A Physical–Mathematical Approach to Climate Change Effects Through Stochastic Resonance

Article A Physical–Mathematical Approach to Climate Change Effects through Stochastic Resonance

Maria Teresa Caccamo 1 and Salvatore Magazù 2,3,*

1 Consiglio Nazionale delle Ricerche (CNR)—Istituto per i Processi Chimico-Fisici (IPCF), Viale Ferdinando Stagno D’Alcontres n◦37, S. Agata, 98166 Messina, Italy; [email protected] 2 Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, Viale Ferdinando Stagno D’Alcontres n◦31, S. Agata, 98166 Messina, Italy 3 Istituto Nazionale di Alta Matematica “F. Severi”—INDAM—Gruppo Nazionale per la Fisica Matematica—GNFM, P.le Aldo Moro 5, 00185 Roma, Italy * Correspondence: [email protected]

Received: 22 November 2018; Accepted: 24 January 2019; Published: 27 January 2019

Abstract: The aim of this work is to study the effects induced by climate changes in the framework of the stochastic resonance approach. First, a wavelet cross-correlation analysis on Earth temperature data concerning the last 5,500,000 years is performed; this analysis confirms a correlation between the planet’s temperature and the 100,000, 41,000, and 23,000-year periods of the Milankovitch orbital cycles. Then, the stochastic resonance model is invoked. Specific attention is given to the study of the impact of the registered global temperature increase within the stochastic model. Further, a numerical simulation has been performed, based on: (1) A double-well potential, (2) an external periodic modulation, corresponding to the orbit eccentricity cycle, and (3) an increased value of the global Earth temperature. The effect of temperature increase represents one of the novelties introduced in the present study and is determined by downshifting the interaction potential used within the stochastic resonance model. The numeric simulation results show that, for simulated increasing values of the global temperature, the double-well system triggers changes, while at higher temperatures (as in the case of the absence of a global temperature increase although with a different threshold) the system goes into a chaotic regime. The wavelet analysis allows characterization of the stochastic resonance condition through the evaluation of the signal-to-noise ratio. On the basis of the obtained findings, we hypothesize that the global temperature increase can suppress, on a large time scale corresponding to glacial cycles, the external periodic modulation effects and, hence, the glacial cycles.

Keywords: stochastic resonance model; climate change effects; temperature increasing; simulation

1. Introduction Climate occurs as a result of processes operating on multiple scales, some of which are slow, as in the Earth’s glacial cycles, and others of which are fast, such as daily weather ﬂuctuations [1]. Glacial cycles, on the order of tens of thousands of years, start with a gradual temperature decrease, which gives rise to an increase of sea ice, polar cap volume, and total global area occupied by ice, increasing Earth’s overall albedo and the amount of the sun’s energy reﬂected away from the Earth. With less energy entering the Earth’s system, temperatures decrease further, creating positive feedback. Furthermore, as more ocean water is converted to ice, the overall water volume of the oceans decreases, allowing continental submerged portions to emerge [2–5]. The equilibrium global average temperature reached by this process represents a lower stationary regime of the overall climate during that period. Glacial ages end with a temperature increase reversing the water transfer from the cryosphere back to

Climate 2019, 7, 21; doi:10.3390/cli7020021 www.mdpi.com/journal/climate Climate 2019, 7, x FOR PEER REVIEW 2 of 16

stationary regime. On the other hand, climate variations can give rise to harsh conditions that are of interest for the study of extremophile organisms [10–13] for biotechnological applications [14–22]. ClimateTemperature2019, 7, 21 determinations performed at the Vostok scientific station (Figure 1) revealed2 of 16a Climatestrong 2019, correlation7, x FOR PEER with REVIEW the Milankovitch cycles [23–32], and are shown in Figure 2 2 of 16 stationarythe oceans regime. [6–9 ].On Equilibrium the other temperaturehand, climate at thesevariatio timesns representscan give rise an upperto harsh stationary conditions regime. that On are of the other hand, climate variations can give rise to harsh conditions that are of interest for the study of interest for the study of extremophile organisms [10–13] for biotechnological applications [14–22]. extremophile organisms [10–13] for biotechnological applications [14–22]. Temperature determinations performed at the Vostok scientific station (Figure 1) revealed a Temperature determinations performed at the Vostok scientiﬁc station (Figure1) revealed a strong strongcorrelation correlation with with the Milankovitch the Milankovitch cycles cycles [23–32], [23–32], and are and shown are in shown Figure 2in. Figure 2

Figure 1. Temperature as a function of time during the last 5,500,000 years [31].

In particular, the cycles correspond to: (1) an eccentricity variation of the Earth’s orbit around the Sun over a period of about 100,000 years; (2) a variation in the inclination of the Earth’s axis from a minimum of 21°55′ to a maximum of 24°20′, over a period of approximately 41,000 years; and (3) a variation of the Earth’s axis during a double-conic motion, called solar-precession, over a period of about 23,000 years [33–41]. FigureFigure 1. Temperature 1. Temperature as as a afunction function of time duringduring the the last last 5,500,000 5,500,000 years years [31]. [31].

FigureFigure 2. 2. ThreeThree Milankovitch Milankovitch cycles. cycles.

SeveralIn particular, mathematical the cycles procedures correspond are to: reported (1) an eccentricity in the literature variation to ofcorrelate the Earth’s in order orbit to around compare the differentSun over data a period sets through of about a 100,000 spectral years; comparison. (2) a variation Wavelet in transform the inclination (WT) ofanalysis the Earth’s is powerful axis from tool a forminimum analysing of 21variations◦550 to a maximumof spectral ofpower 24◦20 within0, over given a period data of series. approximately By decomposing 41,000 years; data andsets (3)into a waveletvariation components, of the Earth’s it axisis possible during to a double-coniccompare the dominant motion, called signal solar-precession, spectral modes overas well a period as their of correlationabout 23,000 degree. years [ 33In– 41particular,]. in order to highlight the correlation between the registered temperaturesSeveral mathematical of Figure 1 and procedures the mentioned are reported Milankovitch in the literature periodicities, to correlate in this in work order the to comparewavelet cross-correlationdifferent data sets (XWT) through approach a spectralFigure has comparison. been 2. Three applied Milankovitch Wavelet [42–44]. transform Furthermore, cycles. (WT) analysis these glacial is powerful cycles toolcan forbe treatedanalysing as variations a slow process of spectral or power low-frequency within given clim dataate series. “signal”. By decomposing At this geological data sets scale, into wavelet daily temperaturecomponents,Several mathematical fluctuations it is possible procedures can to compare be thought are the dominantreportedof as a fa signalinst process,the spectralliterature or modesa much to correlate as higher-frequency well as in their order correlation to signal compare differentsuperimposeddegree. data In sets particular, on through the glacial in a order spectral signal. to highlight comparison.Because the the correlation variation Wavelet in betweentransform solar radiation the (WT) registered over analysis the temperatures Milankovitch is powerful of tool for analysingcyclesFigure alone1 and variations is the not mentioned able of to spectral justify Milankovitch temperature power within periodicities, variation given from indata this interglacial series. work theBy age waveletdecomposing to glacial cross-correlation age, data in 1980, sets into Roberto(XWT) approach Benzi introduced has been appliedthe idea [ 42that–44 the]. Furthermore, high-frequency these temperature glacial cycles fluctuations can be treated were as a aweak slow wavelet components, it is possible to compare the dominant signal spectral modes as well as their periodicprocess orinput low-frequency and a source climate of “noise” “signal”. on the Atglacial this geologicalsignal, creating scale, a dailysmall temperatureresonance that fluctuations amplified correlation degree. In particular, in order to highlight the correlation between the registered thecan signal be thought of the oflower as a glacial fast process, frequency. or a muchHe termed higher-frequency this hypothesis signal “stochas superimposedtic resonance” on the (SR) glacial [45– temperatures52].signal. Figure Because of3 shows Figure the variationa typical1 and curv inthe solar ementioned of radiation output performance overMilankovitch the Milankovitch as a periodicities,function cycles of input alone in noisethis is not workmagnitude. able tothe justify wavelet cross-correlationtemperature variation (XWT) approach from interglacial has been age applied to glacial [4 age,2–44]. in 1980, Furthermore, Roberto Benzi thes introducede glacial cycles the idea can be treatedthat as the high-frequencya slow process temperature or low-frequency fluctuations clim wereate a weak “signal”. periodic At input this and geological a source ofscale, “noise” daily temperatureon the glacial fluctuations signal, creating can be a small thought resonance of as that a fa amplifiedst process, the signalor a ofmuch the lower higher-frequency glacial frequency. signal superimposedHe termed on this the hypothesis glacial signal. “stochastic Because resonance” the variation (SR) [45 –in52 ].solar Figure radiation3 shows over a typical the Milankovitch curve of cyclesoutput alone performance is not able asto a justify function temperature of input noise variation magnitude. from interglacial age to glacial age, in 1980, Roberto Benzi introduced the idea that the high-frequency temperature fluctuations were a weak periodic input and a source of “noise” on the glacial signal, creating a small resonance that amplified the signal of the lower glacial frequency. He termed this hypothesis “stochastic resonance” (SR) [45– 52]. Figure 3 shows a typical curve of output performance as a function of input noise magnitude.

ClimateClimate 20192019, ,77, ,x 21 FOR PEER REVIEW 3 3of of 16 16

FigureFigure 3. 3. TypicalTypical curve curve of of output output performance performance as as aa fu functionnction of of input input noise noise magnitude, magnitude, for for systems systems capablecapable of of stochastic stochastic resonance. resonance.

2.2. Methodology Methodology 2.1. Wavelet Analysis 2.1. Wavelet Analysis In order to highlight the correlation between the registered temperatures of Figure1 and the In order to highlight the correlation between the registered temperatures of Figure 1 and the mentioned Milankovitch periodicities, and to characterize the SR condition, in the present work we mentioned Milankovitch periodicities, and to characterize the SR condition, in the present work we shall evaluate the wavelet cross-correlation (XWT). It is well known that wavelet tranform (WT) allows shall evaluate the wavelet cross-correlation (XWT). It is well known that wavelet tranform (WT) the evaluation of signal frequency and time localization. From a general viewpoint, WT assigns allows the evaluation of signal frequency and time localization. From a general viewpoint, WT to function y(x) another function WT(a, τ), where a, with a > 0 denotes the scale and τ a time 𝑦(x) 𝑊𝑇(𝑎, 𝜏), 𝑎 𝑎0 τ assigns to function another function where , with 1 x −denotesτ the scale and shift [53–57]. Thus, WT decomposes y(x) into wavelets components √ ψ , as follows: a time shift [53–57]. Thus, WT decomposes 𝑦(𝑥) into wavelets componentsa a 𝜓 , as follows: √ Z +∞ 11 ∗ 𝑥−𝜏x − τ WT𝑊𝑇(a,(𝑎,τ) 𝜏)==√ 𝑦y(𝑥x)𝜓ψ∗ 𝑑𝑡dt (1) a −∞ a (1) √𝑎 𝑎 wherewhere 𝜓ψ∗ is is thethe complexcomplex conjugateconjugate ofof thethe waveletwavelet mothermother ψ𝜓[ 58[58–61].–61].

GivenGiven two two signals signals 𝑦(𝑥)y(x) andand 𝑔(𝑥)g(x), let us us indicate indicate 𝑊𝑇WTy((𝑎,a, τ 𝜏)) andand 𝑊𝑇WTg((𝑎,a, ) 𝜏τ )asas their their WTs, WTs, respectively.respectively. It It is is possible possible to to define define the the cross-correlation cross-correlation wavelet wavelet (XWT) (XWT) as: as: ∗ XWT = 𝑊𝑇∗ (𝑎, 𝜏)𝑊𝑇(𝑎, 𝜏) (2) XWT = WTy (a, τ)WTg(a, τ) (2) while the wavelet spectrum (WS) is: while the wavelet spectrum (WS) is: ( ) | ( )| 𝑊𝑆 𝑎 =Z +∞ 𝑊𝑇 𝑎, 𝜏 𝑑𝑡 (3) WS(a) = |WT(a, τ)|2dt (3) − Since XWT allows to evaluate the degree of∞ affinity between two signals 𝑦(𝑥) and 𝑔(𝑥), in order Sinceto extract XWT allowsquantitative to evaluate information the degree from of affinitythe signal, between one twocan signalscalculatey( xthe) and waveletg(x), incross- order correlationto extract quantitativecoefficient (CXWT): information from the signal, one can calculate the wavelet cross-correlation coefficient (CXWT): R 𝑊𝑇(a, τ) CXWT = WTyg(a, τ) (4) CXWT = q𝑊𝑆(𝑎)𝑊𝑆(𝑎) (4) WSy(a)WSg(a) where 𝑊𝑆(𝑎) and 𝑊𝑆(𝑎) are the wavelet spectrum of the signal 𝑦(𝑥) and 𝑔(𝑥), respectively. In orderwhere toWS showy(a) theand effectivenessWSg(a) are theof the wavelet WT approach, spectrum ofthe the XWT signal betweeny(x) and theg (signalx), respectively. and the function In order to show the effectiveness of the WT approach, the XWT between the signal and the function obtained obtained as a sum of three in phase sinusoidal contributions with periodicities 𝑇 = , 𝑇 = 2π 2π as a sum of three in phase sinusoidal contributions with periodicities T = , T = and and 𝑇 = has been evaluated. In such a case, XWT, as reportedM1 in100000 literature,M2 allows41000 us 2π TM3 = has been evaluated. In such a case, XWT, as reported in literature, allows us to highlight to highlight23000 existing correlations, both in terms of spectral power and of relative phase, in the time–

Climate 2019, 7, 21 4 of 16 Climate 2019, 7, x FOR PEER REVIEW 4 of 16 existingfrequency correlations, space. Figure both 4 in reports terms the of spectralresults of power such a and XWT of relativeevaluation phase, that inshows the time–frequencya noticeable level space.of correlation Figure4 reports which theincreases results for of suchthe most a XWT recent evaluation times [62–65]. that shows a noticeable level of correlation which increases for the most recent times [62–65].

FigureFigure 4. 4.Wavelet Wavelet cross-correlation cross-correlation (XWT) (XWT) between between the the temperature temperature signal signal and and the the function function obtained obtained asas a non-weighteda non-weighted sum sum of of three three in in phase phase sinusoidal sinusoidal terms terms with with periodicities periodicitiesT 𝑇= =2π , T, 𝑇= =2π M1 100000M2 41000 = 2π 3 andandTM 𝑇3 =23000 . Period. Period and and time time are are expressed expressed in in 10 10years.3 years. The arrows in the ﬁgure indicate the phase relationship between the two series. The colour bar The arrows in the figure indicate the phase relationship between the two series. The colour bar indicates the strength of the wavelet power. The redder it appears in the scalogram, the more power indicates the strength of the wavelet power. The redder it appears in the scalogram, the more power the spectra contains at that particular scale. In the following the WT approach will be employed to the spectra contains at that particular scale. In the following the WT approach will be employed to characterize the SR condition. characterize the SR condition. 2.2. Stochastic Resonance 2.2. Stochastic Resonance Let us consider now a system that can be modelled as a particle of mass m whose position is x(t) (particleLet us spaceconsider coordinate), now a system moving that incan a be potential modelledV(x as) anda particle on which of mass a friction m whose force position is exerted is x(t) ( ) proportional(particle space to its coordinate), velocity, i.e., movingf =in −a ζpotentialdx t (Stokes V(x) force), and on and which a random a friction force, forcef is= exertedξ(t), f riction dt () random proportional to its velocity, i.e., 𝑓 = −𝜁 (Stokes force), and a random force, 𝑓 = connected with the “heat bath” in which the particle moves. The particle motion equation is: 𝜉(𝑡), connected with the “heat bath” in which the particle moves. The particle motion equation is: d2x(t) dV(x) dx(t) m 𝑑𝑥=(𝑡)− 𝑑𝑉(−𝑥)ζ 𝑑𝑥+(𝑡)ξ(t) (5) dt𝑚2 = −dx −𝜁dt +𝜉(𝑡) (5) 𝑑𝑡 𝑑𝑥 𝑑𝑡 ( ) TheThe term termξ t𝜉(𝑡makes) makes this this differential differential equation equation astochastic a stochastic one. one. We We assume assume the the particle particle moves moves inin the the “high “high friction friction limit” limit” where where the the acceleration acceleration is is negligible, negligible, and and absorbing absorbing the the constant constantγ 𝛾into into the the ( ) potentialpotential function functionV (V(x)x) and and the the random random force forceξ 𝜉t(𝑡, one), one has: has: dx(t𝑑𝑥) (𝑡) dV(𝑑𝑉x)(𝑥) = −= − + ξ+𝜉(t) (𝑡) (6)(6) dt 𝑑𝑡 dx 𝑑𝑥 SRSR is is termed termed a phenomenona phenomenon that that occurs occurs in in nonlinear nonlinear systems systems whereby whereby a weaka weak signal signal is is ampliﬁed amplified andand optimized optimized by by the the assistance assistance of of noise. noise. TheThe effect effect requires requires three three basic basic ingredients: ingredients:

(1) a(1) non-linear a non-linear system system characterized characterized by an energetic by an energetic activation activation barrier or,barrier more or, generally, more generally, by a form by a formof threshold; of threshold; (2) a(2) weak a input;weak input; (3) a source of noise that is inherent in the system, or that adds to the input.

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(3) a source of noise that is inherent in the system, or that adds to the input.

Given these features, the response of the system undergoes to a resonance-like behaviour as a function of the noise level. An example of signal performance parameter is furnished by the output signal-to-noise ratio (SNR). In particular, the SNR against input noise amplitude/intensity, as reported in Figure3, shows a single maximum at a nonzero value similar to frequency-dependent systems that have a maximum SNR, or output response, for some resonant frequency. However, in the case of SR, the resonance is “noise-induced” rather than at a particular frequency [66,67]. It should be stressed that for linear signal-processing systems, the output SNR is maximized in the absence of noise; therefore, noise is not beneficial in a linear system, since only the interactions between nonlinearities and randomness can lead to SR [45–48]. In its classical setting, SR applies to bistable nonlinear dynamical systems subject to noise and periodic forcing [68,69]. Taking into account the Langevin equation, a generic equation form for these types of systems is: dx dV = − + ξ(t) + ε cos(ω t + ϕ) (7) dt dx 0 where the periodic forcing term, having a pulsation ω0, is given by ε cos(ω0t + ϕ); for simplicity, in the following we assume that the phase shift ϕ = 0. To figure out what SR is, it can be useful to consider the canonical example first put forth by McNamara et al. [70] by introducing the symmetric bistable potential well given by: a b V(x) = − x2 + x4 (8) 2 4 q a This potential shows two stable equilibrium points at x± = ±c = ± b , and an unstable equilibrium at x = 0. Let V0 be the height of the potential when ε = 0; V1 = εc be the amplitude of the modulation in barrier height when the periodic force is turned on; and ξ(t) be a zero mean Gaussian white noise with intensity D and an autocorrelation function. A delta function, δ(t), is given by:

< ξ(t)ξ(0) ≥ 2Dδ(t) (9)

When no periodic force is present, i.e., when ε = 0, the behaviour of the system is characterized by a x(t) function which, depending on the involved parameter values, can ﬂuctuate around the minima with a variance proportional to D, making random noise-induced hops from one well to the other. As we shall see, under speciﬁc circumstances, the introduction of a periodic force makes the transition probability of escaping from each well varying in time.

2.3. Slow and Fast Processes of Earth’s Energy Balance Following the Budyko–Sellers model [71–75], the Earth’s temperature time variation can be evaluated by taking into account the energy balance between the incoming (Rin), and outcoming (Rout), radiation: dT c = R − R (10) dt in out where c is the Earth thermal capacity. By Rin and Rout, one can express Equation (18) in terms of a temperature-dependent potential V:

dT ∂V c = − (11) dt ∂T This implies that the stationary solutions of this equation are the minima or maxima of the potential V, occurring in correspondence to stable and to unstable configurations, respectively. It should be noted that Equation (19) does not take into account fast processes connected to meteorological fluctuations due to atmospheric dynamics and circulation (of the order of months or years). We can Climate 2019, 7, 21 6 of 16 consider these fast variables as noise acting on the slow variables (varying on a time scale of the order of thousands of years); under such circumstances, the slow variable is the temperature, whose overall variation is considered over a time window of million years. Finally, including the fast processes, the equation becomes [45–48]: 1 ∂V √ dT = − dt + αdη(t) (12) c ∂T where α represents the intensity of the stochastic noise and η(t) is a stochastic function describing the short time scale processes connected with atmospheric dynamics and circulation. To simulate the behaviour of such an equation, at first we neglect the stochastic noise and build ∂V ∂T , parameterizing Rin and Rout; after this we estimate the intensity of meteorological noise. To take into account the effect of the periodic variation of the eccentricity that varies the solar constant, we use for Rin the following formula:

Rin = Q(1 + A cos ωt) (13) where Q is the solar constant, A is the amplitude of the modulation that as calculated by Milankovi´c, is 5 × 10−4 and corresponds to an overall variation of Q of 0.1% over a period of about 11 years, and ω = 2π/T with T = 100,000 years. The outgoing radiation, Rout, can be parameterized as the sum of two contributions, one corresponding to the reﬂected solar radiation and one corresponding to the component radiated from the Earth according to its temperature, i.e., E(T) = 4πR2σT4:

Rout = µ(T)Rin + E(T) (14) with µ(T) coefﬁcient representing the fraction of incident radiation reﬂected in all directions known as albedo [4]. Replacing Equations (21) and (22) into Equation (18), it follows that:

dT c = Q(1 + A cos ωt)(1 − µ(T)) − E(T) (15) dt Now, we examine some qualitative properties of Equation (15), basing our remarks on the well-established mathematics of Budyko–Sellers models. dT Assuming that A = 0, Equation (15) becomes c dt = Q(1 − µ(T)) − E(T). Let us compare these two contributions in Figure5. Here the dotted-dashed green line represents the black body law, while the continuous pink line represents absorbed radiation, that is, the solar constant modulated by the effect of the albedo. The intersections between the two curves define the equilibrium points of the system in which, by definition dT/dt = 0, while their stability is determined by the potential second derivatives. Let us assume that T3 is the actual planet average temperature, which is about 288.6 K; furthermore, be T1 = 278.6 K and T2 = 283.3 K the glacial state and the average temperature respectively, the stationary solutions will be determined by the condition Q(1 − µ(T)) = E(T). We introduce the ( ) = Q(1−µ(T)) − ( ) = = function γ T E(T)−1 1, defined by the conditions γ T 0 when T Ti with i = 1, 2, 3. Then, in the temperature range under investigation, γ(T) can be modelled as γ(T) = β 1 − T 1 − T 1 − T , where β is a constant parameter. Then, by inserting the obtained T1 T2 T3 quantities into Equation (15), which takes into account the external periodic perturbation, one obtains:

dT E(T) T T T = = β 1 − 1 − 1 − (1 + A cos ωt) + A cos ωt (16) dt c T1 T2 T3 where the potential (see Equation (20) for a comparison) is a time-dependent function. Climate 2019, 7, 21 7 of 16

Finally, by adding the noise contribution [45–48] to Equation (16):

dT E(T) T T T √ Climate 2019=, 7, x FOR =PEERβ REVIEW1 − 1 − 1 − (1 + A cos ωt) + A cos ωt + αdη(t) 7 of(17) 16 dt c T1 T2 T3 it is possible to study, through Equation (17), the total system response.

