Sample Component:

<!--Sample.cmp-->
<aura:component>
    <aura:attribute name="map" type="Map" default="{str1:null,str2:null,obj:null}"/>
    <aura:attribute name="str1" type="String"  default="{!v.map.str1}"/>
    <aura:attribute name="str2" type="String" default="{!v.map.str2}"/>
    <aura:attribute name="obj" type="Contact" default="{!v.map.obj}"/>
    <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
    <p><b>{!v.map.str1}</b></p>
    <p><b>{!v.map.obj.FirstName}</b></p>
    <p><b>{!v.map.str2}</b></p>
    <p><b>{!v.map.obj.LastName}</b></p>
</aura:component>

Sample Component JS Controller:

({
    doInit: function(cmp, event, helper) {
        //Set the attribute value.
        var map = cmp.get("v.map");
        map['str1'] = 'FirstName';
        map['str2'] = 'Lastname';
        var obj = {"FirstName":"Biswajeet","LastName":'Samal'};
        map['obj'] = obj;
        cmp.set("v.map", map);
    }
})

Lightning App:

<aura:application extends="force:slds">
    <c:Example />
</aura:application>

Output: