Stefano Mammola
email: stefanomammola@gmail.com

Elena Piano

Pedro Cardoso

Philippe Vernon

David Domínguez-Villar

David C Culver

Tanja Pipan

Marco Isaia
email: marco.isaia@unito.it@tel.

Climate change going deep: The effects of global climatic alterations on cave ecosystems

Keywords: cave-dwelling species, Global Warming, hypogean habitat, cave meteorology, superficial subterranean habitats, stygobionts, troglobionts, relative humidity, temperature

published or not. The documents may come

I. INTRODUCTION

Climate change is considered to be one of the most challenging concerns for humanity [START_REF] Walther | Ecological responses to recent climate change[END_REF][START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF][START_REF] Scholze | A climate-change risk analysis for world ecosystems[END_REF][START_REF] Williams | The anthropocene biosphere[END_REF]. The complexity of the global climate issue stretches far beyond the currently observed pattern of global temperature increase [START_REF] Santer | Contributions of anthropogenic and natural forcing to recent tropopause height changes[END_REF], as it involves a variety of multifaceted ecological responses to climatic variations, such as shifts in species distribution ranges [START_REF] Chen | Rapid range shifts of species associated with high levels of climate warming[END_REF], phenological displacements [START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF][START_REF] Root | Fingerprints of global warming on wild animals and plants[END_REF], complex interactions among previously isolated species [START_REF] Williams | Novel climates, no-analog communities, and ecological surprises[END_REF][START_REF] Krosby | Climate-induced range overlap among closely related species[END_REF], extinctions (Thomas et al., 2004;Cahill et al., 2013) and other unpredictable cascading effects on different ecosystem components [START_REF] Walther | Ecological responses to recent climate change[END_REF].

In order to minimize confounding effects, scientists have often used isolated ecosystems-and specific organisms within them-as models to unravel ecological responses to recent climate alterations, upscaling results and conclusions to a wider range of systems and organisms. Under this perspective, mountain summits, oceanic islands, lakes and other confined habitats have offered insightful models for determining the effects of climate change on biodiversity in situ [START_REF] Hortal | Perspectives on the use of lakes and ponds as model systems for macroecological research[END_REF][START_REF] Warren | Islands as model systems in ecology and evolution: prospects fifty years after MacArthur-Wilson[END_REF][START_REF] Whittaker | Island biogeography: Taking the long view of nature's laboratories[END_REF][START_REF] Itescu | Are island-like systems biologically similar to islands? A review of the evidence[END_REF][START_REF] Mammola | Finding answers in the dark: caves as models in ecology fifty years after Poulson and White[END_REF]. Even if the potential of subterranean habitats as ideal biological laboratories has been long ago foreseen [START_REF] Barr | Cave ecology and the evolution of troglobites[END_REF][START_REF] Poulson | The cave environment[END_REF]Culver, 1982), little has been written about the specific contribution of cavebased studies to the understanding of patterns and processes in global change biology [START_REF] Mammola | Finding answers in the dark: caves as models in ecology fifty years after Poulson and White[END_REF][START_REF] Sánchez-Fernández | The deep subterranean environment as a model system in ecological, biogeographical and evolutionary research[END_REF]. As a direct consequence, most syntheses focusing on the effect of climate change on ecosystems did not feature terrestrial caves in the potential study systems (e.g. [START_REF] Walther | Ecological responses to recent climate change[END_REF][START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF][START_REF] Scholze | A climate-change risk analysis for world ecosystems[END_REF]. However, there exists an emerging trend in environmental, physical, and ecological studies alike to use caves and other subterranean habitats as models for unraveling current climate change dynamics. Recent relevant studies focused on the role of climatic alterations on the geophysical components of the cave environment [START_REF] Domínguez-Villar | Is global warming affecting cave temperatures? Experimental and model data from a paradigmatic case study[END_REF][START_REF] Colucci | Response of ice caves to weather extremes in the southeastern Alps, Europe[END_REF][START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF], the potential of caves as methane-sinks [START_REF] Fernandez-Cortes | Subterranean atmospheres may act as daily methane sinks[END_REF] and the general effects of global temperature increase on different subterranean animals [START_REF] Colson-Proch | First cellular approach of the effects of global warming on groundwater organisms: a study of the HSP70 gene expression[END_REF][START_REF] Lencioni | Cold resistance in two species of cave-dwelling beetles (Coleoptera: Cholevidae)[END_REF][START_REF] Brandmayr | Hypogean carabid beetles as indicators of global warming?[END_REF][START_REF] Mermillod-Blondin | Thermal tolerance breadths among groundwater crustaceans living in a thermally constant environment[END_REF][START_REF] Rizzo | Lack of evolutionary adjustment to ambient temperature in highly specialized cave beetles[END_REF][START_REF] Sánchez-Fernández | Thermal niche estimators and the capability of poor dispersal species to cope with climate change[END_REF][START_REF] Mammola | Rapid poleward distributional shifts in the European cave-dwelling Meta spiders under the influence of competition dynamics[END_REF][START_REF] Lorenzo | Effect of temperature rising on the stygobitic crustacean species Diacyclops belgicus: does global warming affect groundwater populations?[END_REF]Mammola et al., 2018), and microorganisms [START_REF] Brielmann | Effects of thermal energy discharge on shallow groundwater ecosystems[END_REF][START_REF] Brielmann | Shallow geothermal energy usage and its potential impacts on groundwater ecosystems[END_REF]Gribler et al., 2016).

We hypothesize here that caves represent one of the most informative systems for the study of climate change across its biotic and abiotic components, for multiple reasons: i) they are semi-closed systems that are extensively replicated across the Earth [START_REF] Culver | The Biology of Caves and other Subterranean Habitats[END_REF]; ii) they are buffered from external variations and generally characterised by a remarkable thermal stability, especially in temperate areas [START_REF] Badino | Cave temperatures and global climatic change[END_REF][START_REF] Badino | Underground meteorology. What's the weather underground?[END_REF]; iii) temperature inside caves highly correlate with mean annual temperatures on the surface [START_REF] Moore | Out of phase seasonal temperature fluctuations in Cathedral Cave, Kentucky[END_REF], at least in temperate areas [START_REF] Sánchez-Fernández | The deep subterranean environment as a model system in ecological, biogeographical and evolutionary research[END_REF]; and iv) subterranean organisms have evolved a suite of morphological and physiological adaptations to thrive under the peculiar environmental conditions found in caves, which are often convergent even across phylogenetically distant groups [START_REF] Culver | The Biology of Caves and other Subterranean Habitats[END_REF][START_REF] Juan | Evolution in caves: Darwin's 'wrecks of ancient life' in the molecular era[END_REF]Protas and Jeffrey, 2012).

However, a lack of synthesis on this topic is possibly holding back the establishment of caves as standard models for the study of the effects of climate change in the field. By bringing together literature focusing on cave biology, ecology, and physics that were previously disconnected, we illustrate the rationale supporting the use of subterranean habitats as ideal model for studies of global change biology. We describe the potential effects of the anthropogenic climate alterations on the abiotic and the biotic components of the cave ecosystem, giving special reference to the cave climate, subterranean organisms and the interconnection between caves and other ecosystems (surface and soil). We focused here mostly on terrestrial subterranean habitats and only partly on groundwater systems (see [START_REF] Green | Beneath the surface of global change: impacts of climate change on groundwater[END_REF]Taylor et al., 2013 for more arguments on groundwaters). We did not considered marine caves. A definition of the jargon used in the text is presented in Box 1. Throughout the text, the term "cave" is used to refer to the extended network of fissures and cracks that permeate most substrates, even if their size is not commensurable to the human-scale [START_REF] Romero | Caves as biological space[END_REF].

II. ABIOTIC FEATURES (1) Subterranean climate

A wealth of literature documents the ecological peculiarity of the subterranean realm, and how distinct it is from surface habitats. Whilst the absence of solar radiation is possibly the most crucial factor conditioning the ecology of subsurface habitats [START_REF] Culver | Shifting paradigms of the evolution of cave life[END_REF], the most important feature that corroborates the idea that caves are ideal laboratories for the study of climate change in natural environment is their unique climatic stability. Indeed, deep subterranean habitats have, in most cases, an almost constant temperature over the year and a relative humidity often close to saturation [START_REF] Cigna | Modern trend(s) in cave monitoring[END_REF][START_REF] Badino | Underground meteorology. What's the weather underground?[END_REF].

The cave temperature is coupled with surface atmosphere temperature. The surface ground thermal signal is transferred by conduction through the bedrock, eventually reaching the depth of the cave [START_REF] Domínguez-Villar | Reconstruction of cave air temperature based on surface atmosphere temperature and vegetation changes: implications for speleothem palaeoclimate records[END_REF]. External air penetrating caves rapidly reaches a nearly constant temperature along the entrance sectors due to the buffering effect associated to the increase of relative humidity, and the progressive equilibration with the temperature of the rock [START_REF] Wigley | Geophysical applications of heat and mass transfer in turbulent pipe flow[END_REF][START_REF] Wigley | The physics of Caves[END_REF]. As a result of this process, mean air temperature in cave interiors correspond almost exactly to the average annual value of the outside temperature [START_REF] Moore | Cave temperature[END_REF][START_REF] Moore | Out of phase seasonal temperature fluctuations in Cathedral Cave, Kentucky[END_REF]; Figure 1).