Figure 5. Sketch of incoming energy flux (continuous pink line) and radiated energy flux (dotted-dashed green line) as a function of the average planet temperature for the energy balance Figuremodel. 5. The Sketch intersections of incoming between energy the fl twoux (continuous curves identify pink theline) system and radiated stationary energy configurations, flux (dotted- of dashedwhich T green1 and line) T3 are as stable, a function while of T the2 is average unstable. planet temperature for the energy balance model. The

intersections between the two curves identify the system stationary configurations, of which T1 and Before proceeding, it is possible to evaluate the temperature variation in the absence of noise i.e., T3 are stable, while T2 is unstable. for α = 0; in particular, for numerical values hEi = 8.77 × 10−3 W and c = 4000 J/K, a variation of 0.3 KThen, (compatible in the with temperature the results ofrange other under investigations). investigation, Evidently, 𝛾(𝑇) this can result be ismodelled not compatible as 𝛾(𝑇 with) = the observed variation of 10K. Therefore, one has to conclude that the only periodical perturbation is 𝛽1− 1− 1− , where 𝛽 is a constant parameter. Then, by inserting the obtained not sufficient for the system to pass from one stable state to the another one (i.e., from T3 to T1 and vice quantitiesversa); furthermore, into Equation the astronomical (15), which modulationstakes into account move thethe curves external reported periodic in Figureperturbation,5 very little, one obtains:and therefore the fixed points are kept [45–48]. What emerges𝑑𝑇 is that𝐸(𝑇) from the simulation𝑇 of Equation𝑇 𝑇 (17) and varying the noise, for certain values = =𝛽1− 1− 1− (1+𝐴 cos 𝜔𝑡) + 𝐴 cos 𝜔𝑡 (16) of α the jump from𝑑𝑡 T3 to T𝑐1 and vice versa𝑇 occurs𝑇 in sync𝑇 with the frequency (resonance condition) of the periodic signal; therefore, this result is able to reproduce the characteristic periodicity of the ice where the potential (see Equation 20 for a comparison) is a time-dependent function. ages and the corresponding temperature difference. Furthermore, the noise value is compatible with Finally, by adding the noise contribution [45–48] to Equation (16): the current temperature fluctuations around the climatic average which is about 0.1 K2/year [45–48]. 𝑑𝑇 𝐸(𝑇) 𝑇 𝑇 𝑇 The transition between= the=𝛽1− two stable1− configuration1− rests (1+ is a𝐴 probabilisticcos 𝜔𝑡) + 𝐴 event;cos 𝜔𝑡 this + √ implies𝛼𝑑𝜂(𝑡) that if(17) the transition between𝑑𝑡 the𝑐 two states does𝑇 not occur,𝑇 the𝑇 system has to wait for another cycle to obtain the itsame is possible resonant to condition.study, through Equation (17), the total system response. Before proceeding, it is possible to evaluate the temperature variation in the absence of noise i.e., 3. Results for 𝛼=0; in particular, for numerical values 〈𝐸〉 = 8,77 x 10 W and c = 4000 J/K, a variation of 0.3K (compatibleIn the following with the results we will of present other investigations). at first the outputs Evidently, of a numerical this result simulation is not compatible investigation with the for observeda system characterizedvariation of 10K. by Therefore, a double well one potentialhas to conc tolude which that is the added only aperiodical small periodic perturbation contribution; is not sufficientthe study for is performed the system by to takingpass from into one account stable different state to weightsthe another for theone noise. (i.e., from Then, T3 to to simulate T1 and vice the versa); furthermore, the astronomical modulations move the curves reported in Figure 5 very little, and therefore the fixed points are kept [45–48]. What emerges is that from the simulation of Equation (17) and varying the noise, for certain values of 𝛼 the jump from T3 to T1 and vice versa occurs in sync with the frequency (resonance condition) of the periodic signal; therefore, this result is able to reproduce the characteristic periodicity of the ice ages and the corresponding temperature difference. Furthermore, the noise

Climate 2019, 7, x FOR PEER REVIEW 8 of 16 value is compatible with the current temperature fluctuations around the climatic average which is about 0.1 K2/year [45–48]. The transition between the two stable configuration rests is a probabilistic event; this implies that if the transition between the two states does not occur, the system has to wait for another cycle to obtain the same resonant condition.

3. Results In the following we will present at first the outputs of a numerical simulation investigation for a system characterized by a double well potential to which is added a small periodic contribution; the Climatestudy2019 is performed, 7, 21 by taking into account different weights for the noise. Then, to simulate8 of 16 the effects of the temperature rise in the framework of the stochastic resonance approach, we will show the results of a simulation for a system characterized by a down-shifted double well potential. effects of the temperature rise in the framework of the stochastic resonance approach, we will show Let us first introduce a simple double-well potential which now includes the small periodic the results of a simulation for a system characterized by a down-shifted double well potential. contribution: Let us first introduce a simple double-well potential which now includes the small periodic contribution: 𝑉(𝑥,) 𝑡 =𝑥 −𝑥 + 𝑎𝑥 cos(𝜔𝑡) (18) V(x, t) = x4 − x2 + ax cos(ωt) (18) Then Equation (6) can be written as: Then Equation (6) can be written as: 𝜕𝑉 𝑑𝑥 =− 𝑑𝑡 + 𝛼𝑑𝜂(𝑡) (19) ∂V √ √ dx = − 𝜕𝑥dt + αdη(t) (19) ∂x where 𝛼 and 𝜂(𝑡), describe the stochastic noise. As above stated, depending on the 𝛼 values, thiswhere latter αcontributionand η(t), describe can amplify the stochastic the Milankovic noise. As cycle above signal. stated, The depending simulation on the of αEquationvalues, this (19) has beenlatter performed contribution by varying can amplify the theintensity Milankovic of the cycle noise signal. while The keeping simulation the ofperiodic Equation signal (19) has constant. been As a result,performed we observe by varying that the there intensity is a certain of the noise noise while value, keeping i.e., thea value periodic for signal𝛼, for constant. which the As ajump result, of the systemwe observe from one that therehole isto a the certain other noise is value,synchronized i.e., a value with for α ,the for whichfrequency the jump of the of the external system fromsignal as one hole to the other is synchronized with the frequency of the external signal as reported in Figure6 reported in Figure 6 (i.e., the so-called resonance condition). More precisely, Figure 6 reports the (i.e., the so-called resonance condition). More precisely, Figure6 reports the solution of Equation (19) 𝑎 𝛼 = 0.10, 0.25, 0.4 𝛼 = 0.10, solutionfor a =of 0.3 Equation and α = 190.10, for 0.25, = 0.400.3 and, 0.55. For α = 0.10, the system0, 0.55. is For confined in thethe left system well while is confined for in theα = left0.25 wellit jumps while to for the right𝛼 = well, 0.25 it starting jumps a to periodic the right oscillation well, starting between a the periodic two wells; oscillation for α = 0.40between the andtwo αwells;= 0.55, for the 𝛼 resonance = 0.40 and efficiency 𝛼= 0.55, decreases. the resonance efficiency decreases.

