**Abstract**

Pediatric cancer patients are immunocompromised, and the risks are higher in this population. Confirmed cases are defined as PCR (polymerase chain reaction) positive patients. The severity of infection is divided into four groups: asymptomatic/mild, moderate, severe, and critical, based on the clinical, laboratory, and radiological features. In the pediatric population, the COVID-19 disease has a mild course. Chemotherapy courses can be interrupted according to the symptoms and severity of the disease. Azithromycin, antivirals are used as a single agent or in combination. In critical patients, convalescent plasma, mesenchymal stem cells, tocilizumab, and granulocyte transfusions are administered. In recent studies, having hematological malignancy, stem cell transplantation, a mixed infection, and abnormal computerized tomography findings increase the severity of the disease and the need for an intensive care unit. Therefore, the patients and their families should be aware of a higher risk of severe forms than immunocompetent children.

**Keywords:** chemotherapy, COVID-19, immunocompromised, immunotherapy, pediatric oncology

## **1. Introduction**

Coronaviruses are zoonotic RNA viruses. SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), the novel coronavirus, belongs to the Betacoranovirus subgroup [1]. SARS-CoV-2's incidence in children varies (China: 2-12.3%, Italy: 1.2%, USA: 5%). The infection in pediatric cases is asymptomatic or mild. The median incubation period is 5-7 days. The primary source of transmission is respiratory droplets and direct contact. The primary tool for diagnosis is a real-time polymerase chain reaction test (RT-PCR) on samples. Eighty percent of children had household contact; ten percent were asymptomatic, fifty percent had a fever. Other symptoms are cough, respiratory distress, fatigue, myalgias, vomiting, diarrhea, anosmia, ageusia, sore throat. Children generally recover in 1-2 weeks. The case fatality rate in children is zero percent [2]. This benign course of the disease is related to the immune preparedness of children to a new pathogen. Immunologic mechanisms are also different in children compared with adults [3, 4]. Multisystem Inflammatory Syndrome in Children (MIS-C) is a rare postinfectious complication of SARS-CoV-2 infections; it is RT-PCR negative for SARS-CoV-2 virus but antibody positive [5].

Cancer treatment includes various immunosuppressive drugs [6]. It is well known that immunocompromised children have higher mortality and morbidity rates than the healthy population due to viral respiratory infections [7]. In the

pediatric oncology setting, the mortality rate in COVID-19 is reported up to 4% [8]. In COVID-19 relevant areas, the virus transmission rate is low in children with cancer. Cancer diagnosis, treatment, palliative care, hospital visits are interrupted because of the pandemic. Another concern is delayed cancer diagnosis, chemotherapy shortages, decreased availability of surgery, radiotherapy, supportive treatment, inadequate personal productive equipment, and drugs, especially in lowmiddle income countries [9, 10]. The most common cancer in the pediatric population is acute leukemia. In industrialized countries, the incidence of acute leukemia is a 40-60 age-standardized rate per million, one-third of all childhood cancers. Brain and spinal tumors are the second; lymphomas are the third most common tumors in industrialized countries [11]. Therefore, urgent treatment is critical and life-saving, especially in leukemia and lymphoma induction therapy [12]. Here we present an explanatory review of different approaches and experiences in this unique population.


**2. Vulnerability risks for IoT systems**

Anything that can disrupt the operation, integrity, and availability of an IoT device, or network of IoT devices is a threat. There are different types of threats. There are natural threats, such as floods, earthquakes, or storms. There are unintentional threats that result from accidents or mistakes. Finally, there are deliberate threats that result from ulterior motives. Each of these types of threats can be fatal to the IoT infrastructure.

The IoT risks are divided into four categories (**Figure 1**).

Therefore, the general security requirements for IoT systems (**Figure 2**) must include the following:

**Figure 1.** *IoT risks.*

**Figure 2.**

*IoT security challenges and requirements.*

*Methods for Detection and Prevention of Vulnerabilities in the IoT (Internet of Things) Systems DOI: http://dx.doi.org/10.5772/intechopen.113898*

IoT vulnerability is an integral weakness of the design, configuration, or implementation of the IoT system that weakens protection. Most vulnerabilities can be found in one of these sources:

Although there are only three types of vulnerabilities, they can manifest in many ways. The first security rule is the physical protection of devices and networks. Central hosts and servers should be stored in separate rooms that can only be accessed by authorized personnel (owners). Routers, communications equipment, and portable media (disks, smartcards) should also be stored in secure locations with limited access. As part of this process, individuals and companies must consider the physical and natural environment in which IoT operates. The possibility of earthquakes, fires, floods, and other unforeseen accidents should be considered and properly planned. Accordingly, owners must ensure the security of all media (disks, tapes, smartcards) that contain vital information and make regular data backups.