It is worth noting that, depending on the geophysical characteristics of the ground layers above the cave, thermal differences between the cave and the external mean annual temperature may occur. The main factor affecting diversion of cave temperature from the mean annual atmosphere one in temperate climates is related to the type of vegetation cover of the area above the cave, since different levels of shading may affect ground temperature [START_REF] Domínguez-Villar | Reconstruction of cave air temperature based on surface atmosphere temperature and vegetation changes: implications for speleothem palaeoclimate records[END_REF]. In addition, variations in solar radiation, the presence of long lasting snow cover and the evaporative cooling in soils caused by evaporation, are other factors which may contribute to explain small thermal disequilibria between cave and mean annual atmosphere temperatures [START_REF] Beltrami | An examination of short-and long-term air-ground temperature coupling[END_REF]Yzaki et al., 2013). Finally, strong air circulation or significant water streams can impact the cave temperature by the advection or radiation of heat from the fluids (De [START_REF] Freitas | Cave climate: assessment of heat and moisture exchange[END_REF][START_REF] Covington | Mechanisms of heat exchange between water and rock in karst conduits[END_REF], although in those cases the internal temperature is also linked to the external one [START_REF] Smithson | Inter-relationships between cave and outside air temperatures[END_REF][START_REF] Kranjc | Temperature monitoring in Škocjanske Jame Caves[END_REF].

One may argue about the possible effect of the geothermal gradient, as temperature below the surface slowly increases with the vertical distance under the surface (ca. 30 °C/km away from tectonic plates). However, the geothermal gradient is generally so small that tens to hundreds of meters are required to notice a shift in the thermal equilibrium between the external air temperature and the cave one [START_REF] Luetscher | Temperature distribution in karst systems: the role of air and water fluxes[END_REF]. Moreover, in karst regions the geothermal gradient is further buffered by the advection of groundwater [START_REF] Bögli | Karst hydrology and physical speleolology[END_REF], limiting the sources of heat variability affecting caves [START_REF] Badino | Underground meteorology. What's the weather underground?[END_REF].

In light of the intimate relationship of cave temperature with the surface atmosphere temperature, it is no surprise that changes in surface atmosphere temperature results in shifts in the temperature recorded underground [START_REF] Perrier | Long-term climate change and surface versus underground temperature measurements in Paris[END_REF]; Figure 1).

(2) Climate change and the thermal inertia of caves Theoretical models have predicted that climate warming will impact temperature in caves [START_REF] Badino | Cave temperatures and global climatic change[END_REF][START_REF] Covington | Consider a cylindrical cave: a physicist's view of cave and karst science[END_REF], and model-data comparisons have confirmed such scenario [START_REF] Domínguez-Villar | Is global warming affecting cave temperatures? Experimental and model data from a paradigmatic case study[END_REF][START_REF] Šebela | Cave micro-climate and tourism: towards 200 years (1819-2015) at Postojnska jama (Slovenia)[END_REF][START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF], indicating that anthropogenic climate alterations are currently modifying the subterranean microclimate. Most caves show fairly stable air temperature in their interiors, as a result of the slow mechanism conveying underground the outer thermal signals by conduction.

Consequently, there is lag-time between the air temperature increase recorded at the surface and its record in cave interiors. The delay depends on the cave depth (that is, the thickness of the rock above the cave) and on the duration of the anomaly (Domínguez-Villar, 2012). In the pioneer synthesis of [START_REF] Moore | Out of phase seasonal temperature fluctuations in Cathedral Cave, Kentucky[END_REF], the authors speculated with delay times in the order of hundreds to thousands of years for climate anomalies at the surface to reach the deeper sectors of karst massifs. In a study set in a Slovenian cave located 37 m under the surface, the signal of the onset of global warming was recorded 20 years later [START_REF] Domínguez-Villar | Is global warming affecting cave temperatures? Experimental and model data from a paradigmatic case study[END_REF]. However, it should be noticed that at the same site, cave conduits located closer to the surface were expected to record thermal anomalies earlier, whereas cave conduits located deeper into the karst massif are unlikely to have registered the thermal impact of climate warming yet.

(3) Potential implications of subterranean warming on abiotic features

In the past, caves have suffered climate changes that affected their temperature. Those changes were recorded in cave deposits (e.g., Mangini et al., 2005, Fairchild and[START_REF] Fairchild | Definition of the Anthropocene: a view from the underworld[END_REF], although during last millennia they had limited impact on the geophysical environment of most caves. An increase in cave temperature is associated to a higher relative content of water vapour required to reach saturation (i.e., 100% relative humidity).

However, in most cases this has no environmental implication, since dripping water provides enough moisture to reach saturation. Concentration of CO2 in caves is a major control on the dissolution and precipitation of carbonates. However, at the inter-annual timescale, the CO2 concentration does not depend on the cave temperature but on the soil CO2 production and cave ventilation [START_REF] Fairchild | Speleothem science: from processes to past environments[END_REF]. Therefore, subtle changes in cave temperature are not expected to produce large geophysical changes.

However, the warming rate during the last decades is unusually large compared to changes recorded during previous millennia [START_REF] Moberg | Highly variable Northern Hemisphere temperatures reconstructed from low-and high-resolution proxy data[END_REF], affecting the magnitude of thermal change per unit of time. Due to the required time to transfer the external thermal signal to caves, a thermal decoupling exists between the external temperature and the cave interior temperature during a climate change period. This decoupling affects the seasonal air flow in caves that, in most cases, is driven by gradients in air density, depending, in turn, from temperature differences between external atmosphere and the cave interior. Thus, in most caves, enhanced ventilation occurs when the external temperature is below the cave temperature, whereas limited ventilation is recorded during the rest of the year (e.g., [START_REF] Kowalczk | Cave air ventilation and CO2 outgassing by radon-222 modeling: how fast do caves breathe?[END_REF]. Under a scenario of thermal decoupling, the thermal difference between the external and the cave atmosphere increases during winters, and decreases during summers. This large thermal decoupling may be affecting the duration of air flow regime in caves, causing longer periods of enhanced ventilation and reducing the periods with limited ventilation. Despite their potential implication for cave environments and their importance in determining seasonal concentrations of CO2 and other cave environmental parameters, changes in the duration of seasonal air flows have not been studied in detail yet.

(4) Shallow Subterranean Habitats

Besides caves and the associated networks of fissures, superficial subterranean habitats (SSHs; Box 1) stand out as a group of subterranean habitats which are likely to be affected by the global temperature increase. As the name suggests, these habitats are restricted to areas close to the surface and, compared with caves, have higher energy inputs and higher intrinsic variability, including significant microclimatic variations [START_REF] Gers | Diversity of energy fluxes and interactions between arthropod communities: from soil to cave[END_REF][START_REF] Pipan | Temperature variation and the presence of troglobionts in terrestrial shallow subterranean habitats[END_REF][START_REF] Mammola | Ecology and sampling techniques of an understudied subterranean habitat: the Milieu Souterrain Superficiel (MSS)[END_REF].

The increased flux of energy from the atmosphere to the subterranean environment induced by climate change is expected to be primarily in the SSHs-see physical models in [START_REF] Mammola | Ecology and sampling techniques of an understudied subterranean habitat: the Milieu Souterrain Superficiel (MSS)[END_REF]-as well as in the most superficial sectors of caves [START_REF] Badino | Cave temperatures and global climatic change[END_REF]. It is therefore expected that the temperature increase in these habitats will parallel the external one almost synchronically. Compared to the deep subterranean sectors, effects on the SSH fauna are expected to be more immediate.

III. BIOTIC FEATURES (1) Subterranean organisms

The adaptive morphology of subterranean animals has attracted the attention of evolutionary biologists since the discovery of life in caves. Subterranean obligate species, either terrestrial (troglobionts) or aquatic (stygobionts), have indeed often evolved behavioral, physiological and/or morphological traits to survive the peculiar conditions of the subterranean habitat (Box 1). Morphologically, they often lack eyes and pigmentation, and evolved elongated appendages and an assortment of sensory organs to perceive the environment by senses other than vision. Given the general low energy availability of the subterranean environment, they often exhibit low metabolic rates with consequent slow growth rates, high fasting performances, delayed maturation, and extended longevity when compared to their surface relatives [START_REF] Hervant | Long-term fasting and realimentation in hypogean and epigean isopods: a proposed adaptive strategy for groundwater organisms[END_REF]Mezec et al., 2010;Voituron et al., 2011;[START_REF] Fišer | Coevolution of life history traits and morphology in female subterranean amphipods[END_REF]. A charismatic example is offered by the first cave species ever described, the olm Proteus anguinus Laurenti. This aquatic cave salamander reaches sexual maturity at 16 years, lays eggs every 12 years, has an adult average lifespan of nearly 70 years and a predicted maximum lifespan of over 100 years (Voituron et al., 2011).

(2) Potential species sensitivity to climate change While much attention has been given to the effects of global climate change on surface organisms, communities and ecosystems, studies about the influence of temperature increase on subterranean biota are still at their infancy (Table 1). The contrasting response obtained on different animal groups indicate that the sensitivity of subterranean species to altered climatic conditions is likely to depend on phylogeny, evolutionary history and by the degree of subterranean adaptations or other functional traits.

The most important, yet heterogeneous, evidence about the sensitivity of subterranean species to the ongoing climate change derives from physiological tests. As a direct result of a long evolutionary history in a thermally stable environment, it is theoretically expected that most subterranean species should exhibit a stenothermal profile (sensu [START_REF] Huey | Evolution of thermal sensitivity of ectotherm performance[END_REF], which maximises their physiological performance over a narrow temperature range. However, in a climate change perspective, adaptation to narrow ranges of temperature turns out to be a strong limitation. Indeed, while most invertebrates living in the vicinity of the surface have the capacity to withstand temperature variations, most deep subterranean species lack such thermoregulatory mechanisms [START_REF] Novak | Cold tolerance in terrestrial invertebrates inhabiting subterranean habitats[END_REF][START_REF] Raschmanová | Testing the climatic variability hypothesis in edaphic and subterranean Collembola (Hexapoda)[END_REF]. In certain species, even a positive or negative variation of 2°C in respect to their habitat temperature proved to be fatal [START_REF] Mermillod-Blondin | Thermal tolerance breadths among groundwater crustaceans living in a thermally constant environment[END_REF]. However, the figure remains rather crude, as most studies conducted so far focused on single model taxa and thus lacked a wider phylogenetic perspective (but see [START_REF] Pallarés | Heat tolerance and acclimation capacity in unrelated subterranean arthropods living under common and stable thermal conditions[END_REF]. For instance, tests carried on subterranean beetles are divergent, pointing out a wide thermal niche for numerous species of Cholevidae [START_REF] Rizzo | Lack of evolutionary adjustment to ambient temperature in highly specialized cave beetles[END_REF][START_REF] Sánchez-Fernández | Thermal niche estimators and the capability of poor dispersal species to cope with climate change[END_REF]. In addition, it has been shown that two cold stenothermal cave-dwelling beetles in the genus Neobathyscia (Coleoptera: Catopidae) have the ability to synthesise heat shock proteins, which provide resistance to heat exposure [START_REF] Bernabò | Cold stenothermal cave-dwelling beetles do have an HSP70 heat shock response[END_REF]. The same kind of pattern (inducible HSP70 heat shock response) has been observed in subterranean amphipods in the genus Niphargus (Crustacea: Amphipoda) [START_REF] Colson-Proch | First cellular approach of the effects of global warming on groundwater organisms: a study of the HSP70 gene expression[END_REF]. Also, contrasting results were obtained by different authors focusing on survival and performance patterns in stygobiont crustaceans living in thermally buffered subterranean aquatic habitats. Physiological tests suggested that certain species have lost the mechanisms for withstanding thermal variation, while certain others have not [START_REF] Mathieu | Temperatures letales et acclimatation thermique chez Niphargus longicaudatus (Amphipode, Gammaride)[END_REF][START_REF] Issartel | Behavioural, ventilatory and respiratory responses of epigean and hypogean crustaceans to different temperatures[END_REF][START_REF] Colson-Proch | First cellular approach of the effects of global warming on groundwater organisms: a study of the HSP70 gene expression[END_REF][START_REF] Mermillod-Blondin | Thermal tolerance breadths among groundwater crustaceans living in a thermally constant environment[END_REF].

Moreover, an intra-specific variability in the thermal performance across different populations of Niphargus rhenorhodanensis Schellenberg was demonstrated [START_REF] Colson-Proch | Do current environmental conditions explain physiological and metabolic responses of subterranean crustaceans to cold?[END_REF]. Some of this variability may be the result of different cave habitats. For example, epikarst (an SSH) inhabitants face a wider range of temperatures than do deep cave inhabitants (see [START_REF] Pipan | Temperature variation and the presence of troglobionts in terrestrial shallow subterranean habitats[END_REF]. One would expect species facing this increased range of temperatures to have an increased thermal tolerance, but there are no data to directly bear on this point. It seems clear that thermal tolerance is necessarily speciesspecific, and the resulting bioindicator potential should be evaluated on a case-by-case basis.

Apart from thermal tolerance, it is worth noting that air moisture content (i.e. humidity) is one of the most important limiting factors for terrestrial cave obligate species [START_REF] Howarth | The Zoogeography of specialized cave animals: a bioclimatic model[END_REF][START_REF] Howarth | Ecology of cave arthropods[END_REF][START_REF] Simões | Cave physical attributes influencing the structure of terrestrial invertebrate communities in Neotropics[END_REF]. For example, [START_REF] Howarth | The Zoogeography of specialized cave animals: a bioclimatic model[END_REF] tested in laboratory the longevity of spiderlings of Lycosa howarthi Gertsch (Araneae: Lycosidae) at three distinct values of relative humidity (100%, 95%, and 90%). At 90% the longevity dropped to onefourth of the mean value at 100% (15.4±0.9 versus 61.8±1.3 days), pointing toward a pronounced sensitivity to saturation deficit. Accordingly, the maintenance of high humidity levels appears to be essential for the survival of different troglobionts. This is generally explained by the high cuticular permeability of many species, associated with a low resistance to desiccation [START_REF] Howarth | The Zoogeography of specialized cave animals: a bioclimatic model[END_REF][START_REF] Howarth | Ecology of cave arthropods[END_REF][START_REF] Hadley | Water and metabolic relations of caveadapted and epigean lycosid spiders in Hawaii[END_REF]. As previously explained, in caves from regions with limited water infiltration, relative humidity of cave environments may be reduced as a consequence of the cave temperature increase.

However, the impacts associated to the alteration of relative humidity due to increased temperature is rather limited when compared to role of infiltrating water in karst [START_REF] Eraso | Ideas sobre la climática subterránea[END_REF]. Nevertheless, in many regions of the planet, such as the Mediterranean, it adds on top of the decrease in precipitation [START_REF] Xoplaki | Wet season Mediterranean precipitation variability: influence of large-scale dynamics and trends[END_REF] that is reducing infiltration of water in karst, pushing caves towards lower relative humidity scenarios. Consequently, drops in the relative humidity in some regions of the planet are likely to play a critical role in the species response of subterranean species to climate change [START_REF] Shu | Drought and China's cave species[END_REF]. However, studies observing eco-evolutionary responses mostly concern surface organisms, while subterranean species may not be able to adapt in the same way. As a rule of thumb, subterranean species can theoretically couple with climatic alterations only by shifting their distributional range (i.e. spatial information) or by adapting to the new environmental conditions (i.e. physiological information). It must be added that it is difficult to integrate temporal information as phenology in caves is not directly linked to climatic factors (e.g. light, temperature; Figure 2).

If the dispersal capacity is enough to accompany the spatial change in temperatures, one of the most common response of surface organisms to climate change is the altitudinal and/or latitudinal shift of distribution ranges [START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF][START_REF] Chen | Rapid range shifts of species associated with high levels of climate warming[END_REF]. However, most subterranean systems are geographically isolated, acting as islands for many species (e.g., [START_REF] Cardoso | Diversity and community assembly patterns of epigean vs. troglobiont spiders in the Iberian Peninsula[END_REF][START_REF] Niemiller | Effects of climatic and geological processes during the Pleistocene on the evolutionary history of the northern cavefish, Amblyopsis spelaea (Teleostei: Amblyopsidae)[END_REF][START_REF] Fattorini | Trapped in the web of water: Groundwater-fed springs are island-like ecosystems for the meiofauna[END_REF][START_REF] Rizzo | Substratum karstificability, dispersal and genetic structure in a strictly subterranean beetle[END_REF]. This ecological feature, together with physiological constraints, caused an extreme reduction in the dispersal potential of most troglobionts [START_REF] Bohonak | Dispersal, gene flow and population structure[END_REF]. Indeed, as discussed in [START_REF] Juan | Evolution in caves: Darwin's 'wrecks of ancient life' in the molecular era[END_REF], a breadth of studies uncovered pronounced genetic structuring and low gene flow between populations inhabiting different caves. Thus, long range dispersal events in troglobionts are extremely unlikely and, in fact, available future projections point at reductions in troglobiont ranges rather than range expansions or shifts [START_REF] Sánchez-Fernández | Thermal niche estimators and the capability of poor dispersal species to cope with climate change[END_REF]Mammola et al., 2018). In this respect, due to the same kind of barriers inhibiting local migrations, altitudinal shifts may be equally unlikely. However, it is worth noting that isolation between caves should be theoretically higher for troglobionts than for stygobionts, as subterranean waters present broader connections than subterranean terrestrial habitats [START_REF] Christman | The relationship between cave biodiversity and available habitat[END_REF], often resulting in aquatic subterranean species having wider ranges then terrestrial ones [START_REF] Lamoreaux | Stygobites are more wide-ranging than troglobites[END_REF]. Also, it has been recently shown that adaptation to subterranean habitats is not always a one-way evolutionary path, with subterranean species being occasionally able to recolonise and widely disperse in surface habitats [START_REF] Prendini | Troglomorphism, trichobothriotaxy and typhlochactid phylogeny (Scorpiones, Chactoidea): more evidence that troglobitism is not an evolutionary dead end[END_REF], Copilas-Ciocianu et al., 2018).

At a smaller scale, in response to climate change cave species could theoretically change their spatial distribution inside the cave system itself (Fejér and Moldovan, 2013;[START_REF] Mammola | Seasonal dynamics and micro-climatic preference of two Alpine endemic hypogean beetles[END_REF]Mammola et al., , 2017;;[START_REF] Lunghi | Cave features, seasonality and subterranean distribution of non-obligate cave dwellers[END_REF]. Evidence suggests that a number of subterranean species are indeed able to seasonally migrate toward greater depths and vice versa (e.g., [START_REF] Juberthie | Relations entre le climat, le microclimat et les Aphaenops cerberus dans le grotte de Sainte-Catherine (Ariege)[END_REF][START_REF] Crouau-Roy | Dynamic and temporal structure of the troglobitic beetle Speonomus hydrophilus (Coleoptera: Bathyscimae)[END_REF]Tobin et al., 2013).

However, a permanent shift in the species spatial niche toward greater depths may be less likely, because this would affect the general energetic needs, as the energy input far from the surface is generally scarcer especially in temperate oligotrophic caves [START_REF] Gers | Diversity of energy fluxes and interactions between arthropod communities: from soil to cave[END_REF]Souza-Silva et al., 2011;Venarsky et al., 2014). Moreover, most of specialised subterranean arthropods are confined in caves lacking climatic heterogeneity to be exploited (Figure 2).

If enough genetic variability is encompassed by a population, allowing for a change of dominant traits over time, adaptive evolution of physiology may be the best response to climate warming [START_REF] Bradshaw | Evolutionary response to rapid climate change[END_REF][START_REF] Visser | Keeping up with a warming world; assessing the rate of adaptation to climate change[END_REF][START_REF] Williams | Towards an integrated framework for assessing the vulnerability of species to climate change[END_REF]. In particular, physiological adaptation to increasing temperatures has been reported in surface organisms, resulting in increased metabolic rate and faster growth (Hughes, 2000). Although physiological modifications are hard to be measured or predicted, previous studies suggest that subterranean organisms exhibit decreased metabolic plasticity, i.e., show far less elevation of metabolism than their close surface relatives in response to higher temperatures [START_REF] Dresco-Derouet | Contribution a l'étude de la biologie de deux Crustacés aquatiques cavernicoles: Caecosphaeroma burgundum D. et Niphargus virei[END_REF]Vandel, 1965;[START_REF] Hervant | Respiratory adaptations in epigean and hypogean aquatic crustaceans[END_REF]. Indeed, habitat specialists often present low functional variability due to specialisation (but see [START_REF] Faille | A molecular phylogeny shows the single origin of the Pyrenean subterranean Trechini ground beetles (Coleoptera: Carabidae)[END_REF][START_REF] Juan | Evolution in caves: Darwin's 'wrecks of ancient life' in the molecular era[END_REF][START_REF] Ribera | Ancient origin of a Western Mediterranean radiation of subterranean beetles[END_REF]. This in turn limits the evolutionary potential even within large populations [START_REF] Kellermann | Very low additive genetic variance and evolutionary potential in multiple populations of two rainforest Drosophila species[END_REF]. With a relatively stable habitat and restricted distribution, genetic variability of cave populations is in fact often very low [START_REF] Juan | Evolution in caves: Darwin's 'wrecks of ancient life' in the molecular era[END_REF]; but see [START_REF] Cieslak | Life-history specialization was not an evolutionary dead-end in Pyrenean cave beetles[END_REF]Stern et al., 2017). In addition, cave species typically have low reproductive rates, slow growing/maturing and long life-span (Voituron et al., 2011). These characteristics generally constrain rapid evolutionary change [START_REF] Mckinney | Extinction vulnerability and selectivity: combining ecological and paleontological views[END_REF].

(4) Inter-connection with other systems

The arguments presented in the previous section illustrate the potential eco-evolutionary responses of cave species to climate change within the subterranean domain. However, it is worth noting that most subterranean habitats are semi-closed systems, thus interconnected and strongly dependent of other environments, especially surface habitats.

Being light deficient, the subterranean ecosystem is strongly dependent from the outside ecosystems providing the fundamental energy inputs for its maintenance [START_REF] Gibert | Subterranean ecosystems: a truncated functional biodiversity[END_REF]-although chemolithotrophic organisms may account for primary production within caves [START_REF] Northup | Geomicrobiology of caves: a review[END_REF]. Trophic inputs mainly consist of organic materials passively transported underground, as well as by active migration of animals within some caves. Consequently, energy flow in a typical subterranean habitat is strongly influenced by seasonal fluctuations [START_REF] Culver | The Biology of Caves and other Subterranean Habitats[END_REF] and the supply of organic matter might also increase with current surface land-use changes [START_REF] Wilhelm | Comparison of routine metabolic rates of the stygobite, Gammarus acherondytes (Amphipoda: Gammaridae) and the stygophile, Gammarus troglophilus[END_REF]. Phenological shifts-the advancement or postponing of annual phenomena-in relation to global change have been extensively documented in surface species [START_REF] Peñuelas | Responses to a warming world[END_REF], especially in plants [START_REF] Cleland | Shifting plant phenology in response to global change[END_REF][START_REF] Wolkovich | Warming experiments underpredict plant phenological responses to climate change[END_REF]. It is thus theoretically expected that the amount and timing of allochtonous energy inputs in caves will change accordingly, with direct effects on the subterranean biota.

It is also well documented how climate changes will enhance invasion processes, causing the introduction and spread of alien species [START_REF] Bellard | Will climate change promote future invasions?[END_REF], which are expected to affect cave biology [START_REF] Wynne | Disturbance relicts in a rapidly changing world: the Rapa Nui (Easter Island) factor[END_REF]. Accordingly, several studies have documented the potential recent spread of pathogen fungi in terrestrial subterranean habitats [START_REF] Escobar | Potential for spread of the white-nose fungus (Pseudogymnoascus destructans) in the Americas: use of Maxent and NicheA to assure strict model transference[END_REF] or of alien species in marine caves [START_REF] Gerovasileiou | Alien biodiversity in Mediterranean marine caves[END_REF]. Furthermore, dramatic modification of surface habitats (e.g., aridification and deforestation), may also induce changes in the hydrological regimes of caves, with consequent further degradation of the cave ecosystem (Trajano et al., 2009, Bichuette and[START_REF] Bichuette | Conservation of Subterranean Fishes[END_REF].

From a slightly different perspective, it is likely that some soil and surface organisms may exploit subterranean environments, especially cave entrances and SSHs, as refugial area in a climate change scenario. Indeed, it can be argued that the critical difference between caves or other subterranean habitats and surface habitats is that temperature extremes are buffered. If these extremes are what constitute a physiological challenge, then these habitats may serve as a refuge. For instance, different soil arthropods (e.g. springtails, woodlouse) may find more favourable conditions (e.g. lower desiccation level, lower predation level) in the subterranean habitats that in surface habitats [START_REF] Fernandes | Does the cave environment reduce functional diversity?[END_REF][START_REF] Mammola | Ecology and sampling techniques of an understudied subterranean habitat: the Milieu Souterrain Superficiel (MSS)[END_REF]. Similarly, arthropods adapted to cold environments may find safer places for survival in some subterranean habitats as in karst areas [START_REF] Raschmanová | Community composition and cold tolerance of soil Collembola in a collapse karst doline with strong microclimate inversion[END_REF].

These examples do not intend to be exhaustive, but we rather aim to highlight the importance of considering the deep relation between caves and other ecosystems, and thus to consider the reciprocal interaction between the underground and the above-ground world. However, in lack of specific studies involving the collaboration of cave-based scientists and ecologists operating in other research fields, most of these interactions are still difficult to disclose and predict.

IV. FUTURE DIRECTIONS

We have discussed that caves represent remarkable examples of natural laboratories in which the climatic conditions are as homogeneous as the one that could be obtained in a laboratory. Thus, in contrast with fluctuating surface temperatures, temperature measurements in caves allow researchers to readily detect temperature variations related to climate warming (see, e.g., [START_REF] Fejér | Population size and dispersal patterns for a Drimeotus (Coleoptera, Leiodidae, Leptodirini) cave population[END_REF][START_REF] Šebela | Cave micro-climate and tourism: towards 200 years (1819-2015) at Postojnska jama (Slovenia)[END_REF][START_REF] Domínguez-Villar | Is global warming affecting cave temperatures? Experimental and model data from a paradigmatic case study[END_REF]. Currently monitoring programs in caves mostly involves the analysis of speleothems, which requires a single visit to obtain a sample [START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF]. Indeed, the sheltered environment in which speleothems grow, and their capacity to preserve even sub-annual climatic events, offers a certain potential for their use as climatic archives of the Anthropocene [START_REF] Fairchild | Definition of the Anthropocene: a view from the underworld[END_REF][START_REF] Fairchild | Geochemical records in Speleothems[END_REF][START_REF] Waters | Global Boundary Stratotype Section and Point (GSSP) for the Anthropocene Series: Where and how to look for potential candidates[END_REF]. On the other hand, long-term monitoring programs within karst settings are currently very rare (Brookfield et al., 2016;[START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF]; we thus point out that monitoring programs should be established for relevant abiotic and biotic variables-see, e.g., the notable case of Postojna cave [START_REF] Šebela | Local characteristics of Postojna Cave climate, air temperature, and pressure monitoring[END_REF][START_REF] Šebela | Cave micro-climate and tourism: towards 200 years (1819-2015) at Postojnska jama (Slovenia)[END_REF][START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF].

With small distribution ranges (due to the isolation among subterranean systems), small population size (due to low energy availability), and restricted habitat (by definition), cave organisms often fulfil all forms of rarity [START_REF] Rabinowitz | Seven forms of rarity[END_REF][START_REF] Gaston | Rarity[END_REF].

Species with small range and abundance are expected to have low adaptive potential in the face of environmental change [START_REF] Williams | Towards an integrated framework for assessing the vulnerability of species to climate change[END_REF]. Subterranean species are in fact prevalent on conservation priority lists [START_REF] Martín | Using taxonomically unbiased criteria to prioritize resource allocation for oceanic island species conservation[END_REF][START_REF] Cardoso | Adapting the IUCN Red List criteria for invertebrates[END_REF]IUCN, 2017). We have shown that a significant number of subterranean species cannot accommodate to changing conditions by dispersal or microhabitat use, and the only possibility to cope with climate change will be to persist in situ. However, if natural populations cannot adapt to environmental change by means of adaptive shifts, they should be more prone to local extinction due to the direct effects of climate change than their surface counterparts. We therefore advocate the need for long-term monitoring programs for cave species, namely those with high threat status.

Nevertheless, it is worth noting that not all subterranean species are likely to become extinct in light of a warming climate. Depending on the organisms under consideration and its degree of subterranean adaptation, a variety of possible responses to an altered climate has to be expected. Whilst a great deal of attention is paid to troglobionts, there are high-dispersal subterranean species which may, on the contrary, be able to respond with dispersal to climatic alterations (e.g., [START_REF] Mammola | Rapid poleward distributional shifts in the European cave-dwelling Meta spiders under the influence of competition dynamics[END_REF]. We have also discussed that movements of faunas from surface to subterranean habitats, and vice versa, can be expected. The confrontation of these opposite displacements, even if they are currently rare, might become a research priority in the next future, because of the rapid changes observed in surface habitats and the loss of climatic stability of subterranean ones. In that context, analytical approaches typically used in landscape and island ecology-e.g. source-sink and metapopulations dynamics [START_REF] Pipan | How to protect a diverse, poorly known, inaccessible fauna: identification of source and sink habitats in the epikarst[END_REF]Moldovan et al., 2012;[START_REF] Fattorini | Trapped in the web of water: Groundwater-fed springs are island-like ecosystems for the meiofauna[END_REF]Trajano and de Carvalho, 2017)-will become helpful to document the amplitude of these reciprocal movements.

