the resonant frequency is generically a function of the amplitude of the periodic motion. Commented Feb 21, 2023 at 19:47

Surely resonances are not a property of second-order differential equations in general. The night is dark and dreary and even if we assumed the class of smooth non-linear ODEs for a single unknown function, we would struggle to even understand what is meant by your question.

But this is not Mathematics StackExchange, this is Physics StackExchange and I get what you mean. You are asking, whether second-order ODEs representing physical dynamical systems are generally subject to resonant phenomena when some sort of driving term is applied. To this question the answer is yes, at least as far as the system in question is stable, autonomous, the equations of motion (evolution equations) are continuous, and the dissipation is weak in a sense that will be specified later.

Conservative systems

Generally speaking, an autonomous system of a single degree of freedom can be expressed by evolutionary equations of the form $$\ddot{y} = f(\dot{y},y)$$ where $$f(\dot{y},y)$$ is some smooth, but generally nonlinear function and dots stand for time derivatives. (The autonomy of the system is expressed by the fact that $$f$$ is independent of $$t$$.) Much of the behaviour of this system is characterized by the Poincaré-Bendixson theorem.

The system in question will either be conservative or dissipative. A conservative system means that there is some energy function $$E(\dot{y},y)$$ such that $$\dot{E} = 0$$. Such a system can be turned into a first-order equation inverting the energy function as $$\dot{y} = \dot{y}(E,y)$$. If this system is stable for some set of initial conditions, this means that it stays in some bounded region of $$y$$ at all times. The theory of dynamical system tells you that this motion is also necessarily periodic (an oscillation) with some frequency $$\Omega(E)$$ (which does not need to be the same for all amplitudes of oscillation/energies).

Weakly dissipative systems

Now to finish setting the stage, we also briefly mention weak dissipation. Dissipation means, at least for a stable system that stays in some finite range of $$y$$, that the system evolves to a finite set of limiting points (equilibria, very often just a single one). Weak dissipation means that we can write the equations of motion as a sum of a conservative term and a dissipative term $$\ddot{y} = f_{\rm c}(\dot{y},y) + f_{\rm d}(\dot{y},y)$$ such that the frequency $$\Omega$$ of the unperturbed conservative system $$\ddot{y} = f_{\rm c}$$ decays on some long dissipative time-scale $$\tau_{\rm diss.} = \frac{\Omega}{d\Omega/dt}$$ Now the system is weakly dissipative if $$\tau_{\rm diss.}$$ is longer than the characteristic period of the motion $$\tau_{\rm diss.} \gtrsim 1/\Omega$$ or $$\frac{\Omega^2}{d \Omega/d t} \gtrsim 1$$ Note that this criterion might evaluate differently for different initial conditions, so the dynamical system can be weakly or strongly dissipative only in some parts of the phase space.

The meaning of the weak dissipation criterion is that the system can be understood as a reasonably slowly damped oscillator. In return, there even is a reasonable notion of an oscillation frequency of the system in question. On the other hand, when and if the dissipative time-scale is shorter than the period of the original system, the oscillation gets damped on the scale of less than a period and it is physically meaningless to even talk about a frequency of the system (and thus resonance).

Driving and resonance

Finally, let us talk about driving forces. The driving can be written as $$\ddot{y} = f_{\rm c} + f_{\rm d} + \epsilon F(\dot{y},y,t)$$ where it is assumed that $$\epsilon F/f_{\rm c}$$ is small anywhere of interest and that it is periodic with a frequency $$\Omega_{\rm drive}$$ in $$t$$. Now the response of the system to this driving force can be captured by perturbation theory and will generally scale with $$\epsilon$$ in the $$\epsilon \to 0$$ limit. However, very generally the perturbation theory will diverge in a $$\propto \sqrt{\epsilon}$$ neighborhood of parts of phase space where $$\Omega = \Omega_{\rm drive}$$ or even $$\Omega = q/r \Omega_{\rm drive}$$, where $$q,r \in \mathbb{N}$$. This is related to the so-called small-divisors problem in analytical mechanics (see also the Kolmogorov-Arnold-Moser theorem). In this neighborhood the system is forced to co-oscillate in some sense with the driving frequency. In the case of dissipative systems, it gets "locked" with the driving frequency for a time $$\propto \sqrt{\epsilon}$$. This all comes from rather general considerations about the system.