Communication is the transmission of information through a medium. As such, it is inevitably vulnerable to interception, monitoring, burglary, etc. Owners should also take care of other forms of communication interception (Wi-Fi, antennas, etc.). Network and packet eavesdroppers are common tools that can read network flow.

It is important to note that every network and IoT system has vulnerabilities. Human mistakes, carelessness, laziness, greed, and rage pose the greatest threat to infrastructure with possible high damage. Moreover, human vulnerability and the risks associated with them are the most difficult to defend.


**10. Smart grid architecture model (SGAM)**

The SGAM is a reference architecture intended to show the cases of SG architecturally. This was the result of an EU Mandate M/490 reference working group [37, 38]. SGAM primarily consists of five component layers: business, functionality, content, communication and layers. These are referred to as interoperability layers. Each interoperability layer contains a smart grid plane covering electrical space and knowledge processing areas. The main aim of this model is to show which areas of data management communicate with each other.

Three-layered Architecture- The three-layer IoT-assisted SG architectures have been used extensively in [39], focusing on the IoT-assisted method's characteristics. As seen in **Figure 13**, the architecture consists of three layers, a vision layer, a network and an application layer.

#### **Figure 13.**

*IoT-aided SG structures, with a vision layer, network layer and device layer, have a three-cover architecture.. The layer consists of two sub-layers, a sub-layer for contact expansion, and a perception [36].*

ecosystem in real time based on the information obtained by the network layer. There are several aspects of the IoT-assisted SG systems presented in **Figure 3**. It comprises an infrastructure/middleware frame and various types of records, web and directory resources servers. IoT technology computing, delivery and services are offered through the application/middleware infrastructure. The key components of the application layer are information sharing and secrecy. The deployment layer would particularly increase for SGs who can provide far richer data sets.

#### **Figure 14.**

*IoT-aided SG systems' four-layered architecture built on the SG features. It consists of a terminal layer, a field network layer, a remote connection and a system layer for the main station [36].*

because fixed and stagnant schedules do not fit in with consumer lifestyles or because there are volatile business dynamics. Users are now using their smartphones or PCs to change energy conditions using IoT technology. This allows users to comply with their own schedules by adjusting policy as and where necessary to easily respond to accidents. The IoT interface has been proposed in 2015 for the effective control of smartphones and cloud computing services. This new framework transforms the existing statically energy management system and central control modes, which generally consists of different buildings, to complex and dispersed energy control on the user side of SG. This frame is seen in **Figure 7**, which includes four key elements, (i) energy conservation policy for multiple sources, (ii) telephone tracking and monitoring, (iii) automated access position-based control and (iv) data storage and computing cloud network. The premise of an enterprise consists of a variety of separate sections, including campuses, houses, offices, laboratories and offices, each with different energy needs and regulations, which are essential in terms of energy use management. Each family member often has a preference for energy usage in a single home, which has to be taken into account. As seen in **Figure 15**, there are several layers in different stages in this scheme of energy saving techniques. This management strategy framework (e.g., building, department, lab and room).

6.**Web-enabled SG Architecture-** In [43], for IoT-assisted SG frameworks, Web architectures have been proposed (**Figure 16**). A number of IoT

**Figure 15.** *Network-enabled SG architecture with IoT-advanced web services [42]. There are two different categories of renewable and non-renewable energy sources, all linked to modern energy metres [36].*

computers provide access to online services, which are the guidance of the web browser to these web services. There are several online sites in the Internet. Two forms of energy exist: non-renewable and renewable energy. Nonrenewable energy sources are thermal energy stations like coal and oil fires, which release carbon dioxide into the air and the nuclear power stations. Renewable energy is an environmentally friendly alternative which comprises hydropower, wind turbines, solar, biogas and biological fuel as well as geothermal, tidal/wave fuels.

It consists of three key components, the network of a sensor and drive, the simulator and IoT server. The sensor and actuator network comprises sensor and actuator nodes and IP gates. The IoT system has an IoT message sending computer, an SG database, a data processing server, a software setup, a settings unit, a device log and a safe Access manager.


**2. Materials and Methods**

#### *2.1. Study Area and Dataset*

Different from the previous study [14] that looks at seven climatic divisions in California, the current study focuses on the 10 hydrologic regions (Figure 1 and Table 1) defined by the California Department of Water Resources (DWR) for operational water resources planning and management purposes [5]. These regions include four coastal regions (North Coast, San Francisco Bay, Central Coast, and South Coast), three Central Valley regions (Sacramento River, San Joaquin River, and Tulare Lake), and three Eastern regions (North Lahontan, South Lahontan, and Colorado River). For each of these three categories (Coastal, Central Valley, and Eastern), climate tends to be drier towards the southern regions.

**Figure 1.** Ten hydrologic regions in California: North Coast (NC), San Francisco Bay (SF), Central Coast (CC), South Coast (SC), Sacramento River (SAC), San Joaquin River (SJQ), Tulare Lake (TUL), North Lahontan (NL), South Lahontan (SL), and Colorado River (CR). Dots represent the centroid points of individual climate projection grids (1/16th degree) located in each region.

**Table 1.** Geographic and climatic characteristics of study hydrologic regions.

The North Coast region contains the California Coast Ranges, the Klamath Mountains, and parts of the Modoc Plateau [5]. The eastern side of the region is mostly mountainous with crests around 1800 m (6000 ft) and a few more than 2400 m (8000 ft) in elevation. It is the wettest region in terms of annual precipitation received (1390 mm; Table 1). As such, the region is prone to flooding. Major floods were recorded in 1955, 1964, 1986, 1997, 2006, and 2017. The San Francisco Bay region is the smallest in size. It is bounded by the Pacific Ocean on the west and Coast Ranges on the east where the peaks are above 1200 m (4000 ft) in elevation. The region faces multiple water management challenges including an unreliable water supply, declining water quality and ecosystems, increasing flood risks, and threats posed by sea level rise to coastal areas. The Central Coast region is the most groundwater-dependent region. Groundwater supplies about 80% of its total water usage. The water management challenges of this region include managing groundwater quality and overdraft, sea water intrusion, and flood risks. The South Coast region is the most urbanized and populous region. It accounts for about 7% of the state's total area but accommodates more than half of the state's population. As a result, water supply is always a concern of local water managers. The region is also prone to flooding including debris flows and mud slides, particularly in areas where hillsides have been damaged by wildfires. It is the driest and warmest region in the coastal regions (Table 1).

Central Valley regions are the major water supply sources for the state, of which the Sacramento River region is the primary source. It is the largest and second wettest region (925 mm/year; Table 1) of all 10 hydrologic regions. It contributes a majority portion of the water supplied to the SWP and CVP. The region is bounded by Coast Ranges on the west and Sierra Nevada on the east. In this region, about one in three residents is exposed to a 500-year flood event. The region has approximately \$65 billion of assets, 1.2 million acres of farmland, and over 340 sensitive species [5]. Major floods in the region normally originate from extreme atmospheric river events during the winter. The San Joaquin River region receives less precipitation than the Sacramento River region. It is also bordered by the Sierra Nevada on the east. However, Sierra Nevada watersheds in this region are higher in elevation, making them more dominated by snow compared to Sacramento River region watersheds. Floods in this region come from both winter rainfall and melting Sierra snowpack [5]. The Tulare Lake region is the driest in the Central Valley and one of the driest regions in the state. It is the largest agricultural region in the state heavily relying on groundwater and imported water supply. Groundwater pumping in this region accounts for more than 38% of the state's total annual groundwater extraction. The region is also prone to floods caused by winter rainfall and spring snowmelt.

The eastern regions are the least populous. The North Lahontan region accommodates approximately 0.3% of the state's population. It comprises arid high desert (1200–1500 m in elevation) in the north and the eastern slopes of the Sierra Nevada (up to 3750 m in elevation) in the central and southern portions. It is the coolest region in the state (Table 1). In contrast, the Colorado River region is the hottest. It is also the driest region, receiving only about one tenth of the precipitation received by the North Coast region. The Colorado River region is, however, also subject to flooding which threatens about 38% of its population. Different from all other regions, most flooding events occur from infrequent but high-intensity summer storms in this region. The South Lahontan region is the second driest region in the state. Precipitation for this region comes from both winter storm events and summer thunderstorms.

In general, California has a typical Mediterranean-like climate, with the summer (winter) being dry and warm (cool and wet). This is evident for the 10 regions on the monthly scale (Figure 2). Most of the precipitation occurs during the wet season (November to April). During that period, those regions receive 69% (Colorado River region) to 91% (Central Coast region) of their total annual precipitation. Statewide, 85% of annual precipitation occurs during the wet season. January normally observes the highest amount of precipitation while July is typically the driest month. Meanwhile, January is the coolest month while July has the highest average temperature. South Lahontan and Central Coast regions have the largest (22.1 ◦C) and smallest (10.1 ◦C) variations in monthly temperature, respectively. Across all regions, the Colorado River region is the driest and hottest. The North Lahontan region is the coolest and the North Coast region is the wettest. Those observations are consistent with values shown in Table 1.

This study looked at both the historical and projected precipitation, and maximum and minimum temperature data. The projections for 2020–2099 were based on climate model simulations from the Coupled Model Intercomparison Project Phase 5 (CMIP5) [32], which represents the current state of the climate science. Specifically, 20 individual projections from 10 Climate Circulation Models (GCMs) under two newly developed emission scenarios named Representative Concentration Pathways (RCP) 4.5 and RCP 8.5 [33] were selected for the analyses. These 10 GCMs (Table 2) were chosen by DWR Climate Change Technical Advisory Group and deemed as the most suitable for California climate and water resources assessment [34]. RCP 4.5 (RCP 8.5) assumes low (high) future greenhouse-gas concentrations. These projections were downscaled to a very high spatial resolution at 1/16th degree (approximately 6 by 6 km, or 3.75 by 3.75 miles) to better capture the spatial variability of the climate via the Localized Constructed Analogs (LOCA) method [35]. This dataset is made available for California's Fourth Climate Change Assessment (http://cal-adapt.org/). There are other ways of selecting representative GCMs models [36,37] for water planning analysis. However, they are beyond the scope of this study which exclusively used the GCMs recommended by the CCTAG. These 20 CCTAG-recommended projections have been applied in DWR's and the California Water Commission's planning activities including the Central Valley Flood Protection Plan [38] and the Water Storage Investment Program [39]. There is no consensus that some of those projections are more likely to occur than the remaining projections in the future. As a result, these projections are typically treated equally in planning activities. In this study, we looked at these 20 projections together. When looking at the mean of future projections on the annual scale, however, the 10 RCP 4.5 projections and 10 RCP 8.5 projections were analyzed separately.

**Figure 2.** Long-term (1951–2013) mean monthly precipitation (**a**) and temperature (**b**) of 10 hydrologic regions.

**Table 2.** GCMs Selected for California Water Resources Planning 1.

1 Adapted from Tables 2–4 of [34].

The gridded historical observational dataset of these three variables on daily scale for water years 1951–2013 of Livneh et al. [40] (https://data.nodc.noaa.gov/) were employed as the historical baseline. The spatial resolution (1/16th degree) of this dataset is consistent with that of the LOCA-downscaled climate model projections. This dataset has been applied extensively in hydrologic modeling and drought assessment [41–44], and deemed as the best available historical data at this spatial resolution. In this study, both projected and historical datasets were aggregated from grid scale to (hydrologic) regional scale in the analyses presented below.