A general caveat to consider in discussing this topic is that studies conducted so far have been mostly correlative, meaning that a causal attribution of recent biological trends to climate change in subterranean species is currently lacking. Although it is difficult to overcome this impediment, it is likely that advances can be done both by studying multiple subterranean systems and by combining multiple lines of evidence [START_REF] Mammola | Applying species distribution models to cves and other subterranean habitats[END_REF][START_REF] Pipan | Analyzing climate change and surface-subsurface interactions using the Postojna Planina Cave System (Slovenia) as a model system[END_REF]. For instance, the simultaneous use of physiological data, genetic evidence and forecast derived from statistical projections has a great deal of potential.

There is little doubt that this integrated approach would greatly benefit the study of climate change dynamics in deep subterranean habitats, prompting a fast and significant advance in knowledge.

V. CONCLUSIONS

(1) Due to their intrinsic environmental stability, subterranean ecosystems are unique models for the study of global change biology. However, monitoring programs of caves abiotic conditions are rare and we still have a limited understanding of the mechanisms that underlie survival of the most adapted species to climatic and environmental alterations, even more so in tropical regions.

(2) Most subterranean species are expected to have fine-tuned their thermal physiology to the narrow and stable condition of their habitats, and should have low adaptive potential in the face of environmental change. The contrasting response obtained on different animal groups suggest that the sensitivity of subterranean species to temperature variations is likely to significantly differ depending on phylogeny, evolutionary history and degree of subterranean adaptation.

(3) Subterranean communities are relatively depauperate, with less redundancy in ecological roles among species. Disruption of trophic webs and species interactions is therefore much likely in such communities.

(4) Some external organisms may be able to exploit subterranean environments as refugia in a climate change scenario. However, if caves can be seen as shelters for preadapted surface and soil species in the face of rapid climate change, they may become dead-end traps for their current inhabitants, being characterised by a poor dispersal ability a low adaptation potential.

( 3 )

 3 Eco-evolutionary response to climate change Despite the predicted and observed extinction of different taxa across terrestrial and aquatic biomes (Thomas et al., 2004, Cahill et al., 2013), Bellard et al. (2012) underlined how species can respond to climate change by shifting their climatic niche along three non-exclusive axes: time (e.g. phenology), space (e.g. range) and self (e.g. physiology).

( 5 )

 5 When thinking about the ecology of subterranean ecosystems, it is crucial to consider their interconnections with other environments, especially surface habitats. Studies involving the collaboration of cave-based scientists with ecologists operating in other research fields are likely to produce a more realistic picture of the effects of climate change in the underground world. (6) We encourage renewed effort to better characterise the dynamic processes and challenges associated with global climate change in deep subterranean habitats. Evidence arising from such studies would not exclusively be important for the conservation of the subterranean fauna, but more generally they would contribute to increase our Souza-Silva M, Parentoni MR and Ferreira LR (2011) Trophic dynamics in a neotropical limestone cave. Subterranean Biology 9: 127-138. doi:10.3897/subtbiol.9.2515 Stern DB, Breinholt J, Pedraza-Lara C et al. (2017) Phylogenetic evidence from freshwater crayfishes that cave adaptation is not an evolutionary dead-end. Evolution 71(10): 2522-2532. doi:10.1111/evo.13326 Taylor RG, Scanlon B, Döll P et al. 2013. Ground water and climate change. Nature Climate Change 3: 322-329. doi:10.1038/nclimate1744 Thomas CD, Cameron A, Green RE et al. (2004) Extinction risk from climate change. Nature 427(6970): 145-148. doi:10.1038/nature02121 Tobin BW, Hutchins BT and Schwartz BF (2013) Spatial and temporal changes in invertebrate assemblage structure from the entrance to deep-cave zone of a temperate marble cave. International Journal of Speleology 42(3): 203-214. doi:10.5038/1827-806X.42.3.4 Trajano E and Carvalho MR de (2017) Towards a biologically meaningful classification of subterranean organisms: a critical analysis of the Schiner-Racovitza system from a historical perspective, difficulties of its application and implications for conservation. Subterranean Biology 22: 1-26. doi:10.3897/subtbiol.22.9759 Trajano E, Secutti S and Bichuette ME (2009) Population decline in a Brazilian cave catfish, Trichomycterus itacarambiensis Trajano and Pinna, 1986 (Siluriformes): reduced flashflood as a probable cause. Speleobiology Notes 1: 24-27. doi:10.1590/1982-0224-20170057 Vandel A (1965) Biospeleology. The biology of cavernicolous animals. Oxford: Pergamon Press.

Figure 1 .

 1 Figure 1. The (theoretical) sinusoidal conduction dominating the thermal signal of a

Figure 2 .

 2 Figure 2. Eco-evolutionary response to climate change. Potential shifts in the thermal

Table 1 .

 1 A selection of recent experimental studies investigating the response of subterranean organisms to global climate change. Only articles written in English are reported.

	Area	Model organism(s)	Ecological classification	Method(s)	Observed/predicted effect(s)	Reference
					Expansion of the spatial niche
	Europe	Beetles (various genera)	Troglobiont	Indirect evidence extrapolated from species accumulation curves	of cave species toward the surface-i.e., into superficial cavities and Shallow Subterranean Habitats.	Brandmayr et al., 2013
					i) Most lineage have lost
					some of the thermoregulatory
	Pyrenees (France, Spain)	outgroups) Beetles (gen. Troglocharinus +	Troglobiont	Physiological tests	ii) Broader thermal tolerance mechanisms common in temperate insects	Rizzo et al. 2015
					than expected by habitat
					climatic seasonality
	Pyrenees (France, Spain)	Beetles (Tribe Leptodirini)	Troglobiont	i) Species Distribution Modelling ii) Molecular data iii) Physiological test	A slight future decline in habitat suitability, but a broad thermal tolerance in most subterranean species	Sanchez-Fernandez et al., 2016
					Subterranean species	Colson-Proch et
	Jura Mountains (France)	Crustacean (gen. Niphargus)	Stygobiont	Expression gene profile	maintain the expression of heat shock protein	al. 2010
					i) Future decline in habitat
	Western Alps (Italy)	Spiders (gen. Troglohyphantes)	Troglobiont	Species Distribution Modelling	suitability ii) Potential local extinction in	Mammola et al., 2018
					a number of populations
					i) Some species are sensitive
					to changes in temperature
					(±2°C), although one	Mermillod-
					exhibited a higher thermal
	Jura Mountains (France)	Aquatic isopods (gen. Proasellus)	Stygobiont	Physiological test	tolerance breadth (11°C) ii) Extinction risk of	Blondin et al., 2013.
					groundwater endemics is
					higher than that of widely
					distributed species (inferred).
	Medio Valdarno porous aquifer (Italy)	Aquatic copepod (Diacyclops belgicus Kiefer)	Stygophile/Stygo biont	Physiological test	No significant variations in the oxygen consumptions to a +3 °C change in temperature	Di Lorenzo & Galassi, 2017
	Great Britain	Spiders (gen. Meta)	Troglophile	Species Distribution Modelling	Future poleward shift in the distribution ranges	Mammola 2017
					i) Future poleward shift in the
				Species Distribution	distribution ranges	Mammola &
	Europe	Spiders (gen. Meta)	Troglophile	Modelling	ii) Niche overlap between	Isaia, 2017
					congeneric species

This review was developed in the frame of the research project 'The Dark Side of Climate Change' funded by University of Turin and Compagnia di San Paolo (Grant Award: CSTO162355). SM conceived the idea and led the writing. SM, EP, PC, PV, DCC, TP and MI provided general arguments for all biological sections.

Yazaki T, Iwata Y, Hirota T et al. (2013) Influences of winter climatic conditions on the relation between annual mean soil and air temperatures from central to northern Japan. [START_REF] Curl | On the definition of a cave[END_REF], network of fissures with sizes smaller than the human-scale and artificial subterranean habitats (mines, blockhouses, cellars, etc.) [START_REF] Culver | The Biology of Caves and other Subterranean Habitats[END_REF].

Cold

Troglobiont/Stygobiont. A terrestrial (troglo-) or aquatic (stygo-) species having its source population in the subterranean habitat (Trajano & Carvalho, 2017). Usually shows pronounced adaptation to the subterranean conditions.

Troglophile/Stygophile. A terrestrial (troglo-) or aquatic (stygo-) species forming source populations both in subterranean and surface habitats (Trajano & Carvalho, 2017).

Troglomorphism. The suite of morphological, physiological and behavioral adaptations to the subterranean habitat [START_REF] Christiansen | Morphological adaptations[END_REF]. Examples of troglomorphic traits include pigment loss, eye regression and parallel increase in the development of other receptors and sensory organs, reduction in metabolic rate, increase in life span.


Andrey Polyakov
email: andrey.polyakov@inria.fr

Sergiy Zhuk
email: sergiy.zhuk@ie.ibm.com

Generalized Lyapunov Exponents of Homogeneous Systems

The paper deals the method of Lyapunov exponents for a class of a generalized homogeneous systems. Homogeneous systems may have some sup-exponential and super-exponential grows. In this case, the method of Lyapunov exponents becomes non-informative, e.g. all Lyapunov exponents may equal to zero but the system is globally uniformly asymptotically stable. In this paper we propose an approach which allows us to analyze a behavior of such homogeneous systems by means of the method of Lyapunov exponents.

I. INTRODUCTION AND PROBLEM STATEMENT

The concept of stability introduced by A.M. Lyapunov [START_REF] Lyapunov | The general problem of the stability of motion[END_REF] considers some nominal motion x * (t, x 0 ) of a dynamic system ẋ = f (x), f :

R n → R n (1)
and studies a behavior of the system for perturbed initial condition x 0 + ∆x 0 . If small perturbations always imply small deviations of the perturbed motion x t0,x0+∆x0 (t) from x * t0,x0 (t) then the nominal motion is called stable. The conventional approach to the corresponding analysis is based on linearizion of the system along the trajectory ė = A(t)e + g(t, e),

where e(t) = x(t) -x * (t, x 0 ), A(t) = f (x * (t, x 0 )) and g = (g 1 , ..., g n ) is given by g i (t, e) = e The similar problem appears for observer design for nonlinear systems (see e.g. [START_REF] Frank | A detectability criterion and data assimilation for non-linear differential equations[END_REF]). The method of Lyapunov exponents [START_REF] Bylov | Theory of Lyapunov exponents and its application to problems of stability[END_REF], [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF] is frequently utilized for the stability analysis of the latter system. This method allows stability analysis provided that the system satisfies some regularity property( see e.g. [START_REF] Bylov | Theory of Lyapunov exponents and its application to problems of stability[END_REF], [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF]).

Homogeneity is a kind of symmetry when an object (a function, a vector field, a set etc) remains invariant in a certain sense with respect to a class of transformations called dilations [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. For example, if ∃ν ∈ R : f (e s x) = e νs f (x) for all s ∈ R and for all x the the function f is called standard homogeneous (or homogeneous in Euler's sense). All linear and a lot of nonlinear models of mathematical physics are homogeneous in a generalized sense [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]. Homogeneous models of dynamical systems also appears as local approximation of nonlinear systems if linearization is not informative or simply impossible [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], e.g. asymptotic stability of the zero solution of the system ẋ = -x 3 , x ∈ R cannot be studied by means of linearization at zero, which has the form ẋ = 0. Below we also show that the conventional analysis based on Lyapunov exponents also does not allow us to study stability of this system.

The aim of this paper is to expand/adapt the method Lyapunov exponents to a class of homogeneous systems.

Notation. R, R + ,λ min , λ max , co, sign; a class K function;

A := inf x =0 Ax x , A := sup x =0 Ax x , where A ∈ R n×n , x ∈ R n .

II. GENERALIZED HOMOGENEITY A. Linear Dilations and Canonical Homogeneous Norm

Different generalization of standard (Euler's) homogeneity can be found in the literature [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF]. The generalized homogeneity studied in [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF], [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF], [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF] deals with the group of linear transformations (linear dilations).

Definition 1:

[16] A map d : R → R n×n is called dilation in R n if
S := {x ∈ R n : x = 1}. The dilation d is a continuous group of invertible linear maps d(s) ∈ R n×n , d(-s) = [d(s)] -1 . The matrix G d ∈ R n×n G d = lim s→0 d(s)-I s is known [
ds = G d d(s) and d(s) = e G d s = +∞ i=0 s i G i d i! , s ∈ R. (3) The latter implies d(s 1 )-d(s 2 ) = G d s1 s2 d(s)ds, s 1 , s 2 ∈ R.
The most popular dilations in R n are [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] • uniform (or standard) dilation (L. Euler 17th century) :

d 1 (s) = e s I n , s ∈ R,
• weighted dilation (Zubov 1958, [5]): , s ∈ R, r i > 0, i = 1, ..., n

d 2 (s) =
They satisfy Definition 1 with G d1 = I n and G d2 = diag{r i }, respectively. In fact, any anti-Hurwitz 1 matrix G d ∈ R n×n defines a dilation d(s) = e G d s in R n . The geometric dilation studied in [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF] is more general since it allows the dilation group to be nonlinear. Definition 2: The dilation d is said to be strictly monotone if there exists β > 0 such that d(s) ≤ e βs as s < 0. Obviously, the monotonicity of a dilation may depend on the norm • in R n .

Theorem 1: [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF] Let d be a dilation in R n , then 1) all eigenvalues λ i of the matrix G d are placed in the right complex half-plane, i.e., (λ i ) > 0, i = 1, 2, ..., n; 2) there exists a matrix P ∈ R n×n such that

P G d + G d P 0, P = P 0; (4)
3) the dilation d is strictly monotone with respect to the weighted Euclidean norm • = •, • induced by the inner product x, z = x P z with P satisfying (4). Moreover,

e αs ≤ d(s) ≤ d(s) ≤ e βs if s ≤ 0, e βs ≤ d(s) ≤ d(s) ≤ e αs if s ≥ 0, (5)
where α = 1 2 λ max P

1 2 G d P -1 2 + P -1 2 G d P 1 2 and β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G d P 1 2
. The latter theorem proves that any dilation d is strictly monotone if R n is equipped with the norm x =

√

x P x, provided that the matrix P 0 satisfies (4).

Definition 3: A continuous function p : R n → R + is said to be d-homogeneous norm if p(x) → 0 as x → 0 and p(d(s)x) = e s p(x) > 0 for x ∈ R n \{0} and s ∈ R. Obviously, the d-homogeneous norm is neither a norm nor semi-norm in the general case, since the triangle inequality may not hold. However, many authors (see [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Grune | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF] and references therein) call a function satisfying the latter definition a homogeneous norm. We follow this tradition. The canonical homogeneous norm • d : R n → R + is defined as

x d = e sx where s x ∈ R : d(-s x )x = 1. (6)
The map • d : R n → [0, +∞) is well defined and single-valued for monotone dilations [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF]. In [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] such a homogeneous norm was called canonical because it is induced by the (canonical) norm in R n . Notice that

d(ln x d ) ≤ x ≤ d(ln x d ) for x ∈ R n ,
< 0 if s ∈ R, x ∈ R n \{0} and ∂ x d ∂x = x d ∂ z ∂z | z=d(-s)x ∂ z ∂z | z=d(-s)x G d d(-s)x s=ln x d (7)
1 The matrix

G d ∈ R n is anti-Hurwitz if -G d is Hurwitz.
Below we use the notation • d only for the canonical homogeneous norm induced by the weighted Euclidean norm x =

√

x P x with a matrix P 0 satisfying (4). The unit sphere S is defined using the same norm. In this case the formula [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] becomes

∂ x d ∂x = x d x d(-ln x d )P d(-ln x d ) x d(-ln x d )P G d d(-ln x d )x .

B. Homogeneous vector-fields and flows

Vector fields, which are homogeneous with respect to dilation d, have many properties useful for control design and state estimation of linear and nonlinear plants as well as for analysis of convergence rates [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF].

Definition 4:

[16] A vector field f : R n → R n (resp. a function h : R n → R) is said to be d-homogeneous if there exists ν ∈ R f (d(s)x) = e νs d(s)f (x), ∀x ∈ R n \{0}, ∀s ∈ R. (8) (resp. h(d(s)x) = e νs h(x), ∀x ∈ R n \{0}, ∀s ∈ R.) (9) The number ν ∈ R is called the homogeneity degree of f (resp. h). Let F d (R n ) (resp. H d (R n )) be the set of vector fields R n → R n (resp. functions R n → R) satisfying the identity (8) (resp. (9)), which are continuous on R n \{0}. Let deg F d (f ) (resp. deg H d (h)) denote the homogeneity degree of f ∈ F d (R n ) (resp. h ∈ H d (R n )).
The homogeneity allows local properties (like smoothness) of vector fields (functions) to be extended globally [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF].

Lemma 1 ( [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF]):

The vector field f ∈ F d (R n ) is Lip- schitz continuous (smooth) on R n \{0}
if and only if it satisfies Lipschitz condition (smooth) on the unit sphere S = {x ∈ R n : x P x = 1}, where P satisfies (4). The next results is a corollary of Euler's Homogeneous Function Theorem.

Lemma 2: If f : R n → R n is continuously differentiable on S and f (e s x) = f (x) for all x ∈ R n and all s ∈ R then

f (x)x = 0, ∀x ∈ R n \{0}, e -s f (x) = f (e s x) , ∀x ∈ R n \{0}, ∀s ∈ R.
The result follows, for example, from [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF]Corollary 2].

Homogeneity may simplify the analysis of differential equations. The most important property of d-homogeneous systems is the symmetry of solutions [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Grune | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF].

Theorem 2 ([16]): If ϕ(•, x 0 ) : [0, T ) → R n is a solution to ẋ = f (x), f ∈ F d (R n ) ( 10
)
with the initial condition

x(0) = x 0 ∈ R n , then ϕ(•, d(s)x 0 ) : [0, e -νs T ) → R n defined as ϕ(t, d(s)x 0 ) = d(s)ϕ(te νs , x), s ∈ R is a solution to (10) with the initial condition x(0) = d(s)x 0 , where ν = deg F d (f ).
The next result shows that any generalized homogeneous system is equivalent to a standard homogeneous one. Theorem 3: The systems [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF] and

ż = z 1+deg d (f ) f (z), (11)
f (z) = (In-G d )z zP z P G d z + I n f z z with z =
√ z P z and P satisfying (4), are topologically equivalent on R n \{0}. Moreover, these systems are topologically equivalent on R n if the origin of one of these systems is Lyapunov stable. Proof. Topological equivalence means that there exists a homeomorphism 2 between solution sets of two systems. In [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF] its is shown that the change of variables

z = Φ(x)
where Φ : R n → R n is given by

Φ(x) = x d d(-ln x d )x, x ∈ R n , (12)
transforms the system (10) to the system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Its inverse Φ -1 is as follows 12) is a diffeomorphism (i.e. continuously differentiable homeomorphism) on R n \{0}. This means that a set of solutions of the system (10) defined on R n \{0} homeomorphic to the set of solutions of the system (11) defined on R n \{0}. The latter means that if one of the systems is Lyapunov stable then the another one is also Lyapunov stable. Since Lyapunov stability implies uniqueness of the zero solution, then the stable systems [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF] and [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] are topologically equivalent on R n .

