# Regular and chaotic regimes in Saltzman model of glacial climate

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Regular and chaotic regimes in Saltzman model of glacial climate

Eur. Phys. J. B (2014) 87: 227 DOI: 10.1140/epjb/e2014-50208-0 THE EUROPEAN PHYSICAL JOURNAL B Regular Article Regular and chaotic regimes in Saltzman model of glacial climate dynamics under the influence of additive and parametric noise Dmitry V. Alexandrov1, Irina A. Bashkirtseva1 , Sergei P. Fedotov2 , and Lev B. Ryashko1,a 1 2 Department of Mathematical Physics, Ural Federal University, Lenina ave., 51, 620000 Ekaterinburg, Russia School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK Received 26 March 2014 / Received in ﬁnal form 18 April 2014 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2014 Published online 7 October 2014 – Abstract. It is well-known that the climate system, due to its nonlinearity, can be sensitive to stochastic forcing. New types of dynamical regimes caused by the noise-induced transitions are revealed on the basis of the classical climate model previously developed by Saltzman with co-authors and Nicolis. A complete parametric classiﬁcation of dynamical regimes of this deterministic model is carried out. On the basis of this analysis, the inﬂuence of additive and parametric noises is studied. For weak noise, the climate system is localized nearby deterministic attractors. A mixture of the small and large amplitude oscillations caused by noise-induced transitions between equilibria and cycle attraction basins arise with increasing the noise intensity. The portion of large amplitude oscillations is estimated too. The parametric noise introduced in two system parameters demonstrates quite diﬀerent system dynamics. Namely, the noise introduced in one system parameter increases its dispersion whereas in the other one leads to the stabilization of the climatic system near its unstable equilibrium with transitions from order to chaos. 1 Introduction It is well-known that the variations of the global ice volume of the last million years contain a certain number of pronounced irregular peaks [1,2]. This behavior responsible for the cyclic character of glaciations has been studied by many investigators to describe long-term climatic variability of the limit cycle type [3,4]. Such self-oscillations were shown to appear quite naturally from the coupling between mean ocean temperature and sea ice extent [5]. A closure scrutiny of the variance spectra of the global ice volume shows the presence of random structures superimposed on the irregular peaks. These structures appear due to aperiodic dynamics in climate systems. The simplest case of periodic forcing leading to the system dynamics with a quasi-periodic attractor was previously studied in reference [6]. The presence of several variables and parameters and of the external forcing makes it diﬃcult to disentangle the basic mechanisms responsible for the aperiodic climate dynamics. In recent years it has been pointed out that the climate system is capable of producing irregular sustained self-oscillations [7,8]. This autonomous behavior is connected with possibilities for internal and external feedbacks of both positive and negative sign that tend to produce oscillatory, damping or amplifying dynamics of the earth’s climatic system on many time scales (e.g., Earth orbital radiation and anthropogenic CO2 changes, changes a e-mail: [email protected] in the atmospheric composition and temperature due to unpredictable volcanic eruptions, the natural variability of system variables due to uncertainties in their initial conditions and due to the presence of stochastic noises, etc.). It should be noted that the quasi-periodic and irregular climatic variations dependent of these feedbacks have been revealed in a number of geological records [9–12]. A number of simple climatic feedback oscillator models involving sea-ice extent and mean ocean temperature have been developed earlier by Saltzman et al. [13–16] and Nicolis [6,17]. All of these models are based on a feedback loop arising from the positive insulating eﬀect of sea ice on temperature, and from the negative eﬀect of temperature on sea ice. Because it is too diﬃcult to model completely and accurately the physics