Figure 6. Solution of Equation (19) for a = 0.3 and α = 0.10, 0.25, 0.40, 0.55. In abscissa the displacement Figure 6. Solution of Equation (19) for 𝑎 = 0.3 and 𝛼 = 0.10, 0.25, 0.4 0, 0.55. In abscissa the x(t) in arbitrary units and in ordinate time steps is reported. For α = 0.10, the system is confined in the displacement x(t) in arbitrary units and in ordinate time steps is reported. For 𝛼 = 0.10,the system is left well, while for α = 0.25 it jumps to the right well, starting a periodic oscillation between the two 𝛼 = 0.25 confinedwells; forin theα = left0.40 well, and αwhile= 0.55, for the resonance it efficiency jumps to decreases. the right well, starting a periodic oscillation between the two wells; for 𝛼 = 0.40 and 𝛼=0.55, the resonance efficiency decreases. In order to characterize the stochastic resonance condition, we have evaluated the WT scalograms (seeIn Figureorder7 )to relative characterize to the solutions the stochastic of Equation reso (19)nance for a condition,= 0.3 and α =we0.10 have(a), 0.25evaluated(b), 0.40 the(c) WT scalogramsand 0.55 (d).(see Figure 7) relative to the solutions of Equation (19) for 𝑎 = 0.3 and 𝛼= 0.10 (𝑎),0.25(𝑏),0.4It should be noticed,0 (c) and as 0.55 shown (d). in figure, that the transition from one stable state to the other one is a probabilistic event and hence, also under the resonance condition, the system jump between the

two states is not certain; furthermore, when that the transition between the two states does not occur, the system has to wait for another cycle to get the same resonant condition. ClimateClimate2019 2019,,7 7,, 21x FOR PEER REVIEW 99 of 1616

Figure 7. WT scalograms of the solutions of Equation (19) for a = 0.3 and α = 0.10 (a), 0.25(b), 0.40 (c) andFigure 0.55 7. ( dWT). In scalograms abscissa the of simulationthe solutions in timeof Equation steps and (19) in for ordinate 𝑎 = 0.3 periods and 𝛼=0.10 is reported.( 𝒂),0.25(𝒃),0.40 (c) and 0.55 (d). In abscissa the simulation in time steps and in ordinate periods is reported. Climate 2019, 7,Furthermore, x FOR PEER REVIEW by means of WT it is possible to extract quantitatively the value of the signal-to-noise10 of 16 ratio,It shown should in be Figure noticed,8. What as shown it emerges in figure, is that that the SNRthe transition shows a maximum from one stable for a = state 0.3 and to theα = other0.25 one(b). is a probabilistic event and hence, also under the resonance condition, the system jump between the two states is not certain; furthermore, when that the transition between the two states does not occur, the system has to wait for another cycle to get the same resonant condition. Furthermore, by means of WT it is possible to extract quantitatively the value of the signal-to- noise ratio, shown in Figure 8. What it emerges is that the SNR shows a maximum for 𝑎 = 0.3 and 𝛼 = 0.25 (𝑏) .

Figure 8.Figure Signal-to-noise 8. Signal-to-noise ratio ratio(SNR) (SNR) versus versus noise noise level level for for 𝑎a = = 0.30.3 as as obtained obtained from from the the data data reported reported in Figurein 7. Figure 7.

In the following we shall investigate in this scenery the outcomes of the stochastic model for increasing values of the global planet temperature. More specifically, it is well known that temperature determinations in the Earth’s atmosphere, in oceans, and on our planet’s surface indicate an increased warming process starting from the nineteenth century [76,77]. Figure 9 shows the global temperature trend which shows how, starting from 1970, the slope of the regression line is undergoing a substantial increase, which brings the heating values between 1970 and 2017 to about 0.17 °C/decade. It is therefore meaningful to investigate which effects can be generated by such a global temperature rise in the framework of the above discussed SR model.

Figure 9. Global surface temperature trend from 1880 to 2017 (Data: NASA Goddard Institute for Space Studies. Dataset accessed 14 June 2018 at https://data.giss.nasa.gov/gistemp/).

In order to simulate the effects of the temperature rise in the framework of the stochastic resonance approach, we have performed a new numerical simulation for a system characterized by a down-shifted double-well potential to which a small periodic contribution is added, i.e., 𝑉(𝑥,) 𝑡 = 𝑥 −𝑥 −𝑉 + 𝑎𝑥 cos(𝜔𝑡); again the study is performed by taking into account different weights for the noise.

Climate 2019, 7, x FOR PEER REVIEW 10 of 16

Figure 8. Signal-to-noise ratio (SNR) versus noise level for 𝑎 = 0.3 as obtained from the data reported in Figure 7.

In theClimate following2019, 7, 21 we shall investigate in this scenery the outcomes of the stochastic10 ofmodel 16 for increasing values of the global planet temperature. More specifically, it is well known that temperature determinationsIn the following wein the shall Earth’s investigate atmosphere, in this scenery in oceans, the outcomes and on of our the stochasticplanet’s surface model for indicate an increasedincreasing warming values process of the global starting planet from temperature. the nine Moreteenth speciﬁcally, century it [76,77]. is well known Figure that 9 temperatureshows the global temperaturedeterminations trend which in the Earth’sshowsatmosphere, how, starting in oceans, from and on1970, our planet’sthe slope surface of indicate the regression an increased line is undergoingwarming a substantial process starting increase, from which the nineteenth brings century the heating [76,77]. values Figure9 betweenshows the global1970 and temperature 2017 to about 0.17 °C/decade.trend which It is shows therefore how, startingmeaningful from 1970, to theinve slopestigate of the which regression effects line is can undergoing be generated a substantial by such a increase, which brings the heating values between 1970 and 2017 to about 0.17 ◦C/decade. It is global temperaturetherefore meaningful rise in the to investigateframework which of effectsthe above can be di generatedscussed by SR such model. a global temperature rise in the framework of the above discussed SR model.

Figure 9. Global surface temperature trend from 1880 to 2017 (Data: NASA Goddard Institute for Space Figure 9. Studies.Global Datasetsurface accessed temperature 14 June 2018trend at https://data.giss.nasa.gov/gistemp/from 1880 to 2017 (Data: NASA). Goddard Institute for Space Studies. Dataset accessed 14 June 2018 at https://data.giss.nasa.gov/gistemp/). In order to simulate the effects of the temperature rise in the framework of the stochastic resonance approach, we have performed a new numerical simulation for a system characterized by a down-shifted In order to simulate the effects of the temperature rise in the framework 4of the2 stochastic double-well potential to which a small periodic contribution is added, i.e., V(x, t) = x − x − V0 + resonanceax approach,cos(ωt); again we thehave study performed is performed a new by taking numerical into account simulation different weightsfor a system for the noise.characterized by a down-shiftedFigure double-well 10 shows thepotential solution to of which Equation a small (19) for periodicV0 = 2 when contribution the amplitude is added, of the externali.e., 𝑉(𝑥,) 𝑡 = 𝑥 −𝑥 −𝑉periodic + 𝑎𝑥 perturbation cos(𝜔𝑡); again is a = 0.3the and study the noiseis performed level is α = by0.10, taking 0.40, and into 0.80. account On the different top of the ﬁgureweights for the noise. the potential double-well is shown, while on the respective bottom, the particle position (horizontal axis) as a function of time (vertical axis) is shown. As can be seen, for α = 0.10 (case a), the system is conﬁned in the left well; however, for α = 0.40 (case b), the particle jumps from the one well to

the other one in synchrony with the external perturbation a cos(ωt); a further noise increase, i.e., at α = 0.8, (case c), gives rise to a diminishing of the resonance efﬁciency. Climate 2019, 7, x FOR PEER REVIEW 11 of 16

Figure 10 shows the solution of Equation 19 for 𝑉 = 2 when the amplitude of the external periodic perturbation is 𝑎 = 0.3 and the noise level is α = 0.10, 0.40, and 0.80. On the top of the figure the potential double-well is shown, while on the respective bottom, the particle position (horizontal axis) as a function of time (vertical axis) is shown. As can be seen, for α = 0.10 (case a), the system is confined in the left well ; however, for α=0.40 (case b), the particle jumps from the one well to the Climateother2019 one, 7in, 21 synchrony with the external perturbation a cos(ωt); a further noise increase, i.e., 11at ofα= 16 0.8, (case c), gives rise to a diminishing of the resonance efficiency.