Φ -1 (z) = d(ln z ) z z , z ∈ R n . Proposition 1 implies that Φ defined by (
The next proposition shows that if the vector-field f satisfies a Lipschitz condition on the unit sphere, then certain equivalence can be established between homogeneous and globally Lipschitz system.

Lemma 3: Let us consider the system

ẏ = y f (y), ( 13
)
where f is defined by the formula [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], and let f satisfy Lipschitz condition of the unit sphere S. Then the system (13) has a unique solution y(•, y 0 ) : [0, +∞) → R n for each y 0 ∈ R n and for any y 0 = 0 one has [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] with the initial condition z(0) = y 0 and

z t ∫ 0 ds y(s,y0) deg F d f , y 0 = y(t, y 0 ), t > 0 z (τ, y 0 ) = y τ ∫ 0 z(s, y 0 ) deg F d f ds, y 0 , τ ∈ [0, τ max (y 0 )), where z(•, y 0 ) : [0, T (y 0 )) → R n is a unique solution of
T (y 0 ) = +∞ ∫ 0 y(s, y 0 ) -deg F d f ds.
Proof. Notice that by construction

sup y∈R n f (y) = K < +∞.
Since f satisfies Lipschitz condition on the unit sphere S then due to Lemma 1, the function y → y 1+deg F d f f (y) is Lipschitz continuous on R n \{0} and the function y → 2 Homeomorphism is a continuous invertible mapping with a continuous inverse.

y f (y) is Lipschitz continuous on R n . This means that the system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] has unique solutions on R n \{0} and the system (13) has unique solutions y(t, y 0 ) for all y 0 ∈ R n . Moreover, 0 < y(t , y 0 ) < +∞ for all t > 0 provided that y 0 = 0. Indeed, finite-time blow up is impossible because f ≤ K but Lipschitz continuity at the origin does not allow finitetime extinction.

Let us denote τ = t 0 y(s, y 0 ) -deg F d (f ) ds and show that z(τ, y 0 ) defined above with τ ∈ [0, τ max (y 0 )) is a unique solution of the system [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF] with the initial condition z(0) = y 0 . Indeed,

z(τ, y 0 ) f (z(τ, y 0 )) = y(t, y 0 ) f (y(t, y 0 )) = dy(t,y0) dt = d dt z t 0 y(s, y 0 ) -deg F d f ds, y 0 = dz(τ ) dτ dτ dt = dz(τ ) dτ y(t, y 0 ) -deg F d f = dz(τ ) dτ z(τ, y 0 ) -deg F d f .
The second claimed identity can be proven similarly. The latter lemma guarantees that if some stability result is obtained for the system (13) then the similar result holds for the system [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF] with the only difference is a convergence rate, e.g. the case deg F d f < 0 would correspond to a finite-time convergence. Indeed, the negative homogeneity is necessary and sufficient condition of finite-time stability of an asymptotically stable homogeneous system (see e.g. [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF]). Below we use Lemma 3 in order to define an analog of Lyapunov exponents for homogeneous systems.

Finally, let us recall the following result, which is utilized in the later constructions.

Theorem 4 ( [START_REF]Sliding mode control design using canonical homogeneous norm[END_REF]): The two claims are equivalent 1) The origin of the system (10) is asymptotically stable.

2) For any matrix P ∈ R n×n satisfying (4) there exists a map

Ξ ∈ C ∞ (R n \{0}, R n×n ) such that det(Ξ(z)) = 0, ∂Ξ(z) ∂zi z = 0, Ξ(e s z) = Ξ(z) for z = (z 1 , ..., z n ) ∈ R n \{0}, s ∈ R, i = 1, ..., n z Ξ (z)P Ξ(z) f (z) < 0, ( 14
)
where f is given by [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Notice that the properties of the matrix-valued function Ξ imply that V : R n → [0, +∞) given by

V (z) = z Ξ (z)P Ξ(z)z
is a Lyapunov function for the system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

III. METHOD OF LYAPUNOV EXPONENTS FOR HOMOGENEOUS SYSTEMS

A. Lyapunov exponents

This approach was introduced by A.M. Lyapunov [START_REF] Lyapunov | The general problem of the stability of motion[END_REF] and developed/generalized in later in many works, e.g. [START_REF] Bylov | Theory of Lyapunov exponents and its application to problems of stability[END_REF], [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF].

Definition 5 ([3], page 17): Let G be a set of functions [0, +∞) → R n . An upper (lower) Lyapunov exponent of a function x ∈ G is a number χ(x) defined as follows

χ + (x) = lim sup t→+∞ 1 t ln x(t) , (15)
(resp. χ -(x) = lim inf t→+∞ 1 t ln x(t) ). (16)
For functionals χ + and χ -one holds (see [START_REF] Bylov | Theory of Lyapunov exponents and its application to problems of stability[END_REF])

• if x(t) ≤ y(t) for all t > 0 then χ ± (x) ≤ χ ± (y)

• χ ± (αx) = χ ± (x) for each x ∈ G and α ∈ R\{0};

• χ + (x + y) ≤ max{χ + (x), χ + (y)} for any x, y ∈ G;

• χ + (xy) ≤ χ + (x) + χ + (y) for any x, y ∈ G;

• χ ± (0) = -∞ (normalization property).

• ε > 0, ∃C(ε) > 0 : g(t) ≤ C(ε)e (χ + (g)+ε)t for t ≥ 0;

• ∀ε > 0 one has g(t) e -(χ + (g)+ε)t → 0 as t → +∞. For example, if there exist K > 0 such that the vector field f : R × R n → R n satisfies the inequality

f (t, x) ≤ K x for all x ∈ R n (17)
for some K > 0 and the system

ẋ = f (t, x)
has a solution x(•, x 0 ) : [0, +∞) → R n with any x 0 ∈ R n :

x(0) = x 0 defined on whole (0, +∞),

then |χ + (x(•, x 0 ))| ≤ K. If sup x(•,x0) χ + (x(•, x 0
)) < 0 then the origin of the system is globally exponentially attractive.

B. Lyapunov exponents of linear systems

Notice that for any linear system

ẋ = A(t)x
with A(t) ∈ R n×n , attractivity implies stability, i.e negativeness of Lyapunov exponents of the linear time-varying system implies its exponential stability. For linear systems

ẋ = A(t)x, t > 0 (18)
a number of disjoint Lyapunov exponents is finite [START_REF] Bylov | Theory of Lyapunov exponents and its application to problems of stability[END_REF], [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF]:

{-∞} ≤ χ + 1 < χ + 2 < ... < χ + s , s ≤ n.
and the numbers χ + i can be calculated or, at least, estimated directly from A(t). For each 1 ≤ i ≤ s, define

E i = {x 0 ∈ R n : χ(x(•, x 0 )) ≤ χ + i } Put E 0 = {0}.
It can be easily shown (see e.g. [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF], page 9) that E i is a linear subspace of R n , and

{0} = E 0 ⊂ E 1 ⊂ ... ⊂ E s = R n , E i = E j if i = j.
A collection E = {E i , i = 0, ..., s} satisfying the latter property is called a filtration of R n [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF]. Definition 6: The values χ + i with the multiplicities dim(E i ) -dim(E i-1 ) define a spectrum σ(A) of the system (18) (see [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF] for more details).

Stability of Lyapunov exponents a linear system with respect to perturbation of its parameters is introduced by the following definition.

Definition 7: [25, page 136] The characteristic exponents ξ + i (A) of A are said to be structurally stable if for any ε > 0 the exists δ > 0 such that sup t>0 Q(t) < δ implies

|ξ + i (A) -ξ + i (A + Q)| ≤ ε, i = 1, 2, ..., s ≤ n.

C. Lyapunov exponents of homogeneous systems

In many cases homogeneous systems behave similarly to linear (or, at least, Lipschitz) ones. We may expect that Lyapunov exponents of homogeneous system have some similar properties. The examples given below show that this is not true in the general case and some specific constructions are required in order to use the conventional methodology of Lyapunov exponents in the context of homogeneous systems.

Let x(•, x 0 ) be a solution of the system [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF]. Formally we can define the Lyapunov exponent for x(•, x 0 ) by a formula [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF]. However, we cannot guarantee that the corresponding number is finite, well-defined and provides some information about convergence/stability of the system. For example, the standard homogeneous Cauchy problem

ẋ = -x 3 , t > 0, x(0) = x 0 = 0, x ∈ R ( 19
)
has the unique solution solution

x(t, x 0 ) = sign(x0) √ 2(t+x -2 0 /2)
.

Obviously, the origin of the system is globally asymptotically stable but χ + (x(•, x 0 )) = χ -(x(•, x 0 )) = 0. This happens because Lyapunov exponents "sense" only exponential convergence, while the convergence rate of this system is subexponential close to the origin and hyper exponential close to ∞. Similarly the unstable system

ẋ = x 1 3 , t > 0, x(0) = x 0 = 0, x ∈ R,
has the solution

x(t, x 0 ) = 2 3 t + |x 0 | 3 2 sign(x 0 )
and, again, χ + (x(•, x 0 )) = χ -(x(•, x 0 )) = 0. Therefore, the Lyapunov exponent given by ( 15) is well defined and informative only if deg F d f = 0 (the case of exponential convergence/divergence). Some constructions are required in order to expand the method of Lyapunov exponents to homogeneous systems with deg

F d f = 0.
According to Theorem 3, the system (10) is equivalent to the system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Topological equivalence of systems implies a certain equivalence of Lyapunov exponents [3, Chapter 10]. Moreover, Lemma 3 establishes some relation between solutions of the systems ( 13) and [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Since f (λz) = f (z) for any λ > 0 and any z ∈ R n \{0} then

f (z) ≤ K = sup z =1 f (z)
and the inequality (17) holds for the system [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. Therefore, Lyapunov exponents for solutions of the system (13) are finite and belongs to the interval [-K, K].