### *2.2. Study Method and Metrics*


*2.3.1. Transition to chaos through bifurcation cascades of stable cycles*

At the beginning let us show that the scenario of transition to chaos through the Sharkovskii subharmonic and homoclinic cascades of bifurcations of stable cycles takes place also in many-dimensional dissipative nonlinear systems of ordinary differential equations. For example consider Rikitaki system

$$
\dot{\mathbf{x}} = -\mu \mathbf{x} + y \mathbf{z},
\
\dot{y} = -\mu y + \mathbf{x}u,
\
\dot{z} = 1 - \mathbf{x}y - b\mathbf{z},
\
\dot{u} = 1 - \mathbf{x}y - cu,\tag{18}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 147

autonomous systems of ordinary differential equations takes place also in manydimensional systems. So, appearance of three-dimensional torus is not necessary condition for generation of chaotic dynamics in dissipative many-dimensional systems of differential

**Figure 13.** Projections of sections of two-dimensional invariant tori of period one (a), two (b) and three

Let us show now that the FSM scenario of transition to chaos is realized also in infinitelydimensional nonlinear autonomous dissipative systems of ordinary differential equations, namely in nonlinear ordinary differential equations with delay arguments. For instance such scenario of transition to chaos through the Sharkovskii subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order (3) takes

( ) () () . ( )

 is increasing, then at first a stable cycle is appearing in phase space of the equation from the stable stationary state as a result of Andronov-Hopf bifurcation. After that the period two stable cycle is appearing from this original singular cycle as a result of double period bifurcation. That is the beginning of the Feigenbaum cascade of period doubling

cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order takes place in the Mackey-Glass equation. Projections of some main stable

Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes transition to dynamical chaos in all nonlinear dissipative systems of ordinary differential equations. Scenario of transition to chaos consists of subharmonic and homoclinic (heteroclinic) cascades of bifurcations of stable cycles or stable

cycles and singular attractors of the Mackey-Glass equation are presented in Fig. 14.

*x t x t ax t*

*n n n*

is small, then Mackey-Glass equation has unique stable stationary state. When

*x t*

(20)

is a bifurcation parameter. When a value of

, all subharmonic

(c) and one of the toroidal singular attractor (d) in complex Lorenz system (19).

*2.3.3. Transition to chaos in nonlinear equations with delay argument*

place in well-known Mackey-Glass equation (Mackey & Glass, 1977).

bifurcations. Then, after further increasing of bifurcation parameter

0

In this equation the delay argument

parameter

two- or many-dimensional tori.

equations.

modelling a change in dynamics of magnetic poles of the Earth. Some main cycles of subharmonic cascade of bifurcations in the system of Eqs. (18) and some singular attractors are presented in Fig. 12.

**Figure 12.** Projections of original singular cycle (a), cycle of period two (b), Feigenbaum attractor (c) and two more complex singular attractors in the Rikitaki system (d)-(e).

### *2.3.2. Transition to chaos through bifurcation cascades of stable two-dimensional tori*

Besides described above mechanism of transition to chaos in accordance with subharmonic and homoclinic cascades of bifurcations of stable cycles, in many-dimensional dissipative nonlinear systems of ordinary differential equations there exists a scenario of transition to chaos through subharmonic and homoclinic cascades of bifurcations of stable twodimensional or many-dimensional tori along any one or several frequencies simultaneously. The mechanism of this cascade of bifurcations has the same above considered FSM nature, and presently there not discovered really any other scenarios of transition to chaos in manydimensional nonlinear systems of ordinary differential equations. Such a scenario of transition to chaos takes place in complex five-dimensional Lorenz system

$$\dot{X} = -\sigma X + \sigma Y, \quad \dot{Y} = -XZ + rX - aY, \quad \dot{Z} = -bZ + (X^\*Y + XY^\*)/2 \tag{19}$$

of two complex variables *X x ix* 1 2 and *Y y iy* 1 2 and one real variable *Z* . If values of parameters *a b*, , and Re*r* are fixed and the value of parameter Im*r* is decreasing, then at first a stable invariant torus is appearing from the stable cycle as a result of Andronov-Hopf bifurcation. After that the period two invariant torus is appearing from this original singular saddle torus as a result of double period bifurcation (Fig. 13). That is the beginning of Feigenbaum cascade of period doubling bifurcations. Then, after further decreasing of bifurcation parameter Im*r* , all subharmonic cascade of bifurcations of stable twodimensional tori with arbitrary period in accordance with the Sharkovskii order (3) takes place in the complex Lorenz system. Projections of sections of period one, two and three two-dimensional invariant tori and one of the toroidal singular attractor are presented in Fig. 13.