If you would like to learn more about the methods leading to these conclusions, I recommend the 2012 book Multiple scale and singular perturbation methods by Kevorkian & Cole.

• +1 Amazing answer, much better than mine and by far the best one so far! Thanks for pointing out KAM. Commented Feb 20, 2023 at 20:56

The term "resonance" is used in physics when a dynamical system, having periodic oscillations at some frequencies $$a_i$$, shows a marked response when subject to a forcing containing frequencies near at least one of the $$a_i$$ (ser this for some intuition on the phenomenon). The prototypical example is that of the forced harmonic oscillator.

"Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts." From Wikipedia.

Understanding if an ordinary differential equation allows for oscillating solutions is the aim of oscillation theory. Therefore, resonance is not a general property of second-order differential equations.

Simple second-order systems - Examples of systems that show no resonance ($$a$$ indicates the natural frequency or "inverse timescale" of the system): $$\ddot{x}(t) = a^2 \, x(t) \quad \rightarrow \quad x(t)=c_1 e^{a \,t} + c_2 e^{-a\,t}$$ $$\ddot{x}(t) = -a^2 \, x(t) \quad \rightarrow \quad x(t)=c_1 \sin(a \,t) + c_2 \cos(a \,t)$$ ...there is no resonance simply because there is no applied force! Similarly, $$\ddot{x}(t) = f(t)$$ shows no resonance as there is no natural frequency $$a$$. A minimal model for resonance is the forced harmonic oscillator: $$\ddot{x}(t) = -a^2 \, x(t) +A\cos(b t) \quad \rightarrow \quad x(t)=c_1 \sin(a \,t) + c_2 \cos(a \,t)+\frac{A \cos(b t)}{a^2-b^2}$$ ... you see that the denominator goes to zero for $$a= \pm b$$, when the natural and forcing frequencies are the same. Note that if we take $$a\rightarrow i a$$, namely $$a^2 \rightarrow - a^2$$, in the above system, we have: $$\ddot{x}(t) = a^2 \, x(t) +A\cos(b t) \quad \rightarrow \quad x(t)=c_1 e^{a \,t} + c_2 e^{-a\,t} -\frac{A \cos(b t)}{a^2+b^2}$$ Again, this system does not show any resonance (the denominator is always larger than zero).

Nonlinear systems - Nonlinear resonance, namely the occurrence of resonance in a nonlinear system, is also possible (but, again, not a general property). Unfortunately, nonlinearity hinders the use of the Fourier transform as a tool to study the frequency response of the system. Finally, there is also the phenomenon of stochastic resonance: adding white noise to a nonlinear system can boost a weak signal since the noise frequencies resonate with the signal ones, see also this question.

First-order systems - Finally, how about first-order systems? Typically, a first-order system will not oscillate unless the forcing function is also oscillating or if you are over the complex field, e.g. $$x'(t)=ix(t)$$. For the sake of completeness, let's consider the equation: $$\dot{x}(t) = i a \, x(t) +A\cos(b t) \quad \rightarrow \quad x(t) = -\frac{A b \sin(b t)}{a^2 - b^2} + \frac{i a A \cos(b t)}{a^2 - b^2} + c_1 e^{i a t}$$ ...this shows resonance for $$a\approx \pm b$$. Remove the $$i$$ in front of $$a$$ in the above equation to kill the possibility of resonance. See also this MathSe question for resonance in first-order systems.

Oscillation theory - An introduction to oscillation theory is given in section 5.5 of Ordinary Differential Equations and Dynamical Systems by G. Teschl, see also The Tiler's answer.