The key idea is to define Lyapunov exponents for the equivalent system [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF].

For example, the system ( 19) is d-homogeneous of degree 1 with d(s) = e s , s ∈ R. The equivalent system (13) has the form ẏ = -y, t > 0, y ∈ R and its Lyapunov exponents are χ + (y(•, y 0 )) = χ -(y(•, y 0 )) = -1 also characterize stability of the original (equivalent) system [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]. This truck can also be utilized for multidimensional systems.

Theorem 5: Let f ∈ F d be a continuously differentiable on the unit sphere S = {x ∈ R n : x = 1}, x = √

x P x, where P satisfies (4). The origin of the system (10) is globally uniformly asymptotically stable if and only if

sup y0∈S χ + (y(•, y 0 )) < 0,
where y(•, y 0 ) denote a solution of the system (13) with y(0) = y 0 . Proof. Sufficiency. The system ( 13) is d-homogeneous of degree zero with the dilation d(s) = e s I n , s ∈ R. Theorem 2 implies that

χ + (y(•, e s y 0 )) = χ + (y(•, y 0 )), ∀s ∈ R.
Therefore, the condition χ + (y(•, y 0 )) < 0 for any y 0 ∈ S implies χ + (y(•, y 0 )) < 0 for any y 0 ∈ R n , i.e. each trajectory of the system (13) converge to the origin exponentially. Theorem 3, Lemma 3 implies that the trajectories of the equivalent system (10) converge to the origin asymptotically. Finally, taking into account that asymptotic attractivity implies stability for homogeneous systems [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] we finish the proof.

Necessity. Equivalence between systems ( 13) and ( 10) is established above. We just need to demonstrate that asymptotic stability of ( 13

y(t, y 0 ) 2 ≤ V (y(t,y0)) c1 ≤ e -c 3 c 2 t V (y0) c1 ≤ e -c 3 c 2 t c2 y0 2 c1
.

The latter means that sup y0∈S χ + (y(•, y 0 )) ≤ -c3 2c2 . The proven result implies the following conclusion. Together with homogeneity degree the Lyapunov exponents of the system (13) completely characterize the asymptotic stability/instability and convergence rages of the original homogeneous system [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF].

The key question now : is it possible to calculate these Lyapunov exponents using solutions of the original system? The next proposition positively answers this question.

Proposition 2: Let f ∈ F d be continuously differentiable on the unit sphere S = {x ∈ R n : x = 1}, x = √

x P x, where P satisfies (4). Let x(•, x 0 ) and y(•, y 0 ) be solutions of the systems (10) and (13) with initial conditions x(0) = x 0 = 0, y(0) = y 0 := Φ(x 0 ), respectively. Then

ξ + (y(•, y 0 )) = lim sup τ →T (x0) ln x(τ, x 0 ) d τ 0 x(δ, x 0 ) deg F d d dδ , where T (x 0 ) = +∞ if 0 < x(τ, x 0 ) d < +∞ for τ > 0 or T (x 0 ) < +∞ is the first instant of time such that lim τ →T (x0) x(τ, x 0 ) d = 0 or lim τ →T (x0) x(τ, x 0 ) d = +∞}.
The proof immediately follows from Lemma 3, the identity x(τ, x 0 ) d = z(τ, Φ(x 0 ) and the definition of the Lyapunov exponent, where Φ is, as before, given by (12).

D. "Linearization" of homogeneous systems around a trajectory

In some cases it is important to investigate a stability of a concrete (e.g. periodic) solution x * (t) of a nonlinear system [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF]. The usual approach in this case is a linearizion of the system around the corresponding trajectory (see introduction) combined with an analysis of negativeness of Lyapunov exponents for A in (2). In the previous section we have discovered that this conventional approach could be too conservative.

Theorem 6: Let f ∈ F d be twice continuously differentiable on the unit sphere S = {x ∈ R n : x = 1}, x =

√

x P x, where P satisfies (4). Let a matrix-valued function A : R n → R n×n be defined as follows

Ã(y) = (deg F d f +1) f (y)y P y + f y y , y ∈ R n , ( 20
)
where f is given by [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Let x(•, x * ) and x(•, x 0 ) be solutions of ( 10) with x * = 0, x 0 = 0 and y(•, y * ) and y(•, y 0 ) be solutions of the system (13) with y * = Φ(x * ) and y 0 = Φ(x 0 ). Then the difference

e(t) = y(t, y 0 ) -y(t, x * ) = Φ x t ∫ 0 ds y(s,y0) deg F d f , x 0 -Φ x t ∫ 0 ds y(s,y * ) deg F d f , x * , x 0 ∈ R n satisfies the equation ė = Ã(y(t, y * ))e + g(y(t, y * ), e), (21)
where g = (g 1 , .., g n ) is given by g i (y(t, y * ), e) = e

Proof. Since f is twice continuously differentiable on S then F is twice continuously differentiable on R n \{0} and continuous on R n . According to Lemma 3 the functions y(•, y * ) and y(•, y 0 ) satisfy ẏ = F (y).

Moreover, it is easy to see that Ã(y) = F (y). Hence, the error e satisfy [START_REF] Grune | Homogeneous state feedback stabilization of homogeneous systems[END_REF] provided that y(t, y * )+τ se(t) = 0 for all t > 0 and all s, τ ∈ [0, 1]. The origin is excluded because it is the only possible discontinuity point of F . If the maximum characteristic exponent from the spectrum (see Definition 6) of Ã(y(t, y * )) is negative then under some additional restrictions [START_REF] Barreira | Lyapunov Exponents and Smooth Ergodic Theory[END_REF] a local exponential stability of the solution y(•, x * ) can be proven. Indeed, the matrix-valued function à is globally bounded, so the Lyapunov exponents of the system (21) with g = 0 are finite and g = O( e 2 ) for all e from a sufficiently small neighborhood of 0. provided that ( 23) is fulfilled. This means that g ∞ admits the linear estimate g(y(t, y * ), e(t)) ∞ ≤ σ(F max )C e(t) , and from χ max < 0 and structural stability of Lyapunov exponents we conclude that e(t) → 0 as t → ∞ for sufficiently small F max .

IV. CONCLUSIONS

In this paper we propose a possible approach for application of the well-known method of Lyapunov exponents to a generalized homogeneous systems. The key difficulty is that transitions of homogeneous systems may be sub-or super exponential while the analysis by means of Lyapunov exponents "sense" only an exponential growth.

The method of Lyapunov exponents is utilized, in particular, for the design of observers for nonlinear systems [START_REF] Frank | A detectability criterion and data assimilation for non-linear differential equations[END_REF].

The interesting direction of future research is application of the developed technique for observers/controllers design for some classes of nonlinear homogeneous systems.

  x * (t, x 0 )+τ se) ds dτ e, i = 1,..., n.

  it satisfies • Group property: d(0) = I n and d(t+s) = d(t)d(s) = d(s)d(t) for all t, s ∈ R; • Continuity property: d is a continuous map, i.e., ∀t ∈ R, ∀ε > 0, ∃δ > 0 : |s-t| < δ ⇒ d(s)-d(t) ≤ ε; )x = +∞ uniformly on the unit sphere

e r 1

 1 s 0 ... 0 0 e r 2 s ... 0 ... ... ... ... 0 0 ... e rn s

  ) implies χ + (y(•, y 0 )) < 0, ∀y 0 ∈ S. For this purpose let us consider the Lyapunov function given by V (y) = y Ξ (y)P Ξ(y)y, where Ξ and P are defined in Theorem 4. Obviously, thatc 1 y 2 ≤ V (y) ≤ c 2 y 2 , y ∈ R n where c 1 = inf y∈S λ min Ξ (y)P Ξ(y) > 0, c 2 = sup y∈S λ max Ξ (y)P Ξ(y) < +∞,where λ min (•) and λ max (•) denote minimal and maximal eigenvalues of a symmetric matrix. According to Theorem 4 for the system (13) we derive V (y) = y y Ξ (y)P Ξ(y) f (y) < 0.Since Ξ and f are continuous on S and Ξ(e s y) = Ξ(y), f (e s y) = f (y), s ∈ R, y ∈ R n \{0} then c 3 = inf y∈R n -y y Ξ (y)P Ξ(y) f (y) > 0 and V (y) ≤ -c 3 y 2 ≤ -c3 c2 V (y). Hence,

1 ∫ 0 1 ∫0

 11 sF i (y(t, y * )+τ se) ds dτ e, and F i (y) = y fi (y) with i = 1,..., n, provided that 0 / ∈ co{y(t, y * )), y(t, y 0 )}, ∀t > 0.

Corollary 1 :

 1 Let all conditions of Theorem 6 hold and χ max := max χ∈σ( Ã(y(s,y * ))), χ < 0, where σ denotes the spectrum of a time-varying matrix (see Definition 6). If Lyapunov exponents of à (y(s, y * ))) are structurally stable and sup t) ds τ y(t, y * )+τ se(t) = C < +∞ (23) then e(t) converge to zero exponentially provided that the number F max = max i∈{1,2,...,n} sup y∈S F i (y) is sufficiently small. Proof. Since F (e s y) = F (y) for all s ∈ R and all y ∈ R n \{0} then from Lemma 2 we derive F i (y) = F i y y y = 1 y F i y y and e F i (y(t, y * )+τ se) e = e F i y(t,y * )+τ se y(t,y * )+τ se e y(t,y * )+τ se . Since F i is continuous on S then there exists is a class K function σ such |g i (y(t, y * ), e(t))| ≤ σ(F max )C e(t) , ∀t > 0,

the Government of Russian Federation.