This example shows that the FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario of transition to dynamical chaos in two-dimensional nonautonomous and three-dimensional autonomous systems of ordinary differential equations takes place also in manydimensional systems. So, appearance of three-dimensional torus is not necessary condition for generation of chaotic dynamics in dissipative many-dimensional systems of differential equations.

**Figure 13.** Projections of sections of two-dimensional invariant tori of period one (a), two (b) and three (c) and one of the toroidal singular attractor (d) in complex Lorenz system (19).

### *2.3.3. Transition to chaos in nonlinear equations with delay argument*

146 Nonlinearity, Bifurcation and Chaos – Theory and Applications

are presented in Fig. 12.

*x x yz y y xu z xy bz u xy cu*

modelling a change in dynamics of magnetic poles of the Earth. Some main cycles of subharmonic cascade of bifurcations in the system of Eqs. (18) and some singular attractors

**Figure 12.** Projections of original singular cycle (a), cycle of period two (b), Feigenbaum attractor (c)

*2.3.2. Transition to chaos through bifurcation cascades of stable two-dimensional tori*

transition to chaos takes place in complex five-dimensional Lorenz system

\* \* *X X Y Y XZ rX aY Z bZ X Y XY*

parameters *a b*, ,

Fig. 13.

Besides described above mechanism of transition to chaos in accordance with subharmonic and homoclinic cascades of bifurcations of stable cycles, in many-dimensional dissipative nonlinear systems of ordinary differential equations there exists a scenario of transition to chaos through subharmonic and homoclinic cascades of bifurcations of stable twodimensional or many-dimensional tori along any one or several frequencies simultaneously. The mechanism of this cascade of bifurcations has the same above considered FSM nature, and presently there not discovered really any other scenarios of transition to chaos in manydimensional nonlinear systems of ordinary differential equations. Such a scenario of

of two complex variables *X x ix* 1 2 and *Y y iy* 1 2 and one real variable *Z* . If values of

first a stable invariant torus is appearing from the stable cycle as a result of Andronov-Hopf bifurcation. After that the period two invariant torus is appearing from this original singular saddle torus as a result of double period bifurcation (Fig. 13). That is the beginning of Feigenbaum cascade of period doubling bifurcations. Then, after further decreasing of bifurcation parameter Im*r* , all subharmonic cascade of bifurcations of stable twodimensional tori with arbitrary period in accordance with the Sharkovskii order (3) takes place in the complex Lorenz system. Projections of sections of period one, two and three two-dimensional invariant tori and one of the toroidal singular attractor are presented in

This example shows that the FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario of transition to dynamical chaos in two-dimensional nonautonomous and three-dimensional

, , ( )/2 (19)

and Re*r* are fixed and the value of parameter Im*r* is decreasing, then at

and two more complex singular attractors in the Rikitaki system (d)-(e).

, ,1 , 1 , (18)

> Let us show now that the FSM scenario of transition to chaos is realized also in infinitelydimensional nonlinear autonomous dissipative systems of ordinary differential equations, namely in nonlinear ordinary differential equations with delay arguments. For instance such scenario of transition to chaos through the Sharkovskii subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order (3) takes place in well-known Mackey-Glass equation (Mackey & Glass, 1977).

$$\dot{\mathbf{x}}(t) = -a\mathbf{x}(t) + \beta\_0 \frac{\theta^n \mathbf{x}(t-\tau)}{\theta^n + \mathbf{x}^n(t-\tau)}.\tag{20}$$

In this equation the delay argument is a bifurcation parameter. When a value of parameter is small, then Mackey-Glass equation has unique stable stationary state. When is increasing, then at first a stable cycle is appearing in phase space of the equation from the stable stationary state as a result of Andronov-Hopf bifurcation. After that the period two stable cycle is appearing from this original singular cycle as a result of double period bifurcation. That is the beginning of the Feigenbaum cascade of period doubling bifurcations. Then, after further increasing of bifurcation parameter , all subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order takes place in the Mackey-Glass equation. Projections of some main stable cycles and singular attractors of the Mackey-Glass equation are presented in Fig. 14.

Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes transition to dynamical chaos in all nonlinear dissipative systems of ordinary differential equations. Scenario of transition to chaos consists of subharmonic and homoclinic (heteroclinic) cascades of bifurcations of stable cycles or stable two- or many-dimensional tori.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 149

freedom and by examples of simply conservative but not Hamiltonian systems. The stability domains of cycles of such a system with zero dissipation become tori of a conservative (Hamiltonian) system around its elliptic cycles into which the stable cycles themselves go. Complicated separatrix heteroclinic manifolds spanned by unstable singular cycles of the dissipative system become (for zero dissipation) even more complicated separatrix manifolds of the conservative (Hamiltonian) system along which the motion of a trajectory is treated as chaotic dynamics. Thus, it becomes clear why the order of the tori alternation in conservative (Hamiltonian) systems can differ from the Sharkovskii order existing in

Let's consider generally nonlinear conservative system of ordinary differential equations

( ), , ( ) 0 *n x f x x R div f x* (21)

Any Hamiltonian system is a special case of system of Eqs. (21)-(22) at even value of dimension *n* and at the given integral of movement (22) generating system of Eqs. (21). Movement in

>

possesses following properties: 1) the only solutions of system of Eqs. (21)-(22) are solutions

system of Eqs. (23) is dissipative system on its solutions laying in neighborhoods of solutions of system of Eqs. (21)-(22). Then attractors of dissipative system of Eqs. (23) at

So, for application of the offered approach to the analysis of conservative and, in particular, Hamiltonian systems it is necessary to construct an extended dissipative system, satisfying the

solutions and their cascades of bifurcations according to the FSM scenario in extended

conditions, satisfying the equality (22). Areas of stability of the found simple regular solutions

Eqs. (21)-(22), and areas of stability of complex cycles and singular attractors and also heteroclinic separatrix manifolds will generate chaotic solutions. By the same method in the

are as much as exact approximations of solutions of conservative system of Eqs.

system of Eqs. (21) occurs in 1 *n* -dimensional subspace, set by the equation (22).

**Theorem.** Let two-parametrical system of ordinary differential equations

(22)

(23)

; 2) at all 0

the

 at 0

one should to find numerically all stable

tends to zero, starting from the various initial

0 regular solutions (tori) of original conservative system of

systems with strong dissipation.

with a smooth right part

small 0

*3.1.1. Theoretical basis of bifurcation approach*

which variables are connected by some equation

1 ( ,..., ) . *Hx xn*

( , , ), , *n x gx x R*

of system of Eqs. (23) with initial conditions 10 0 ( ,..., ) *Hx xn*

(21)- (22) (see proof in (Magnitskii, 2008; Magnitskii, 2011)).

properties 1) and 2). Then for everyone 0

dissipative system of Eqs. (23) when

(simple cycles) will generate at

**Figure 14.** Projections of period one (a), two (b) and three (d) cycles, Feigenbaum attractor (c) and one of more complex singular attractor (e) in the Mackey-Glass equation (20).


**6. miRNA therapeutics: strategies**

There are at least two main strategies to target miRNA expression for prevention and potential treatment of disease. The first strategies is the use of oligonucleotides or virus-based constructs to either directly block the expression of a diseaseassociated signature miRNA or to directly substitute for the loss of expression of the miRNA. The second strategies is indirectly employing drugs to alter miRNA expression by targeting their transcription and processing. Blocking miRNA expression can be achieved by the use of antisense oligonucleotides, miRNA sponges, miRNAmask and small RNA inhibitors. Restoring downregulated miRNA expression can be achieved by using synthetic miRNA (miRNA mimic) or by inserting genes coding for miRNA into viral constructs. At the transcriptional level, small-molecule miRNA inhibitors can be employed to prevent the transitions from DNA transcript to pri-miRNA and pre-miRNA. Antisense oligonucleotides can be employed at the mature miRNA level to induce degradation or revert the mature miRNA into a duplex form with the antisense oligonucleotide. At the functional level, miRNA masks can bind complementarily to the 3' UTR region of target mRNA, competing for bindings with endogenous miRNAs for the specific target. miRNA sponges can be employed to bind target miRNA via complementary mRNA binding sites, decreasing expression levels of target miRNAs (see **Figure 1**).


**Author details**

Ahmed A. Hussein

Address all correspondence to<EMAIL_ADDRESS>
Chemistry Department, Cape peninsula University of Technology, Bellville Campus, Western Cape, South Africa