FigureFigure 10. 10.Solution Solution of of Equation Equation (19) (19) for fora 𝑎= 0.3 = 0.3 and andα =𝛼0.10, = 0.10, 0.19, 0.19, 0.40,0, 0.40.55. 0.55. For Forα =𝛼0.10, = 0.10,thethe system system is conﬁnedis confined in the in leftthe wellleft well (a) while (a) while for α =for0.19 𝛼it =jumps 0.19 it tojumps the right to the well right (b). Forwellα (=b).0.40 Forand 𝛼α = 0.4=00.55 and, the𝛼= resonance0.55, the resonance efﬁciency efficiency decreases decreases (c). (c).

Figure 11a shows a comparison between the two analysed conﬁgurations, i.e., for V0 = 0 on the Figure 11a shows a comparison between the two analysed configurations, i.e., for 𝑉 = 0 on the Climateleft 2019 and, 7V, 0x =FOR 2 onPEER the REVIEW right for a = 0.3. In particular while for V0 = 0 the jump threshold occurs12 of 16 at a left and 𝑉 = 2 on the right for 𝑎 = 0.3. In particular while for 𝑉 = 0 the jump threshold occurs at a noise level of α = 0.21, for V0 = 2 the jump threshold occurs at α = 0.19. noise level of 𝛼 = 0.21, for 𝑉 = 2 the jump threshold occurs at 𝛼= 0.19.

Figure 11. Comparison between the two analysed conﬁgurations, i.e., for V0 = 0 on the top and V0 = 2 Figure 11. Comparison between the two analysed configurations, i.e., for 𝑉 = 0 on the top and 𝑉 =

2 onon the the bottom bottom when when the the amplitude amplitude of of the the external external periodic periodic perturbation perturbation is 𝑎 is =a 0.3.= 0.3. As Ascan can be beseen, seen, while for V0 = 0 the threshold jump occurs at a noise level of α = 0.21, for V0 = 2 the threshold jump while for 𝑉 = 0 the threshold jump occurs at a noise level of 𝛼 = 0.21,for 𝑉 = 2 the threshold jump α = occursoccurs at at𝛼 = 0.19.0.19. The present study clearly shows that the registered planet temperature increase could give rise The present study clearly shows that the registered planet temperature increase could give rise to effects similar to those connected with a high noise value. Therefore, in the framework of the SR to effects similar to those connected with a high noise value. Therefore, in the framework of the SR approach, the global temperature increase corresponds to a downshift of the system potential energy, which in turn can justify the registered increase in the frequency of the planet extreme weather events, such as extreme heat, intense precipitation, and drought, which have become more frequent and severe in recent decades. In particular, the evidence of global climate change based on the analysis of the long-term variability that affects the different components of the climate system (atmosphere, oceans, cryosphere, etc.) indicates unequivocally that since the nineteenth century the planet has undergone a warming process. In fact, in recent decades, soil and ocean temperatures have increased not homogeneously, but with greater warming near polar latitudes, especially in the Arctic region. The ice and snow coverage has decreased in different areas of the planet. The atmospheric water vapor content has increased everywhere due to the higher temperature of the atmosphere while sea level has also increased. Furthermore, changes in other climatic indicators have also been observed, such as seasonal elongation in many places on the planet, often accompanied by an increase in the tendency of extreme events such as heat waves or heavy rainfall and a general decrease in extreme cold events. Most of the warming observed globally over the last 50 years, as well as the increase of the registered extreme events, could be explained if one considers the effect produced by human activities and in particular both the emissions from fossil fuels and the deforestation process in the framework of the stochastic resonance effect in the presence of augmented thermal energy on the planet.

Climate 2019, 7, 21 12 of 16 approach, the global temperature increase corresponds to a downshift of the system potential energy, which in turn can justify the registered increase in the frequency of the planet extreme weather events, such as extreme heat, intense precipitation, and drought, which have become more frequent and severe in recent decades. In particular, the evidence of global climate change based on the analysis of the long-term variability that affects the different components of the climate system (atmosphere, oceans, cryosphere, etc.) indicates unequivocally that since the nineteenth century the planet has undergone a warming process. In fact, in recent decades, soil and ocean temperatures have increased not homogeneously, but with greater warming near polar latitudes, especially in the Arctic region. The ice and snow coverage has decreased in different areas of the planet. The atmospheric water vapor content has increased everywhere due to the higher temperature of the atmosphere while sea level has also increased. Furthermore, changes in other climatic indicators have also been observed, such as seasonal elongation in many places on the planet, often accompanied by an increase in the tendency of extreme events such as heat waves or heavy rainfall and a general decrease in extreme cold events. Most of the warming observed globally over the last 50 years, as well as the increase of the registered extreme events, could be explained if one considers the effect produced by human activities and in particular both the emissions from fossil fuels and the deforestation process in the framework of the stochastic resonance effect in the presence of augmented thermal energy on the planet.

4. Conclusions Climate change phenomena connected to the registered global increase of temperature within the framework of the SR approach are analysed by means of XWT and numeric simulations and then discussed. In particular, a XWT analysis was performed to highlight the correlation between the planet’s temperature and the 105-, 41 × 103-, and 23 × 103-year periodicities of the Milankovitch orbital cycles. Then, the well-known SR model applied to the interpretation of climate cycles is discussed. In this framework, the WT analysis allows us to characterize the stochastic resonance condition through the evaluation of the signal-to-noise ratio. Furthermore, a speciﬁc study of the feedbacks induced by a global temperature increase on SR previsions has been performed by taking into account a downshift of the system’s potential well within the SR model. In particular, a mathematical simulation was performed taking into account a double-well potential, an external periodic modulation corresponding the Milankovitch’s cycle, and increasing values of thermal energy values for the planet. The study puts into evidence different triggering between the two regimes and suggests that the higher Earth temperature values, mainly attributed to human activities, could induce an increase both in the number and intensity of extreme weather events. Future directions of study will concern the global warming of the Earth’s climate system due to human activities, i.e., global warming, when triggered by internal forcing mechanisms, i.e., periodic internal parameter variations.

Author Contributions: Formal analysis, M.T.C. and S.M.; Writing—original draft, M.T.C. and S.M.; Writing—review & editing, M.T.C. and S.M. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Benzi, R. Stochastic resonance: From climate to biology. Nonlinear Proc. Geophs. 2010, 17, 431–441. [CrossRef] 2. Jacka, T.H.; Budd, W.F. Detection of temperature and sea-ice-extent changes in the Antarctic and Southern Ocean, 1949–1996. Ann. Glaciol. 1998, 27, 553–559. [CrossRef] 3. Caccamo, M.T.; Castorina, G.; Colombo, F.; Insinga, V.; Maiorana, E.; Magazù, S. Weather forecast performances for complex orographic areas: Impact of different grid resolutions and of geographic data on heavy rainfall event simulations in Sicily. Atmos. Res. 2017, 198, 22–33. [CrossRef] Climate 2019, 7, 21 13 of 16

4. Grinsted, A.; Moore, J.C.; Jevrejeva, S. Reconstructing sea level from paleo and projected temperatures 200 to 2100 AD. Clim. Dyn. 2010, 34, 461–472. [CrossRef] 5. Calabrò, E.; Magazù, S. Correlation between increases of the annual global solar radiation and the ground albedo solar radiation due to desertiﬁcation-A possible factor contributing to climatic change. Climate 2016, 4, 64. [CrossRef] 6. Monnin, E.; Indermuhle, A.; Dallenbach, A.; Fluckiger, J.; Stauffer, B.; Stocker, T.F.; Raynaud, D.; Barnola, J.-M.