involved in the complex feedbacks, this incompleteness is considered as a stochastic process (noise). Also note that some of the high-frequency phenomena that act in the climatic system can be represented by stochastic perturbations [18]. The examination of noise-induced transitions between possible climatic system attractors is the principal goal of the present article. It is worth noting that due to nonlinearity, noise can induce new dynamical regimes which have no analogue in the deterministic case. Noise-induced phenomena in biophysics, electronics, mechanics, population dynamics, geophysics attract an attention of many researchers [19–26]. Potentially, the long-term climatic changes in the form of aperiodic self-sustained oscillations may arise from the Page 2 of 10 Eur. Phys. J. B (2014) 87: 227 interaction between surface energy balance and mass balance of the cryosphere. The sea-ice coverage (dependent on the ocean temperature) can either prevent the loss of heat from the ocean in high latitudes by its insulating effect, thereby warming the bulk ocean (negative feedback), or cool the ocean by the ice-albedo eﬀect (positive feedback). In addition, positive feedback is also caused by the water vapor and CO2 eﬀects on ocean temperature. A principal model describing the interaction between these processes is the one proposed by Saltzman et al. [13–16]. This model consists of the following coupled autonomous equations governing the departures of the sine of latitude of sea ice extent η and of the mean ocean temperature θ from their equilibrium values dη = φ1 θ − φ2 η , dτ dθ = −ψ1 η + ψ2 θ − ψ3 η 2 θ , dτ (1) where positive constants φ1 , φ2 , ψ1 , ψ2 and ψ3 are detailed previously by Saltzman et al. The linear terms in the right-hand sides of system (1) describe damped harmonic oscillations arising due to the “ice-insulator” eﬀect (negative feedback) and a destabilizing eﬀect caused by the positive feedback between atmospheric CO2 concentration and mean temperature of the ocean. A non-linear restorative mechanism that is dominant when the system is far from its equilibrium is described by the term proportional to η 2 θ (the detailed explanation of this term is given in Ref. [15]). Let us introduce the following scaling transformations √ ψ3 φ1 ψ3 φ1 ψ1 ψ2 x= η, y= θ , t = φ2 τ, a = 2 , b = φ 3/2 φ2 φ 2 φ 2 2 according to references [6,17]. In this case, system (1) becomes dx = y − x, dt dy = −ax + by − x2 y. dt (2) Now we see that the deterministic dynamics is entirely determined by two dimensionless parameters a and b only. Let us summarize the main features of the aforementioned model previously analyzed by Saltzman with coauthors and Nicolis in a narrow domain of system parameters. For any values of a and b, system (2) has a trivial equilibrium M0 (0, 0) stable for b < 1 and b < a. The system behavior in this region is quite simple: all trajectories are attracted to the stable equilibrium M0 . Then, when the system parameters pass through the line a > 1, b = 1 due to Andronov-Hopf bifurcation, the equilibrium M0 loses its stability so that the stable auto-oscillations are born. Note that this sustained oscillation regime was used by Saltzman for theoretical explanation of the global climatic changes. For ﬁxed parameters a = 6.4, b = 4, these oscillations forced by additive noise were studied in reference [15] by means of direct numerical simulations. Then, an analytical investigation of sustained oscillations with additive noise near Andronov-Hopf bifurcation border has been developed in reference [17] with the help of the normal form technique. The eﬀect of a periodic forcing in the right-hand side of dimensionless system (2) in the vicinity of the homoclinic bifurcation point a = b = 1 was studied in reference [6]. Thus, the deterministic model (2) of the “mean ocean temperature – sea-ice” system was investigated only for severely limited parametric zones in spite of the fact that possible estimates of system parameters a and b may lie out of the analyzed interval (see their diﬀerent estimates given in Refs. [14–16]). In the case of stochastic dynamics, only the inﬂuence of additive random noise was previously considered in references [15,17]. Therefore, the “mean ocean temperature – sea-ice” model behavior should be