We put the equation in the form $$\frac{d}{dx}\left(K(x)\frac{dy}{dx} \right)+G(x)y=0\;\;\;\;\;(1)$$
with $$\;\;K(x)=p(x)\;,\;G(x)=q(x)-\lambda w(x)$$

The coefficients $$K(x)$$ and $$G(x)$$ in the equation being supposed to be continuous and bounded in the interval $$a\leq x \leq b$$, let the upper bounds of $$K(x)$$ and $$G(x)$$ in this interval be $$\mathbf{k}$$ and $$\mathbf{g}$$ and their lower bounds $$k$$ and $$g$$ respectivel.

Thus, throughout $$(a, b)$$: $$\mathbf{k}\geq K(x)\geq k>0\\\mathbf{g}\geq G(x)\geq g$$ In particular, a sufficient condition that the equation $$(1)$$ should possess a solution which oscillates in $$(a, b)$$ is that $$\frac{\mathbf{k}}{\mathbf{g}}\geq\frac{\pi^{2}}{(b-a)^{2}}$$

Demonstration (I changed a sign to have the same form of the equation on the two links I gave)

Theorem: If $$r(x)$$ is a continuous function on a closed and finite interval $$a\leq x \leq b$$ and if $$r(x)\leq 0$$ on this interval, all the solutions of the equation $$y''+r(x)y=0\;\;\;\;\;\;(2)$$ do not oscillate in this interval (Demonstration , page 99) (*)

The general type equation $$y''+p(x)y'+q(x)x=0$$ can be put in the form $$(2)$$ by performing the substitution for $$y(x)$$ of the new sought function $$u(x)$$ $$y(x)=e^{-\frac{1}{2}\int p(x)} \;u(x)$$ which gives $$r(x)=q(x)-\frac{1}{4}[p(x)]^{2}-\frac{1}{2}p'(x)$$ The general theorem that deals with this question ( oscillation ) is the theorem of Sturm–Picone.

(*) The English version does not contain this part.

• Commented Feb 18, 2023 at 14:05
• Thanks for pointing out the Sturm–Picone result, interesting stuff! This is also a simple but useful result: en.wikipedia.org/wiki/… Commented Feb 19, 2023 at 14:45
• Quillo: Thank you also for reporting another result that is related to the subject, I think this result has something to do with the condition I mentioned $r(x)=q(x)-\frac{1}{4} p^{2}-\frac{1}{2}p '$ Commented Feb 19, 2023 at 15:33

Resonance is associated with maximizing something. Consider the system defined by:

$$y''(t) + 2 \xi \omega_0 y'(t) +\omega_0^2 y(t) = x(t)$$

Now, let your input be a simple sinusoidal, i.e for simplicity $$x(t) = X_0 \exp(j\omega t)$$. We are supposing that we left the transient and the system behaves under the stable state response. In this case, that solution is given by $$y_{ss}(t) = K \exp(j \omega t)$$ (is an eighenfunction of the system with $$K$$ its eighenvalue). Now if you replace in the ODE you will found $$K$$ and you conclude that:

$$y_{ss}(t) = \frac{X_0\exp(j \omega t)}{\omega_0^2 - \omega^2 + j 2 \xi \omega_0 \omega}$$

But now you have to ask yourself: "¿What do I want to maximize?":

1. If you want to maximize the amplitude, just do:

$$|y(t)| = \frac{|X_0 \exp(j \omega t)|}{|\omega_0^2 - \omega^2 + j 2 \xi \omega_0 \omega|} = \frac{X_0}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2 \xi \omega_0 \omega)^2}}$$

Now, you know that function maximizes when the denominator minimizes, so find $$\omega_a$$ such that:

$$\frac{d}{d\omega} ((\omega_0^2 - \omega_a^2)^2 + (2 \xi \omega_0 \omega_a)^2) = 0 \quad \implies \quad \omega_a = \omega_0 \sqrt{1 - 2\xi^2}$$

And since $$\omega_a \in \Re+$$, you need $$\xi < \sqrt{1/2}$$ (even not all underdamped systems will have this resonance). If that occurs, then $$\omega_a$$ is your amplitude resonance frequency. If it looks weird that I used a complex exponential, this just was with the purpose to make the process faster. But you could take the real part $$RE(y(t))$$ with the same result.