Atmospheric CO2 Concentrations over the Last Glacial Termination. Science 2001, 291, 112–114. [CrossRef] [PubMed] 7. Nakamura, K.; Doi, K.; Shibuya, K. Estimation of seasonal changes in the ﬂow of Shirase Glacier using JERS-1/SAR image correlation. Polar Sci. 2007, 1, 73–84. [CrossRef] 8. Davey, M.C. Effects of continuous and repeated dehydration on carbon ﬁxation by bryophytes from the maritime Antarctic. Oecologia 1997, 110, 25–31. [CrossRef] 9. Caccamo, M.T.; Calabrò, E.; Cannuli, A.; Magazù, S. Wavelet Study of Meteorological Data Collected by Arduino-Weather Station: Impact on Solar Energy Collection Technology”. MATEC Web Conf. 2016, 55, 02004. [CrossRef] 10. Raper, S.C.; Wigley, T.M.; Jones, P.D.; Salinger, M.J. Variations in surface air temperatures: Part 3. The Antarctic, 1957–1982. Mon. Weather Rev. 1984, 112, 1341–1353. [CrossRef] 11. Babalola, O.O.; Kirby, B.M.; Le Roes-Hill, M.; Cook, A.E.; Cary, S.C.; Burton, S.G.; Cowan, D.A. Phylogenetic analysis of actinobacterial populations associated with Antarctic Dry Valley mineral soils. Environ. Microb. 2009, 11, 566–576. [CrossRef][PubMed] 12. Jones, P.D. Recent variations in mean temperature and the diurnal temperature range in the Antarctic. Geophys. Res. Lett. 1995, 22, 1345–1348. [CrossRef] 13. Jones, P.D.; Mann, M.E. Climate Over Past Millennia. Rev. Geophys. 2004, 42, RG2002. [CrossRef] 14. Bailey, D.M.; Johnston, I.A.; Peck, L.S. Invertebrate muscle performance at high latitude: Swimming activity in the Antarctic scallop Adamussium colbecki. Polar Biol. 2005, 28, 464–469. [CrossRef] 15. Stark, J.S.; Snape, I.; Riddle, M.J. Abandoned Antarctic waste disposal sites: Monitoring remediation outcomes and limitations at Casey Station. Ecol. Manag. Restor. 2006, 7, 21–31. [CrossRef] 16. Moberg, A.; Sonechkin, D.M.; Holmgren, K.; Datsenko, N.M.; Karlén, W. Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data. Nature 2005, 433, 613–617. [CrossRef][PubMed] 17. Azam, F.; Beers, J.R.; Campbell, L.; Carlucci, A.F.; Holm-Hansen, O.; Reis, F.M.H.; Karl, D.M. Occurrence and metabolic activity of organisms under the Ross Ice Shelf, Antarctica, at station J9. Science 1979, 203, 451–453. [CrossRef][PubMed] 18. Colombo, F.; Caccamo, M.T.; Magazù, S. Trehalose Based Extremophiles in Harsh Environments. In Trehalose: Sources, Chemistry and Applications; Nova Publishers: New York, NY, USA, 2019. 19. Lokotosh, T.V.; Magazù, S.; Maisano, G.; Malomuzh, N.P. Nature of Self-Diffusion and Viscosity in Supercooled Liquid Water. Phys. Rev. E 2000, 62, 3572–3580. [CrossRef] 20. Magazù, S.; Migliardo, F.; Telling, M.T.F. Structural and dynamical properties of water in sugar mixtures. Food Chem. 2008, 106, 1460–1466. [CrossRef] 21. Migliardo, F.; Caccamo, M.T.; Magazù, S. Elastic Incoherent Neutron Scatterings Wavevector and Thermal Analysis on Glass-forming Homologous Disaccharides. J. Non-Cryst. Solids 2013, 378, 144–151. [CrossRef] 22. Magazù, S.; Migliardo, F.; Caccamo, M.T. Innovative Wavelet Protocols in Analyzing Elastic Incoherent Neutron Scattering. J. Phys. Chem. B 2012, 116, 9417–9423. [CrossRef][PubMed] 23. Barnola, J.-M.; Raynaud, D.; Korotkevich, Y.S.; Lorius, C. Vostok ice core provides 160,000-year record of

atmospheric CO2. Nature 1987, 329, 408–441. [CrossRef] 24. Jouzel, J.; Lorius, C.; Petit, J.R.; Genthon, C.; Barkov, N.I.; Kotlyakov, V.M.; Petrov, V.M. Vostok ice core: A continuous isotopic temperature record over the last climatic cycle (160,000 years). Nature 1987, 329, 403–408. [CrossRef] 25. Pepin, L.; Raynaud, D.; Barnola, J.-M.; Loutre, M.F.Hemispheric roles of climate forcings during glacial-interglacial transitions as deduced from the Vostok record and LLN-2D model experiments. J. Geophys. Res. 2001, 106, 31:885–31:892. [CrossRef] Climate 2019, 7, 21 14 of 16

26. Petit, J.R.; Basile, I.; Leruyuet, A.; Raynaud, D.; Lorius, C.; Jouzel, J.; Stievenard, M.; Lipenkov, V.Y.; Barkov, N.I.; Kudryashov, B.B.; et al. Four climate cycles in Vostok ice core. Nature 1997, 387, 359–360. [CrossRef] 27. Basile, I.; Grousset, F.E.; Revel, M.; Petit, J.R.; Biscaye, P.E.; Barkovd, N.I. Patagonian origin dust deposited in East Antarctica (Vostok and Dome C) during glacial stages 2, 4 and 6. Earth Planet. Sci. Lett. 1997, 146, 573–589. [CrossRef] 28. Becker, J.; Lourens, L.J.; Hilgen, F.J.; van der Laan, E.; Kouwenhoven, T.J.; Reichart, G.J. Late Pliocene climate variability on Milankovitch to millennial time scales: A high-resolution study of MIS100 from the Mediterranean. Palaeogeogr. Palaeocl. 2005, 228, 338–360. [CrossRef] 29. Lachniet, M.; Asmerom, Y.; Polyak, V.; Denniston, R. Arctic cryosphere and Milankovitch forcing of Great Basin paleoclimate. Sci. Rep. 2017, 7, 12955. [CrossRef] 30. Petit, J.R.; Jouzel, J.; Raynaud, D.; Barkov, N.I.; Barnola, J.-M.; Basile, I.; Benders, M.; Chappellaz, J.; Davis, M.; Delayque, G.; et al. Climate and atmospheric history of the past 420,000 years from the Vostok ice core. Antarctica. Nat. 1999, 399, 429–436. 31. Lisiecki, L.E.; Raymo, M.E. A Pliocene-Pleistocene stack of 57 globally distributed benthic d18O records. Paleoceanography 2005, 20, PA1003. 32. Milankovitch, M. Canon of Insolation and the Ice Age Problem; Zavod za Udz benike i Nastavna Sredstva: Belgrade, Serbia, 1941. 33. Petit, J.R.; Mounier, L.; Jouzel, J.; Korotkevitch, Y.S.; Kotlyakov, V.I.; Lorius, C. Paleoclimatological implications of the Vostok core dust record. Nature 1990, 343, 56–58. [CrossRef] 34. Waelbroeck, C.; Jouzel, J.; Labeyrie, L.; Lorius, C.; Labracherie, M.; Stiévenard, N.; Barkov, N.I. A comparison of the Vostok ice deuterium record and series from Southern Ocean core MD 88-770 over the last two glacial-interglacial cycles. Clim. Dyn. 1995, 12, 113–123. [CrossRef] 35. Suwa, M.; Bender, M.L. Chronology of the Vostok ice core constrained by O2/N2 ratios of occluded air, and its implication for the Vostok climate records. Quaternary Sci. Rev. 2008, 27, 1093–1106. [CrossRef] 36. Landais, A.; Barkan, E.; Luz, B. Record of delta δ18O and 17O-excess in ice from Vostok Antarctica during the last 150,000 years. Geophys. Res. Lett. 2008, 35, L02709.

37. Bopp, L.; Kohfeld, K.E.; Le Quere, C.; Aumont, O. Dust impact on marine biota and atmospheric CO2 during glacial periods. Paleoceanography 2003, 18, 1046. [CrossRef] 38. Farquhar, G.D.; Lloyd, J.; Taylor, J.A.; Flanagan, L.B.; Syvertsen, J.P.; Hubick, K.T.; Wong, S.C.; Ehleringer, J.R.