studied more precisely in a broad region of possible values of the system parameters in both cases of deterministic and stochastic dynamics. Here it is necessary to investigate the inﬂuence not only additive but inevitably present parametric noise. This goal explains the outline of the present study. First, in Section 2, a full classiﬁcation of all typical dynamic regimes for the classical two-dimensional deterministic climate model (2) is given via bifurcation analysis. Here, qualitative changes of the phase portraits are discussed. Note that in previous papers only particular regimes were studied. This classiﬁcation of deterministic regimes serves as a basis for understanding the probabilistic mechanisms of the stochastic phenomena. Second, the inﬂuence of additive noise on these dynamical regimes is studied in Section 3, where diﬀerent types of noise-induced mixed-mode oscillations are discussed. It is shown how due to noise-induced transitions between deterministic attractors (equilibria and cycle) a new stochastic regime with intermittency between smalland large-amplitude stochastic oscillations is formed. For this model, two new stochastic phenomena are shown: a generation of large-amplitude stochastic oscillations in a parametric zone where the deterministic model has stable equilibria only; a generation of small-amplitude stochastic oscillations in a parametric zone where the deterministic model has a stable cycle only. Furthermore, some unexpected peculiarities of the system response on parametric noise are studied in Section 4. Note that the inﬂuence of parametric noise on this model has never been documented previously. In our paper, we present a new phenomenon of the localization of system states near the unstable equilibrium due to parametric noise. An interesting relationship of such phase localization with transition to chaos is ﬁrst established and discussed. 2 Overview of deterministic dynamics It is well-known that a mutual arrangement of equilibria and cycles of the deterministic system (2) plays an important role for the understanding of the variety of its dynamical regimes. Qualitatively diﬀerent types of system dynamics are shown in Figure 1. Let us initially consider the Eur. Phys. J. B (2014) 87: 227 a = 0.7 Page 3 of 10 a = 0.76 a = 0.86 a = 1.5 Fig. 1. Dynamics of the deterministic model for b = 2: stable equilibria (blue circles), unstable equilibria (red circles), stable cycles (blue solid curves), unstable cycles (red dotted curves), phase trajectories (black curves). case b > a, when system (2), along with the trivial equilibrium M0 , has a pair of nontrivial symmetric √ equilibria M1 (x̄1 , √ ȳ1 ) and M2 (x̄2 , ȳ2 ). Here x̄1 = ȳ1 = b − a, x̄2 = ȳ2 = − b − a. In the case b < 1 only two simple diﬀerent types of phase portraits are possible (the case without cycles): (1) for 0 < a < b, the equilibrium M0 is unstable and M1 , M2 are stable; (2) for a > b, system (2) has a single stable equilibrium M0 . Now consider in detail the case b > 1. Let us ﬁx b = 2 for deﬁniteness and change the second system parameter a. As a result, one can mark four bifurcation values: a1 ≈ 0.714, a2 ≈ 0.775, a3 = 1, and a4 = 2. These points deﬁne the following intervals of the structural stability: A (0 < a < a1 ), B (a1 < a < a2 ), C (a2 < a < a3 ), D (a3 < a < a4 ), E (a > a4 ). For any zone, system (2) exhibits an intrinsic type of dynamics. Dynamics of system (2) in zone A is determined by the presence of the unstable equilibrium M0 and two stable equilibria M1 , M2 . Here, a phase trajectory tends to M1 or M2 depending on the initial state. As the parameter a passes through the value a1 , a new attractor (stable cycle) appears through a saddle-node bifurcation. This stable limit cycle embraces all equilibria and is separated from these equilibria by the unstable cycle. In zone B, this unstable cycle divides attraction basins between a stable cycle and stable equilibria M1 , M2 . At the next bifurcation point a2 this unstable cycle splits into two separate unstable cycles. In zone C, each of these unstable cycles embraces the corresponding stable equilibrium. At the bifurcation point a3 a subcritical Hopf bifurcation occurs: these unstable cycles merge with equilibria M1 , M2 so that these equilibria become unstable