1. If you want to maximize the power, then the frequency depends on the definition of power you use. If it's a mass-spring system and $$x(t)$$ is an external force and $$y(t)$$ is the position of the object, we could define $$x(t) = F(t) / m$$, where $$m$$ is the mass of the object, then the instant power could be:

$$p(t) = m \cdot RE(x(t)) \cdot \frac{d}{dt} \:RE(y(t)) = F_0 \cos(\omega t) \cdot \frac{d}{dt} \frac{X_0\cos(\omega t - \delta)}{|K|} = \frac{F_0}{m} \cos(\omega t) \frac{-\omega X_0 \sin(\omega t - \delta)}{|K|}$$

Now, compute the averege power $$

= \frac{1}{T} \int_0^T p(t)dt$$

with $$T = \frac{2\pi}{\omega}$$, you should end with:

$$

= \frac{\omega F_0²}{m|K|} (sin(\delta) \cdot \frac{1}{2} - cos(\delta) \cdot 0) = \frac{F_0^2}{4 \xi \omega_0 m} \frac{(2\xi \omega_0 \omega)²}{(\omega_0 - \omega)² + (2\xi \omega_0 \omega)²}$$

¿Where does this function maximize? At $$\omega_p = \omega_0$$, it doesn't matter if is underdamped or overdamped. That is the power resonance frequency

Conclusions:

1. Finding the resonance frequency respect to some magnitude means finding the frequency that maximizes that magnitude.
2. Despite that most of the times isn't specified explicitly with respect of which magnitude, the resonant frequency is always resonant respect to some specific magnitude.
3. Depending on the system, you will find that you have to put some restriction for the existence of a certain type of resonance. As I showed, amplitude resonance exists for some underdamped systems. Power resonance exists for almost all the second order systems, since you will find that it is proportional to the derivative of $$RE(y)$$ in most of the cases. There's some cases where there doesn't exist a resonance like in a parallel RLC circuit, beacuse at $$\omega = \omega_0$$ the $$LC$$ part opens and the power will minimize, instead of maximizing. This is why it's called a blocking frequency instead of a resonance frequency. But even in this case $$\omega_0$$ does mean a critical point for the power dissipation of the system.

P. S. You could try to analyze the system with Lapalce/Fourier and see if you reach to the same conclusions

Mathematically, resonance (not resonant)* frequency is not a general property of second order differential equations. A resonance frequency corresponds to an eigenvalue of a set of linear equations (or matrix equation). The corresponding eigenvector is known as "mode shape" in physical systems. If a physical system can be represented by a set of vectors, then the mode shapes are basis of that vector space. This means a linear combination of those mode shapes (eigen vectors) can reproduce any state of that physical system. The resonance frequencies (eigenvalues) are frequencies at which each of the mode shapes occur.

*Resonant frequency is used to describe when a system is being driven at a resonance frequency.

The differential equation in time domain is:

$$\ddot x+\omega_0^2\,x(t)=A\,\cos(\Omega\,t)$$

to obtain the "Amplitude Resonanz" we transfer the Ode to Laplace domain

$$X(s)\left(s^2+\omega_0^2\right)=A\,\frac{s}{s^2+\Omega^2}\quad\Rightarrow$$ $$\frac{X}{A}=\frac{1}{s^2+\omega_0^2}\frac{s}{s^2+\Omega^2} \tag 1$$

from here with $$~s\mapsto i\,\omega~$$ to frequency domain $$~(\omega=2\,\pi\,f~)$$

$$\frac{X(i\omega)}{A}=\Re(...)+i\Im(...)$$ $$\left|\frac{X}{A}\right|= \left| {\frac {\omega}{ \left( {\omega}^{2}-{{\omega_0}}^{2} \right) \left( -{\omega}^{2}+{\Omega}^{2} \right) }} \right|$$

thus in frequency domain you obtain two "Amplitude Resonanz" for $$~\omega=\omega_0~$$ and $$~\omega=\Omega~$$