Vegetation effects on the isotope composition of oxygen in atmospheric CO2. Nature 1993, 363, 439–442. [CrossRef] 39. Bargagli, R.; Agnorelli, C.; Borghini, F.; Monaci, F. Enhanced deposition and bioaccumulation of mercury in Antarctic terrestrial ecosystems facing a coastal polynya. Environ. Sci. Technol. 2005, 39, 8150–8155. [CrossRef] 40. Richardson, A.J.; Schoeman, D.S. Climate impact on ecosystems in the northeast Atlantic. Science 2004, 305, 1609–1612. [CrossRef] 41. Imbrie, J.; Boyle, E.A.; Clemens, S.C.; Duffy, A.; Howard, W.R.; Kukla, G.; Toggweiler, J.R. On the Structure and Origin of Major Glaciation Cycles 1. Linear Responses to Milankovitch Forcing. PALOC 1992, 7, 701–738. [CrossRef] 42. Li, H.; Nozaki, T. Wavelet cross-correlation analysis and its application to a plane turbulent jet. Int. J. Ser. B 1997, 40, 58–66. [CrossRef] 43. Caccamo, M.T.; Magazù, S. Ethylene Glycol—Polyethylene Glycol (EG-PEG) Mixtures: Infrared Spectra Wavelet Cross-Correlation Analysis. Appl. Spectrosc. 2017, 71, 401–409. [CrossRef] 44. Grinsted, A.; Moore, J.C.; Jevrejeva, S. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Proc. Geoph. 2004, 11, 561–566. [CrossRef] 45. Benzi, R.; Sutera, A.; Vulpiani, A. The mechanism of stochastic resonance. J. Phys. A 1981, 14, L453–L457. [CrossRef] 46. Bulsara, A.R.; Dari, A.; Ditto, W.L.; Murali, K.; Sinha, S. Logical stochastic resonance. Chem. Phys. 2010, 375, 424–434. [CrossRef] 47. Benzi, R.; Parisi, G.; Sutera, A.; Vulpiani, A. Stochastic resonance in climatic change. Tellus 1982, 34, 10–16. [CrossRef] Climate 2019, 7, 21 15 of 16

48. Hänggi, P.; Inchiosa, M.E.; Fogliatti, D.; Bulsara, A.R. Nonlinear stochastic resonance: The saga of anomalous output-input gain. Phys. Rev. E 2000, 62, 6155–6163. [CrossRef] 49. Gammaitoni, L.; Hänggi, P.; Jung, P.; Marchesoni, F. Stochastic resonance: A remarkable idea that changed our perception of noise. Eur. Phys. J. B 2009, 69, 1–3. [CrossRef] 50. Berdichevsky, V.; Gitterman, M. Stochastic resonance in linear systems subject to multiplicative and additive noise. Phys. Rev. E 1999, 60, 1494–1499. [CrossRef] 51. Li, J.H.; Han, Y.X. Phenomenon of stochastic resonance caused by multiplicative asymmetric dichotomous noise. Phys. Rev. E 2006, 74, 051115. [CrossRef] 52. Gammaitoni, L.; Bulsara, A.R. Nonlinear sensors activated by noise. Phys. A 2003, 325, 152–164. [CrossRef] 53. Caccamo, M.T.; Magazù, S. Variable mass pendulum behaviour processed by wavelet analysis. Eur. J. Phys. 2017, 38, 15804. [CrossRef] 54. Prokoph, A.; Patterson, R.T. Application of wavelet and discontinuity analysis to trace temperature changes: Eastern Ontario as a case study. Atmos. Ocean 2004, 42, 201–212. [CrossRef] 55. Mallat, S.G. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 1989, 11, 674–693. [CrossRef] 56. Caccamo, M.T.; Cannuli, A.; Magazù, S. Wavelet analysis of near-resonant series RLC circuit with time-dependent forcing frequency. Eur. J. Phys. 2017, 38, 015804. [CrossRef] 57. Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 1990, 36, 961–1005. [CrossRef] 58. Morlet, J.; Arens, G.; Fourgeau, E.; Glard, D. Wave propagation and sampling theory; Part I, Complex signal and scattering in multilayered media. Geophysics 1982, 47, 203–221. [CrossRef] 59. Caccamo, M.T.; Magazù, S. Variable length pendulum analyzed by a comparative Fourier and wavelet approach. Revista Mexicana de Fisica E 2018, 64, 81–86. 60. Kronland-Martinet, R.; Morlet, J.; Grossmann, A. Analysis of sound patterns through wavelet transform. Int. J. Pattern Recogn. 1987, 1, 97–126. [CrossRef] 61. Caccamo, M.T.; Magazù, S. Thermal restraint on PEG-EG mixtures by FTIR investigations and wavelet cross-correlation analysis. Polym. Test. 2017, 62, 311–318. [CrossRef] 62. Crucifix, M. Oscillators and relaxation phenomena in Pleistocene climate theory. Philosoph. Trans. R. Soc. A 2012, 370, 1140–1165. [CrossRef] 63. Razdan, A. Wavelet Correlation Coefficient of ‘strongly correlated’ financial time series. Phys. A 2004, 333, 335–342. [CrossRef] 64. Caccamo, M.T.; Magazù, S. Tagging the oligomer-to-polymer crossover on EG and PEGs by infrared and Raman spectroscopies and by wavelet cross-correlation spectral analysis. Vib. Spectrosc. 2016, 85, 222–227. [CrossRef] 65. Veleda, D.; Montagne, R.; Araujo, M. Cross-wavelet bias corrected by normalizing scales. J. Atmos. Ocean. Technol. 2012, 29, 1401–1408. [CrossRef] 66. Gitterman, M. Harmonic oscillator with fluctuating damping parameter. Phys. Rev. E 2004, 69, 041101. [CrossRef][PubMed] 67. Wellens, T.; Shatokhin, V.; Buchleitner, A. Stochastic resonance. Rep. Prog. Phys. 2004, 67, 45–105. [CrossRef] 68. Douglass, J.K.; Wilkens, L.; Pantazelou, E.; Moss, F. Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 1993, 365, 337–340. [CrossRef][PubMed] 69. Wiesenfeld, K.; Moss, F. Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs. Nature 1995, 373, 33–36. [CrossRef] 70. McNamara, B.; Wiesenfeld, K. Theory of stochastic resonance. Phys. Rev. A 1989, 39, 4854–4869. [CrossRef] 71. Walsh, J.; Rackauckas, C. On the Budyko-Sellers Energy Balance Climate Model with Ice Line Coupling. Discrete Cont. Dyn. B 2015, 20, 10–20. [CrossRef] 72. Widiasih, E. Dynamics of the Budyko energy balance model. SIAM J. Appl. Dyn. Syst. 2013, 12, 2068–2092. [CrossRef] 73. McGehee, R.; Widiasih, E. A quadratic approximation to Budyko’s ice-albedo feedback model with ice line dynamics. SIAM J. Appl. Dyn. Syst. 2014, 13, 518–536. [CrossRef] 74. Walsh, J.A.; Widiasih, E. A dynamics approach to a low-order climate model. Disc. Cont. Dyn. Syst. B 2014, 19, 257–279. [CrossRef] 75. Budyko, I. The effect of solar radiation variation on the climate of the Earth. Tellus 1969, 5, 611–619. Climate 2019, 7, 21 16 of 16

76. Luterbacher, J.; Dietrich, D.; Xoplaki, E.; Grosjean, M.; Wanner, H. European seasonal and annual temperature variability, trends, and extremes since 1500. Science 2004, 303, 1499–1503. [CrossRef][PubMed] 77. Hansen, J.; Ruedy, R.; Sato, M.; Lo, K. Global Surface Temperature Change. Rev. Geophys. 2010, 48, RG4004. [CrossRef]

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