too. In zone D, the stable cycle encloses three unstable equilibria. As the parameter a increases, equilibria M1 , M2 become closer to M0 and merge with it at the pitchfork-bifurcation value a4 . So, in the zone E this system has a stable cycle embracing the single unstable equilibrium M0 . Dynamics of system (2) in zone E was studied in references [15,17]. For other zones, typical dynamic regimes are presented in Figure 1. Let us especially emphasize that the climate system tends to a stable cycle or stable equilibria depending on initial data and dimensionless parameters. Note that unstable cycles plotted in Figure 1 (zones B, C) divide attraction basins between a stable cycle and stable equilibria M1 , M2 so that if the system is placed in the vicinity of a stable cycle (equilibrium) it attracts to it in course of time. 3 The influence of additive noise This kind of noises is connected with stochastic ﬂuctuations of the main system dynamical variables such as the mean ocean temperature. In this paper, we focus on the analysis of the inﬂuence of such ﬂuctuations on the mean ocean temperature because this variable is more sensitive to random perturbations. Therefore, let us consider the climatic model (2) forcing by the additive noise dx = y − x, dt dy = −ax + by − x2 y + εξ(t), dt (3) where ξ(t) is uncorrelated white Gaussian zero-mean noise term describing ﬂuctuating forces on the mean ocean temperature, and ε is an intensity of random disturbances. In this paper, for the numerical simulation of stochastic models, we used the Euler-Maruyama scheme [27]. For the modeling of appropriate stochastic components of Gaussian random disturbances in this scheme, the standard Box-Muller transform was used. Overview of possible stochastic regimes in zones A, B, C and D is presented in Figure 2. Here, phase trajectories are shown in the left column, time series are plotted in the middle column, and corresponding stationary probability densities are demonstrated on the right. Under the stochastic disturbances, random trajectories leave deterministic stable equilibria or cycles (attractors) and form a probabilistic distribution around them. A shape of this distribution essentially depends on the noise intensity and a geometry of phase portrait of the initial deterministic model. Page 4 of 10 Eur. Phys. J. B (2014) 87: 227 x y p 1 1 5 0 0 0 5 −1 −1 y 0 A −2 ε = 0.1 a = 0.7 −1 0 1 x 0 100 200 300 400 t −5 −2 −5 −2 −5 −2 −5 −2 0 x 2 0 x 2 0 x 2 0 x 2 x y p 2 1 0.4 0 0 −2 −1 0 5 y 0 −2 −4 −2 −1 0 x 1 0 100 200 300 400 t ε = 0.3 x y p 2 1 0 0 −2 −1 3 0 5 y 0 B −2 ε = 0.1 a = 0.76 −1 0 1 x 0 200 400 t x y p 2 1 0 0 −2 −1 0.5 0 5 y 0 −2 −1 0 1 x 0 200 400 t ε = 0.2 Fig. 2. Dynamics of the model forced by additive noise: trajectories starting from the stable equilibrium (black colour), trajectories starting from the stable cycle (blue colour); phase trajectories (left column), time series (middle column), probability density (right column). For weak noise, random trajectories are concentrated nearby deterministic attractors illustrated in Figure 1. As noise intensity increases, random trajectories can cross separatrices between attraction basins and exhibit new dynamical regimes which have no analogue in the deterministic case. Scenarios of such noise-induced transitions depend on the peculiarities of the mutual arrangement of deterministic attractors and separatrices on the phase plain. Zone A. For weak noise (ε = 0.1), random trajectories are concentrated nearby stable equilibria: the climate system exhibits small amplitude stochastic oscillations (SASO). Here, the stationary probability density function has two sharp peaks. An increasing of noise implies noiseinduced transitions between these equilibria and a generation of large amplitude stochastic oscillations (LASO). As a result one can observe an intermittency of SASO and LASO. Here, along with two anticipated peaks corresponding to the stable equilibria, the climate system additionally acquires an unexpected closed ridge embracing these peaks. Note that this ridge does not have a deterministic cycle as an underlying reason. A key for understanding this phenomenon is a non-uniformity of the phase portrait of the initial deterministic system. It is clearly seen that the deterministic trajectory going to the equilibrium Eur. Phys. J. B (2014) 87: 227 Page 5 of 10 x y p 1 2 0.4 C 0 0 −2 −1 −4 −2 0 y −2 ε = 0.1 a = 0.86 0 5 −1 0 x 1 200 400 0 2 x 2 x 2 x 2 x 0 t −5 −2 x y 2 p 2 1 0 0 0.2 0 5 −1 −2 y −2 −4 −2 −1 0 x 1 0 200 400 0 0 t −5 −2 −5 −2 ε = 0.2 y4 x 2 1 2 p 0.2 D 0 0 −2 −1 −4 −2 −2 ε = 0.2 a = 1.1 −1 0 x 1 0 0 5 y 200 400 0 0 t x y 2 5 p 1 0 0.05 0 0 −1 5 −5 y −2 −2 −1 0 1 2 x 0 200 400 t 0 −5 0 −2 ε=1 Fig. 2. Continued. passes a zone of so-called “transient attractor” 1 mm with temporary local stabilization. Thus it can be interpreted as noise-induced generation of the stochastic cycle in a zone where the deterministic system has stable equilibria only. Note that with increase of noise a percentage of LASO in a common mixed-mode oscillation process increases too. This percentage (portion) can be estimated ) by the value k = n(T where n(T ) is a quantity of interT sections of the mixed-mode process x(t) on the interval [0, T ] with x = 0. Here, time of observation T is suﬃciently large. A dependence of the value k on noise intensity ε and parameter a is presented in Figure 3 (left panel). The value k is a mean inverse time between zero-crossings. In the deterministic case (ε = 0), in the parameter zone A attractors are stable equilibria, hence there are no such crossings and k = 0. As noise intensity increases, a probability of such crossings increases too, and one can see that k(ε) monotonically grows. In our numerical simulations of phase trajectories and time series, we used the Euler-Maruyama method with the time step Δt = 10−3 . To calculate probability density functions, we used a grid 100 × 100 on the plane xOy, and simulation time T = 105 . Due to ergodicity [28], one can change the ensemble averaging over the set of realizations of random trajectories by the time averaging over the single random trajectory on the suﬃciently large interval [0, T ]. Here, we follow this widely used constructive approach and handle 108 random states. The same parameters Δt = 10−3 and T = 105 were used for k in Figure 3. Page 6 of 10 Eur. Phys. J. B (2014) 87: 227 −3 −3 x 10 k 2 x 10 k a=0.7 a=0.5 a=0 a=1.1 a=1.5 a=2 5 1.5 4 1 0.5 3 0 0 0.2 0.4 0.6 0.8 ε 0 0.2 0.4 0.6 0.8 ε Fig. 3. Plots k(ε) for zone A (left) and for zone D (right). We have checked that a decrease of time step Δt and increase of T do not change results essentially. Zone B. Here the deterministic system (2) has a stable cycle separated by the single unstable cycle from two stable equilibria M1 , M2 (see Fig. 1). For weak noise, random trajectories starting from the equilibria demonstrate SASO, and random trajectories starting from the cycle execute LASO. As noise intensity increases (ε = 0.1), random trajectories starting from the equilibria are localized nearby these initial points, but a trajectory starting from the cycle, after several turns along it passes into the vicinity of the equilibrium. Here system (3) exhibits a transition “cycle → equilibrium”. With further increase of noise (ε = 0.2), the reverse transitions “equilibrium → cycle” 1 mm begin and the climatic system executes the intermittency of SASO and LASO. As one can see from Figure 2, in these mixed-mode oscillations, SASO dominate. Zone C. Here the deterministic system (2) has a stable cycle separated by the pair of unstable cycles from two stable equilibria. In C, in contrast to zone B, at ﬁrst noiseinduced transitions “equilibrium → cycle” 1 mm appear. Further increase of noise (ε = 0.2) implies intermittency of LASO and SASO with dominating LASO. Zone D. For weak noise, random trajectories with LASO are concentrated nearby initial attractor – deterministic cycle. For increasing noise (ε = 1), smallamplitude scrolls appear near unstable equilibria (see Fig. 2). Plots of k(ε) representing the percentage of LASO in a common mixed-mode process for three values of the parameter a are shown in Figure 3 (right panel). Here, there is a well-deﬁned deterministic limit of k: inverse halfperiod of the corresponding deterministic limit cycle. In contrast to the zone A (see Fig. 3, left panel), k(ε) is reduced unessentially with increasing noise. As a result we can summarize that additive noise blurs the qualitative distinctions in deterministic dynamics in considered zones. In all zones, an increase of noise implies a generation of mixed-mode stochastic oscillations. The resulting system parameters a and b can change a percentage ratio between LASO and SASO only. 4 The influence of parametric noise and transitions to chaos This kind of noises is caused by uncertainties in determination of the reduced system parameters a and b representing diﬀerent physical mechanisms governing the climate dynamics. Parametric noises also allow to take into account the possible random disturbances of the system parameters. Therefore, let us consider the climatic model (2) forcing by additive and parametric noises in the form dx = y − x, dt dy = −(a + εa ξa (t))x + (b + εb ξb (t))y − x2 y + εξ(t). dt (4) Here ξa (t), ξb (t), ξ(t) are uncorrelated white Gaussian zero-mean noise terms, εa , εb are intensities of parametric noises corresponding to two system parameters a and b, and ε is an intensity of additive noise. In this section, we restrict ourselves to the demonstration of the speciﬁc character of the parametric noise inﬂuence in zone A for ﬁxed parameters a = 0.7, b = 2 and small background additive noise ε = 0.01. The deterministic system exhibits unstable equilibrium M0 (0, 0) and two stable equilibria M1 (1.14, 1.14) and M2 (−1.14, −1.14). Let us consider the inﬂuence of a-noise (εa = 0, εb = 0) and b-noise (εa = 0, εb = 0) separately. Figure 4 demonstrates some changes in stochastic dynamics of system (4) induced by increasing parametric noises (top panel for a-noise, and bottom panel for b-noise). Here, random trajectories start from the equilibrium M1 . For weak parametric noise, random trajectories with SASO are concentrated in a small vicinity of M2 (Fig. 4, left column). As parametric noise increases, LASO occur (Fig. 4, middle column). This behavior is similar to the case of the inﬂuence of additive noise only. Further growth of parametric noise implies completely diﬀerent reply of system (4) on a-noise and b-noise. Indeed, an increase of a-noise (εa = 2) enlarges the amplitude Eur. Phys. J. B (2014) 87: 227 Page 7 of 10 y a-noise b-noise y y 5 5 5 0 0 0 −5 −5 −5 −10 −2 0 2 y x −10 −2 0 2 y x −10 20 20 20 10 10 10 0 0 0 −10 −10 −10 −20 −2 0 2 x −20 −2 0 2 x −2 0 2 y −20 −2 0 2 x x Fig. 4. Dynamics of system under the parametric a-noise (top panel) and b-noise (bottom panel) for a = 0.7 and ε = 0.01. Top panel for εb = 0: εa = 0.1 (left); εa = 0.5 (middle); εa = 2 (right); bottom panel for εa = 0: εb = 0.1 (left); εb = 0.5 (middle); εb = 2 (right). and dispersion of stochastic oscillations. On the contrary, an increase of b-noise (εb = 2) leads to the concentration of the random trajectories near the unstable equilibrium M0 . Additional details of this diﬀerence are demonstrated in Figure 5 by time series of x(t) and the probability density functions. As one can see from left column in Figure 5, time series and the probability density functions for a-noise and b-noise are similar. For increasing a-noise, the stationary probability density function holds the crater-like shape with two peaks above the stable equilibria M1 and M2 . On the contrary, as b-noise increases, this function transforms to the single-peak shape. Such counterintuitive behavior of system (4) with b-noise-induced concentration of the random trajectories near unstable equilibrium does not occur in the systems forced by additive noise only. Along with the analysis of the changes of the probability density function form, consider also a deformation of dynamics of the stochastic ﬂows. A standard quantitative measure of this dynamics is a largest Lyapunov exponent Λ. In Figure 6, for the ﬁxed additive noise ε = 0.01, this function is plotted for changing a-noise (left) and b-noise (right). For the calculation of the largest Lyapunov exponent, a standard Benettin method [29] was used. For weak parametric noise, Λ is negative for both cases. As noise intensity increases, this function becomes positive. This transition from negative to positive values of Λ is traditionally interpreted as a noise-induced chaotization. With further growth of the parametric noise intensity, this function demonstrates absolutely diﬀerent behavior for a-noise and b-noise. Indeed, as a-noise increases, Λ decreases and becomes negative again. An increasing of b-noise implies a monotonous growth of Λ. In spite of the localization of phase random trajectories near M0 , dynamics of the system forced by b-noise remains chaotic. Let us discuss this phenomenon. Under the random disturbances, the stochastic phase trajectory passes zones of local convergence and divergence, so a sign of the largest Lyapunov exponent is deﬁned by the balance of the probabilities of the belonging to these zones. If in the distribution the zones of convergence dominate then the Lyapunov exponent is negative, otherwise, the Lyapunov exponent becomes positive. For weak noise, random trajectories of system (4) are localized near the stable equilibrium M1 in a convergence zone, so the Lyapunov exponents are negative. For increasing noise, random trajectories spend most of the time near the deterministically unstable equilibrium M0 and the Lyapunov exponent increases and becomes positive for both a- and b-noise. A diﬀerence in the further behavior of the function Λ(ε) for increasing a- and b-noise can be explained by the diﬀerence in the probability densities. Indeed, under a-noise random states are widely dispersed and the system is arranged in the convergence zones far from unstable M0 . In contrast, under b-noise, random states are concentrated nearby the unstable M0 in a divergence zone, so Λ(ε) is positive. 5 Conclusion The present work is concerned with a systematic study of diﬀerent dynamical regimes of the deterministic Saltzman model reﬂecting the main principles of the climate dynamics. Previously this model was analyzed by Saltzman et al. [13–16] and Nicolis [6,17] only in those parametric regions where the climate system tends either to equilibria or to regular periodic oscillations. However, some estimates of the system parameters show that their values may lie out of the analyzed interval [14–16]. The typical regimes possible in the case of purely deterministic dynamics are plotted in Figure 1 and discussed in detail in Section 2. Our calculations show that the climate system is localized in the vicinity of stable attractors (cycle or Page 8 of 10 a-noise Eur. Phys. J. B (2014) 87: 227 x x 2 2 0 0 −2 −2 0 50 100 t 150 0 p 50 100 t 150 p 0.1 0.02 0 10 0 10 y 0 x 0 y 0 4 −10 −4 b-noise 4 x 0 −10 −4 x x 2 2 0 0 −2 −2 0 50 100 t 150 0 50 100 t 150 p p 0.1 5 0 10 0 10 y 0 0 −10 −3 x 3 y 0 0 x 3 −10 −3 Fig. 5. Time series and probability density functions for the system under the parametric a-noise (top panel) and b-noise (bottom panel) for a = 0.7 and ε = 0.01. Top panel for εb = 0: εa = 0.5 (left); εa = 2 (right); bottom panel for εa = 0: εb = 0.5 (left); εb = 2 (right). Eur. Phys. J. B (2014) 87: 227 Page 9 of 10 Λ Λ 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 0 1 2 εa 0 1 2 εb Fig. 6. Plots of the largest Lyapunov exponent for system (4) under the parametric noise for a = 0.7 and ε = 0.01; with εb = 0 (left) and εa = 0 (right). points of equilibria) depending on the initial point of the phase trajectory. In addition, unstable cycles (revealed for the ﬁrst time and shown in Fig. 1 by the red lines) divide the basins of attraction formally representing two zones – internal (where all initial trajectories tend to stable equilibria) and external (where they lead the system to a stable cycle). A complete analysis of the model under consideration in the presence of additive noises (caused by ﬂuctuations of the main system parameters – the mean ocean temperature and the sea ice extent) demonstrates a new type of stochastic dynamics of the Saltzman model – the mixed-mode oscillations representing small and large amplitude stochastic ﬂuctuations nearby stable attractors of the deterministic model. A typical illustration of possible stochastic dynamics in this case is demonstrated in Figure 2 and discussed in Section 3. If the noise intensity is rather weak, all random trajectories are localized in the vicinity of deterministic attractors. Increasing the noise intensity leads to a jump-like behavior of the phase trajectories crossing separatrices of attraction basins (unstable cycles of the deterministic model). On the whole, the climate system undergoes the noise-induced transitions “equilibrium → cycle” and “cycle → equilibrium” and exhibits an intermittency of SASO and LASO. Note that the generation of LASO can occur even in a zone where the deterministic climate system has no any limit cycle. An important point is that the portion of LASO increases or decreases with increasing the noise intensity in diﬀerent regions of system parameters (Fig. 3). An inﬂuence of parametric noises in both system parameters a and b on the non-linear stochastic dynamics is studied (Figs. 4–6). In the case of weak parametric noises, the system is localized in the vicinity of the stable equilibrium. Then, the large amplitude stochastic oscillations dominate with increasing the noise intensity (middle column in Fig. 4). The subsequent dynamics of the climate system with increasing a and b parametric noises is essentially diﬀerent. So, the amplitude of ﬂuctuations and the dispersion of phase trajectories grow with increasing a-noise intensity. However, the enhancement of b-noise intensity concentrates all phase trajectories near unstable equilibrium so that the probability density function has a sharp peak in its vicinity (Fig. 5). Such noise-induced deformations of the phase random trajectories are accompanied by the transitions from order to chaos (Fig. 6). Thus, the complex non-linear behavior of the model under consideration may help to clarify the intrinsic mechanisms of the noise-induced climate shifts [30] and irregular chaotic dynamics of the climate systems [1,2]. The irregular and chaotic dynamic regimes of the climate system demonstrated on the basis of two-parametric Saltzman’s model (1) in the present study can be a part of more complex dynamics of multi-parametric models too. So, results of the present paper can be useful for the explanation of the complex stochastic behavior of paleoclimate records [30,31]. This work was supported by the Ministry of Education and Science of the Russian Federation under the project N 315. References 1. K.G. Miller, G.S. Mountain, J.D. Wright, J.V. Browning, Oceanogr. 24, 40 (2011) 2. B. de Boer, R.S.W. van de Wal, L.J. Lourens, R. Bintanja, T.J. Reerink, Clim. Dyn. 41, 1365 (2013) 3. A.M. Selvam, Chaotic Climate Dynamics (Luniver Press, UK, 2007) 4. M. Cruciﬁx, Philos. Trans. R. Soc. A 370, 1140 (2012) 5. B. Saltzman, Dynamical Paleoclimatology (Academic Press, San Diego, 2002) 6. C. Nicolis, Tellus 39A, 1 (1987) 7. J. Jouzel, V. Masson-Delmotte, WIREs Clim. Change 1, 654 (2010) 8. N. Scafetta, J. Atm. Solar Terrest. Phys. 72, 951 (2010) 9. R.B. Alley, J. Marotzke, W.D. Nordhaus, J.T. Overpeck, D.M. Peteet, R.A. Pielke Jr., R.T. Pierrehumbert, P.B. Rhines, T.F. Stocker, L.D. Talley, J.M. Wallace, Science 299, 2005 (2003) Page 10 of 10 10. J. Thurow, L.C. Peterson, U. Harms, D.A. Hodell, H. Cheshire, H.-J. Brumsack, T. Irino, M. Schulz, V. MassonDelmotte, R. Tada, Sci. Drill. 8, 46 (2009) 11. J.W.C. White, R.B. Alley, J. Brigham-Grette, J.J. Fitzpatrick, A.E. Jennings, S.-J. Johnsen, G.H. Miller, R.S. Nerem, L. Polyak, Quarter. Sci. Rev. 29, 1716 (2010) 12. J. Holmes, J. Lowe, E. Wolﬀ, M. Srokosz, Glob. Planet. Change 79, 157 (2011) 13. B. Saltzman, Adv. Geophys. 20, 183 (1978) 14. B. Saltzman, R.E. Moritz, Tellus 32, 93 (1980) 15. B. Saltzman, A. Sutera, A. Evenson, J. Atm. Sci. 38, 494 (1981) 16. B. Saltzman, Tellus 34, 97 (1982) 17. C. Nicolis, Tellus 36A, 1 (1984) 18. C. Nicolis, J. Stat. Phys. 70, 3 (1993) 19. L. Arnold, Random Dynamical Systems (Springer-Verlag, 1998) 20. W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1984) 21. S. Fedotov, I. Bashkirtseva, L. Ryashko, Phys. Rev. E 66, 066310 (2002) Eur. Phys. J. B (2014) 87: 227 22. S. Fedotov, I. Bashkirtseva, L. Ryashko, Phys. Rev. E 73, 066307 (2006) 23. L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Eur. Phys. J. B 69, 1 (2009) 24. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001) 25. I. Bashkirtseva, L. Ryashko, Chaos 21, 047514 (2011) 26. D.V. Alexandrov, I.A. Bashkirtseva, A.P. Malygin, L.B. Ryashko, Pure Appl. Geophys. 170, 2273 (2013) 27. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Diﬀerential Equations (Springer, Berlin, 1992) 28. P. Walters, An Introduction to Ergodic Theory (Springer, 1982) 29. G. Benettin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Meccanica 15, 9 (1980) 30. P.D. Ditlevsen, M.S. Kristensen, K.K. Andersen, J. Climate 18, 2594 (2005) 31. P.D. Ditlevsen, H. Svensmark, S. Johnsen, Nature 379, 810 (1996)